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If a function f is defined by an equation and no domain is specified, then the domain will be the largest set of real numbers for which the equation defines a real number. Often, when looking for a domain, it is useful to consider which numbers could not be in the domain. Exclude all values of x that do not generate real numbers, and the domain is what remains. The following facts are useful in such a search.
  • Division by zero does not produce a real number
  • Even roots (square root, for example) of negative numbers are not real
  • The logarithm of a negative number or of zero is not real

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The absolute value of x, denoted |x|, is the distance of x from zero.

The statement |-2| = 2, means that -2 is two units to the left of zero.

Another definition of absolute value is: |A| = A if A > 0 and |A| = - A if A < 0.

To remove the absolute value notation from an algebraic statement, such as

|x - 5| take into consideration whether the value of x - 5 is greater than or less than zero. For example, if x - 5 > 0, then |x - 5| = x - 5 but if x - 5 < 0, then

|x - 5|=- (x - 5).

 

 

 

 

 

 

 

 

 

 

 

The absolute value of x, denoted |x|, is the distance of x from zero.

The statement |-2| = 2, means that -2 is two units to the left of zero.

Another definition of absolute value is: |A| = A if A > 0 and |A| = - A if A < 0.

To remove the absolute value notation from an algebraic statement, such as

|x - 5| take into consideration whether the value of x - 5 is greater than or less than zero. For example, if x - 5 > 0, then |x - 5| = x - 5 but if x - 5 < 0, then

|x - 5|= - (x - 5) = -x + 5 = 5 - x.

 

 

 

 

 

 
























































For more information relating to integer exponents, visit the links below.

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For more information relating to integer exponents, visit the links below.

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For more information relating to rational exponents and radicals, visit the links below.

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For more information relating to rational exponents and radicals, visit the links below.

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To add and subtract polynomials you will remove the parentheses to combine like terms. Remember to distribute the negative sign to each term that is being subtracted.

 























To multiply polynomials you will apply the distributive property of multiplication over addition. That is, multiply each term from the first polynomial by each term of the second polynomial.























To multiply polynomials you will apply the distributive property of multiplication over addition. That is, multiply each term from the first polynomial by each term of the second polynomial.























To divide a polynomial by a monomial (one term in denominator), you can divide each term of the polynomial in the numerator by the monomial. Remember to simplify numerical values and follow the laws of exponents.





































































































































A rational expression is an algebraic fraction for which there is a polynomial in both the numerator and the denominator. To simplify an algebraic fraction means to reduce it to lowest terms. This is done by dividing out the common factors in the numerator and the denominator.

The rule for multiplying algebraic fractions is the same as multiplying fractions; you multiply the numerators and multiply the denominators. If possible, reduce each the expressions by canceling common factors before you multiply.

 























A rational expression is an algebraic fraction for which there is a polynomial in both the numerator and the denominator. To simplify an algebraic fraction means to reduce it to lowest terms. This is done by dividing out the common factors in the numerator and the denominator.

The rule for dividing algebraic fractions is the same as dividing fractions; you multiply by the reciprocal of the divisor. In other words, change the division sign to multiplication and invert the second fraction. If possible, reduce each the expressions by canceling common factors before you multiply.

 























A rational expression is an algebraic fraction for which there is a polynomial in both the numerator and the denominator. To find the domain of a rational expression, it is useful to consider which numbers could not be in the domain. Exclude all values of x that do not generate real numbers and the domain is what remains. For rational expressions the domain restriction is found by determining which values of x will result in division by zero. That is, determine the values of x that will make the denominator equal to zero.

To simplify an algebraic fraction means to reduce it to lowest terms. This is done by dividing out the common factors in the numerator and the denominator.

The rule for multiplying algebraic fractions is the same as multiplying fractions; you multiply the numerators and multiply the denominators. If possible, reduce each the expressions by canceling common factors before you multiply.























For more information relating to simplifying complex fractions, visit the links below.

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For more information relating to simplifying complex fractions, visit the links below.

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Algebra Review – Chapter 2

 

An algebraic equation is a statement involving the equal sign. Algebraic equations typically contain polynomial, rational, or radical expressions. A solution to an equation is any number that results in a true statement. An algebraic equation can have no solutions, a countable number of solutions (one, two, three,…), or an infinite number of solutions.

 

 

Solving Linear Equations (#201)

 

A linear equation in the variable x is any algebraic equation that can be written in the form ax + b = 0, where a is not zero. Linear equations come in many different types, but the critical steps in solving these equations is the same – you must undo what has been done to the variable. That is, isolate the variable on one side of the equal sign and the constants on the other side. Be sure to check your answer by plugging the result back into the original equation to see if a true statement is produced.

 

 

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Solving Equations with Rational Expressions (#202-203, #209)

 

The procedure for solving an equation that involves rational expressions (fractions where the numerator and denominator are polynomial expressions) can be simplified into solving linear equations by eliminating the denominators from the equation. This procedure is provided in the guidelines below.

1.  Find the least common denominator (lcd) of the rational expressions.

2.  To eliminate the denominators by multiplying both sides of the equation by the lcd. Remember to use the distributive property and multiply all terms by the lcd.

3.  Solve the resulting linear or quadratic equation.

4.  Check your result to verify that you have obtained a true solution. Since division by zero is undefined, the numerical values that make the lcd equal to zero are not possible solutions

 

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Literal Equations

 

Mathematical equations or formulas that contain several different variables are often referred to as literal equations. These formulas occur in many applied problems in mathematics. Often it is necessary to solve for a specific variable in terms of the remaining variables in the formula. That is, you want the specified variable to stand alone on one side of the equal sign. Although there is more than one variable, the procedures are is identical to solving algebraic equations.

 

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Applied Problems

 

There are many different applied (word) problems in mathematics that can be solved using the same procedures for solving algebraic equations. To solve these problems, you must translate the words into an algebraic equation by using variables to represent unknown quantities and identifying the relationships that involve these variables. 

 

The general guidelines for setting up an applied problem include:

1.  Read the problem carefully. Identify unknown quantities or quantities that change. Determine what the problem is asking you to find and, if possible, estimate a possible solution.

2.  If appropriate, draw a picture or make a table.

3.  Assign a variable to represent the unknown quantity that you are looking for. If necessary, denote the remaining unknown variables in terms of this variable.

4.  List all known facts or formulas that represent the relationship between the unknown quantities.

5.  Formulate and solve the algebraic equation that represents this relationship.

6.  Check your answer.

 

There are several general types of applied problems that can be solved with linear equations. These types include:

1.  Averages

2.  Geometry (e.g., perimeter)

3.  Simple Interest

4.  Mixture

5.  Distance

6.  Work

 

 

 

 

 

 

 

 

The guidelines for solving these general types of applied problems are outlined below

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Average

 

Applied problems that can be solved using a linear equation include solving word problems related to the average or arithmetic mean of a set of data. To solve these types of problems, recall that the average test score is found by adding all the test scores and dividing by the total number of tests.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Geometry

 

Applied problems that can be solved using a linear equation include solving word problems associated with perimeter. To solve these types of problems, recall that the perimeter of a geometric figure is the sum of all the side lengths of that figure. In a circle, the perimeter is called the circumference which can be found using the formula C = 2πr, where r is the radius of the circle.

 

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Simple Interest:

 

Applied problems involving simple interest are those problems for which interest is earned or charged on an annual basis (often referred to as compounded annually). The simple interest formula is given below.

 

If a principal of P dollars is invested/borrowed at a simple interest rate r (expressed as a decimal), then the interest earned/charged at the end of t years is I = Prt.

 

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Mixture Problems

Applied problems that combine two substances are called mixture problems. In the typical mixture problem two solutions with different values (different amounts and/or different percent of concentration) are mixed together. To solve these types of problem, the first step is to understand what these different values imply.

 

Consider a glass that contains 20 milliliters of a 15% acid solution. This implies that 15% of the solution is pure acid and 85% of the solution is water. The amount of pure acid in this glass can be determined by finding 15% of 20 mL or 0.15(20) = 3 mL.

 

The resulting linear equation needed to solve a mixture problem is often found by finding the relationship between the resulting expressions in the yellow cells with the expression in the gray cell.

 

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Distance = rate x time

 

Distance problems are often referred to as uniform motion problems. These types of applied problems involve objects that move at a constant velocity (rate or speed) over a given amount of time. The key formula needed to solve uniform motion problems is distance is equal to the rate times the time traveled or simply d = rt.

 

The resulting equation needed to solve a distance problem is often found by setting the resulting expression in the yellow cell equal to the expression in the gray cell.

 

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Work Problems

Work problems typically involve two people or things combining efforts to complete a single task. Typical work problems involve two people painting a house or two pumps filling a pool. The critical element of these types of problem is that if a job can be done by a person or thing in t units of time, then 1/t of the job is done in one unit of time. That is, if Jon can paint a house alone in 8 hours, then he can paint 1/8 of the room in one hour.

 

The standard equation for work problems involving tow objects is given below.

 

(Part/fraction work done by first object in one unit of time) +

(Part/fraction work done by second object in one unit of time) =

(Part/fraction work done together in one unit of time, or 1/t)

 

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Solving Quadratic Equations A quadratic equation can be written in standard form as ax2 + bx + c = 0, where a is not equal to zero. These types of equations are often referred to as 2nd degree equations and can be solved using several techniques including factoring, the square root method, and using the quadratic formula. A quadratic equation can have two real solutions, one real solution, or no real solutions (two imaginary solutions).

 

Solve by Factoring: Note, not all quadratic expressions are factorable!

1.  Rewrite the quadratic equation to set it equal to zero.

2.  If possible, factor the 2nd degree polynomial into the product of two 1st degree (linear) polynomials.

3.  Apply the Zero Product Principle by setting each factor equal to zero and solving the resulting linear equation(s).

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Solving Quadratic Equations

A quadratic equation can be written in standard form as ax2 + bx + c = 0, where a is not equal to zero. These types of equations are often referred to as 2nd degree equations and can be solved using several techniques including factoring, the square root method, and using the quadratic formula. A quadratic equation can have two real solutions, one real solution, or no real solutions (two imaginary solutions).

Square Root Method: The square root method is an efficient way to solve a quadratic equation that can be written in the form ax2 + c = 0, where a is not equal to zero. To solve this type of quadratic equation you will isolate the variable, x2, on one side of the equal sign and then take the square root of both sides. Remember to use the ± symbol. If x2 = 9, then image .

 

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Completing the Square:

An extension of the square root method is the procedure for rewriting a quadratic equation in the form ax2 + bx + c = 0 into the form (x + d)2 = e. This procedure is called completing the square. The resulting equation can then be solved by taking the square root of both equations. The procedure for completing the square and solving the resulting quadratic equation is given below.

1.  Write the quadratic equation in the form ax2 + bx + c = 0.

2.  Move the constant term to the other side of the equal sign.

3.  If necessary, divide each term by the leading coefficient, a whatever is multiplied on the squared term.

4.  Take half of the coefficient of the x-term, and square it. Add this square to both sides of the equation.

5.  Factor the resulting trinomial into the product of two identical linear factors or (x + d)(x+d) = (x + d)2.

6.  Take the square root of both sides, remembering the ± symbol. If necessary, simplify under the square root.


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The Quadratic Formula:


The quadratic formula can be used to solve any quadratic equation regardless if it is factorable. To apply the quadratic formula, you must first set the quadratic equation equal to zero.

 

If ax2 + bx + c = 0, then the solution(s) are given by image .


When using the quadratic formula it is often necessary to simplify the result. This can be done by a) reducing the fraction to lowest terms; b) simplifying the real number on the radical; or c) simplifying the imaginary number under the radical.

 

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Operations with Complex Numbers

 

A complex number is any number in the form  a + bi. The real number a is called the real part and the real number b is called the imaginary part. The imaginary unit is defined as i = image  or equivalently i2 = -1.

 

When the imaginary unit, i, is raised to different powers a cyclic pattern emerges. The powers of i repeat in the pattern: i, -1, -i, 1. To add and subtract complex numbers you combine like terms. That is, add/subtract the real part and the imaginary part.

 

When dividing complex numbers you will need to rewrite the quotient without an imaginary part in the denominator by rationalizing the denominator. This procedure is accomplished by multiplying both numerator and denominator by the conjugate of the denominator.

 

The conjugate of the complex number a + bi is abi

 

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Quadratic Equations with Complex Solutions

A complex number is any number in the form a + bi. The real number a is called the real part and the real number b is called the imaginary part. The imaginary unit is defined as i = image  or equivalently i2 = -1.

 

Complex numbers can also be solutions to quadratic equations that have no real numbers as solutions. When solving quadratic equations using the quadratic formula, imaginary solutions will occur whenever there is a negative under the square root sign. To simplify the imaginary solutions, you must take out the imaginary unit i since the problem will contain a square root of -1.

 

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Solving Radical Equations

An equation where the variable is under the radical sign (e.g., square root, cube root, etc.) is called a radical equation. The common procedure is to “get rid” of the radical as outlined in the guidelines below:

1. Isolate the radical sign on one side of the equation.

2. Eliminate the radical sign by squaring, cubing, and so on to both sides.

3. Solve the resulting linear or quadratic equation by following the standard procedures.

4. Verify which answers are solutions to the original equation.

 

In solving radical equations it is important to check all answers because possible solutions may not be actual solutions to the equation. These values are called extraneous solutions and typically result in taking the square root of a negative number when plugged back into the equation.

 

For more information relating to simplifying complex fractions, visit the links below.Regents Exam Prep Center (here), or Purplemath (here).

 

 

 

 

Solving Absolute Value Equations

 

An equation that involves the absolute value sign can be solved as two separate equations. The solutions to the absolute value equation given by  |x – 5| = 3, you would need to solve the two linear equations x – 5 = 3 and  x – 5 = -3. Prior to writing the two equations, the critical step is to isolate the absolute value on one side of the equation.

 

 

For more information relating to simplifying complex fractions, visit the links below.

 

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Solving Equations with Rational Exponents

  To solve an equation that has a variable raised to a fractional exponent is similar to the procedure for solving radical equations. The goal is to make the exponent equal to 1 by raising both sides of the equation to the same rational exponent. If the rational exponent is 2/5, then the inverse operation is to raise both sides to the 5/2 power.  That is, image

The procedure is given below:

1.  Isolate the base with the rational exponent.

2.  Eliminate the rational exponent by recognizing that the inverse operation to a rational exponent is to raise it to the reciprocal of that exponent.

3.  Solve the remaining equation.

4.  Check for extraneous solutions. When solving equations with rational exponents, extra solutions may come up when you raise both sides to an even power. 

 

 

Solving Quadratic-Type Equations

 

A degree-four trinomial is of quadratic-type if it is in the form Ax4 + Bx2 + C. This type of polynomial may be factorable using the same techniques as described for quadratic expressions. That is, try to find two quadratic binomials that when multiplied together result in Ax4 + Bx2 + C.

 

Example: x4 – x2 – 12 = (x2 – 4)(x2 + 3) = (x – 2)(x + 2)(x2 + 3)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solving Linear Inequalities

 

Methods for solving linear inequalities are similar to the strategies as solving linear equations. It is important to remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

 

Solutions to a linear inequality can be expressed in inequality notation (-1 ≤ x < 3) or in interval notation [-1, 3).

 

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http://www.purplemath.com/modules/ineqlin.htm

 

 

 

 

 

 

 

 

 

Solving Absolute Value Inequalities

 

The critical step in the procedure for solve inequalities with absolute values is to re-write the inequality without the absolute value notation. There are two cases to consider.

Case 1: Given |expression| < c, then rewrite the inequality as the compound inequality –c < |expression| < c. For example, |x| < 3 means that the values of x that satisfy the inequality are within three units from zero on the number line and are given by -3 < x < 3.

 

Case 2: Given |expression| > c, then rewrite the inequality as two different inequalities: |expression| < -c  OR |expression| > c. For example, |x| > 3 means that the values of x that satisfy the inequality are more than three units from zero on the number line and are given by  x < -3 OR x > 3.

 

For more information relating to solving absolute value inequalities, visit the links below.

 

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Polynomial Inequalities

 

Inequalities involving polynomials of degree 2 or higher can be solved either graphically or algebraically. The procedure for solving these types of inequalities is to express the polynomial in factored form. We then determine the sign (positive or negative) of each factor over certain intervals. This is done by testing a value in a given interval.

 

1.  Write the polynomial inequality with all terms on one side of the inequality and zero on the other side.

2.  Completely factor the polynomial into the product of polynomials.

3.  To form the intervals, identify the values where the polynomial is equal to zero.

4.  Test a value within each interval to determine sign of each factor and the resulting sign of the original polynomial.

 

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Rational Inequalities

 

Inequalities involving rational expressions can be solved either graphically or algebraically. The procedure for solving these types of inequalities is to express the polynomials in both numerator and denominator in factored form. We then determine the sign (positive or negative) of each factor over certain intervals. This is done by testing a value in a given interval.

 

1.  Write the rational inequality with all terms on one side of the inequality and zero on the other side.

2.  Completely factor the numerator and denominator in the rational expression.

3.  To form the intervals, identify the values where the polynomials in the numerator and denominator are equal to zero.

4.  Test a value within each interval to determine sign of each factor and the resulting sign of the original polynomial.

5.  Remember, the final interval solution can not contain values for which the denominator is equal to zero because the rational expression would not be defined.

 

The Regents Exam Prep Center has additional information relating to solving rational inequalities. Click here to see this information in pop-up window.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Distance Formula

 

The distance formula provides a simple method for computing the distance between two points on the coordinate plane.

 

The distance, d, between two points (x1, y1) and (x2, y2) is given by

 

For more information relating to the distance formula, visit the links below.

 

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Midpoint

 

The midpoint of a line segment is given by a coordinate pair that is equal distance from each endpoint.

 

The midpoint M = (x,y) of a line segment with endpoints (x1, y1) and (x2, y2) is given by

image

 

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Slope

 

Given a line, the ratio of the vertical rise to the horizontal run is called the slope, m. It is a numerical measure of the steepness and direction of the line.  Another way to interpret the slope of a line that passes through two points is the ratio of the vertical change (change in y-values) to the horizontal change (change in x-values).

 

The slope of a line that passes through the points (x1, y1) and (x2, y2) is given by

image

 

A vertical line has no run. The slope is undefined since x2 – x1 = 0.

A horizontal line has no rise. The slope is m = 0 since y2 – y1 = 0.

 

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Applied Problems with Pythagorean’s Theorem

  Distance problems are often referred to as uniform motion problems. These types of applied problems involve objects that move at a constant velocity (rate or speed) over a given amount of time. The key formula needed to solve uniform motion problems is distance is equal to the rate times the time traveled or simply d = rt.

 

To solve this applied problem is helpful to draw a picture. You will discover that the distance between the two objects at time, t, is equal to the length of the hypotenuse of a right triangle. To set-up the equation necessary to solve this problem, use the Pythagorean Theorem. This theorem states that in a right triangle, the sum of squares of the lengths of the legs is equal to the square of the length of the hypotenuse (the longest side).  If c is the length of the hypotenuse of a right triangle and a and b are the lengths of the legs of a right triangle, then c2 = a2 + b2.

 

 

 

 

Sketching Graphs

 

To sketch a graph of an equation in two variables you can plot several points to determine the overall shape of the graph.

 

There are two types of points that are important to graphing. These are the x-intercept(s) and the y-intercept(s).

 

An x-intercept is a point on the coordinate plane where the graph of the equation/function crosses or touches the x-axis. The x-intercept is in the form (a, 0) occurs when the y-value is equal to zero. To find the x-intercept, set y = 0 and solve the resulting equation. An x-intercept is sometimes referred to as the zero of the graph of an equation or as the root of an equation.

 

A y-intercept is a point on the coordinate plane where the graph of the equation/function crosses the y-axis. The y-intercept is in the form (0, b) occurs when the x-value is equal to zero. To find the y-intercept, set x = 0 and solve the resulting equation.




 

Another helpful tool for sketching a graph is symmetry. There are three important types of symmetry that aid in graphing.

1.  Symmetric with respect to the y-axis – this means the graph has a mirror image to the left and bright of the y-­axis. That is for every point (x,y) on the graph, the point (-x,y) is also a point. To test for this type of symmetry, replace x by –x in the equation and simplify. If an equivalent equation results, the graph is symmetric about the y-axis.

2.  Symmetric with respect to the x-axis – this means the graph has a mirror image above and below the x-axis. That is, for every point (x,y) on the graph, the point (x,-y) is also a point. To test for this type of symmetry, replace y by –y in the equation and simplify. If an equivalent equation results, the graph is symmetric about the x-axis.

3.  Symmetric with respect to the origin – this means that the graph has been reflected across the y-axis and then the x-axis. Another way to describe this type of symmetry is that the graph has been rotated by 180 degrees around origin. That is, for every point (x,y) on the graph, the point (-x,-y) is also a point. To test for this type of symmetry, replace x by –x AND y by –y in the equation and simplify. If an equivalent equation results, the graph is symmetric about the origin.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Vertices of Right Triangle

 

To show that the triangle is a right triangle, we need to show that the lengths of the three sides of the triangle satisfy the Pythagorean Theorem.

 

This theorem states that in a right triangle, the sum of squares of the lengths of the legs is equal to the square of the length of the hypotenuse (the longest side).  If c is the length of the hypotenuse of a right triangle and a and b are the lengths of the legs of a right triangle, then c2 = a2 + b2.

 

Let c=d. The distance, d, between two points (x1, y1) and (x2, y2) is given by

 

 

 

 

 

 

Circles

A circle is the set of all points (x,y) that lie equal distant from a fixed point (h,k) . The fixed distance is the radius, r, of the circle and the fixed point is the center.

 

The standard form of an equation of a circle with radius, r, and center

(h,k) is derived from the distance formula and given by

r2 = (x – h)2 + (y – k)2.

 

To derive the standard form of a circle given the center (h,k) and a point (x,y) on the circle, solve for the radius by plugging in the appropriate values into the formula.

 

The general form of a circle is given by Ax2 + By2 + Cx + Dy + E = 0. To identify the center and radius of a circle in general form, use the method of completing the square to put the equation in standard form.

 

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Lines: The slope of a line that passes through the points (x1, y1) and

(x2, y2) can be found using the formula:image

There are three useful forms for equations of nonvertical lines. These include:

1.  Slope-Intercept Form with slope m and y-intercept b is given by y = mx + b.

2.  Point-Slope Form with slope m and contains the point (x1,y1) is given by (yy1) = m(xx1).

3.  The General Form (or Standard Form) of a line is given by Ax + By = C, where A, B, and C are real numbers and A and B are not both 0.

Horizontal lines have a slope m =0 and are described by the equation y = b. Vertical lines have undefined slopes and are described by the equation x = c.

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Parallel and Perpendicular Lines

 

Two lines are parallel if they do not intersect and have the same slope. Two lines are perpendicular if they intersect at a right angle and have slopes that are opposite reciprocals (e.g., m and -1/m). In other words, the slopes of perpendicular lines multiply to -1.

 

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Function Notation

  A function is a relation between a set of input elements (the domain) and another set of output elements (the range) for each element in the domain corresponds with exactly one element in the range. That is, for each input there is only one output. Functions can be represented in tables, graphs, or equations.

 

In mathematics, we use function notation such as f(x) or g(x) to represent a function. We refer to f(x) as the value of the function at the input value x.

 

To evaluate a function with an algebraic expression, substitute all occurrences of the input variable with this expression. Then simplify the result. For example if f(x) =x2 + 2x – 1, f(a + 1) = (a+1)2 + 2(a+1) – 1.

 

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Find the Domain
If a function f is defined by an equation and no domain is specified, then the domain will be the largest set of real numbers for which the equation defines a real number.

Often, when looking for a domain, it is useful to consider which numbers could not be in the domain. Exclude all values of x that do not generate real numbers, and the domain is what remains. The following facts are useful in such a search.

Division by zero does not produce a real number

Even roots (square root, for example) of negative numbers are not real

The logarithm of a negative number or of zero is not real

 

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Even and Odd Functions

 

The terms even and odd functions describe the symmetry in the graph of the given function.  An even function is symmetric with respect to the y-axis which means the graph has a mirror image to the left and right of the y-axis. An odd function is symmetric with respect to the origin which means that the graph has been reflected across the y-axis and then the x-axis. Another way to describe this type of symmetry is that the graph has been rotated by 180 degrees around origin.

 

To test for symmetry, replace x by –x in the equation and simplify.

1.  If f(-x) = f(x), then the function is even.

2.  If f(-x) = -f(x), then the function is odd.

 

 

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Transformations

If we know the graph of a function f(x), then we can use common transformation to sketch related graphs. These transformations include vertical or horizontal shifts, stretching and compressing, or reflections. The chart below summarizes a few important transformations.

Type of Transformation

Given y = f(x), to graph:

Effect on Graph

Vertical Shift

y = f(x) + c, c >0

The graph of f is shifted up c units.

y = f(x) – c, c >0

The graph of f is shifted down c units.

Horizontal Shift

y = f(x+d), d >0

The graph of f is shifted left d units.

y = f(x–d), d >0

The graph of f is shifted right d units.

Horizontal Stretch/Compress

y = f(bx), b > 1

The graph of f is horizontally compressed by a factor of b.

y = f(bx), 0<b<1

The graph of f is horizontally stretched by a factor of 1/a.

 

Transformations

If we know the graph of a function f(x), then we can use common transformation to sketch related graphs. These transformations include vertical or horizontal shifts, stretching and compressing, or reflections. The chart below summarizes a few important transformations.

Type of Transformation

Given y = f(x), to graph:

Effect on Graph

Vertical Shift

y = f(x) + c, c >0

The graph of f is shifted up c units.

y = f(x) – c, c >0

The graph of f is shifted down c units.

Horizontal Shift

y = f(x+d), d >0

The graph of f is shifted left d units.

y = f(x–d), d >0

The graph of f is shifted right d units.

 

Reflection

y = -f(x)

The graph of f is reflected across the x-axis.

y = f(-x)

The graph of f is reflected across the y-axis.

 

Transformations

If we know the graph of a function f(x), then we can use common transformation to sketch related graphs. These transformations include vertical or horizontal shifts, stretching and compressing, or reflections. The chart below summarizes a few important transformations.

Type of Transformation

Given y = f(x), to graph:

Effect on Graph

Vertical Shift

y = f(x) + c, c >0

The graph of f is shifted up c units.

y = f(x) – c, c >0

The graph of f is shifted down c units.

Horizontal Shift

y = f(x+d), d >0

The graph of f is shifted left d units.

y = f(x–d), d >0

The graph of f is shifted right d units.

Horizontal Stretch/Compress

y = f(bx), b > 1

The graph of f is horizontally compressed by a factor of b.

y = f(bx), 0<b<1

The graph of f is horizontally stretched by a factor of 1/a.

 

 

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Piecewise Functions

 

A piecewise function is a function that is defined by different equation on different parts of its domain. To sketch the graph of a piecewise function, graph each piece of the function. Make sure to include open circles and closed circles appropriately as endpoints of each piece.

 

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Quadratic Functions

 

A quadratic function in general form is given by f(x) = ax2 + bx + c where a, b, and c are real numbers and a is not equal to zero. The graphs of quadratic functions are parabolas. If the leading coefficient a > 0, then the parabola opens up and has a maximum point at the vertex. If the leading coefficient a < 0, then the parabola opens down and has a minimum point at the vertex.

 

If the quadratic function is in general form, then the vertex of the parabola is given by the point image .

 

A quadratic function with vertex (h,k) in standard form (vertex form) is given by f(x) = a(x - h)2 + k. To convert from general form to standard form, follow the method of completing the square.

 

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Operations with Functions

 

Functions like numbers and algebraic expressions can be added, subtracted, multiplied, and divided. These operations are defined below.

 

Addition: (f + g)(x) = f(x) + g(x)

Subtraction: (f  g)(x) = f(x) –  g(x)

Multiplication: (fg)(x) = f(x) g(x)

Division: (f/g)(x) = f(x)/g(x)

 

The domain of the sum, difference, or product consists of the numbers that are in the domain of both functions. The domain of the quotient consist all values in the domain of both functions where g(x) is not equal to zero.

 

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Composition of Functions

 

The composition of two function f(x) and g(x) is defined by (f ◦ g)(x) = f(g(x)). The domain of fg is the set of all x in the domain of g such that

g(x) is in the domain of f. Note, (g ◦ f)(x) = g(f(x)).

 

 

The composition of function is similar to evaluating a given function with an algebraic expression. For example, if f(x) = x2 + 2x – 1 and g(x) = 2x + 1, then (f ◦ g)(x) = f(g(x)) =f(2x + 1).

 

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Graphing Polynomial Functions

  A polynomial function is given by f(x) = anxn + an-1xn-1 + … a2x2 + a1x + a0 where n is the degree of the polynomial and an is the leading coefficient.

  To help graph a polynomial function f, follow the guidelines below.

1.  If possible, factor f into the product of linear or quadratic terms.

2.  Find the y-intercept of f by solving f(0). All polynomial functions have exactly one y-intercept.

3.  Find the x-intercept(s) of f. The x-intercepts are the zeros of f and are found by setting f(x) = 0 and solving the resulting equation. A polynomial function can have at most n x-intercepts.

4.  Between the x-intercepts (zeros of f), the graph of f is either above the x-axis (f(x)>0) or below the x-axis (f(x)<0). Use test values within the intervals between the x-intercepts to determine the behavior.

5.  If (x – c)m is a factor and m is an even number, then the graph touches the graph at x = c. If m is an odd number, then the graph passes through the graph at x = c.

6.  The end behavior of the graph is determined by the degree n and sign of the leading coefficient an.

a.   If n is even, then the ends of the graph point in the same direction.

b.  If n is odd, then the ends of the graph point in opposite directions.

 

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Dividing Polynomials

  When you divide a polynomial function f(x) by a divisor p(x), you get a quotient polynomial q(x) and a remainder polynomial r(x) where r(x) = 0 or its degree is less than the degree of the divisor.

We write this relationship as image  or f(x) = p(x)q(x) + r(x).

 

To find the quotient and remainder function, you may use the methods of long division or synthetic division. If the remainder function r(x) = 0, then we say that p(x) and q(x) are factors of the original polynomial function.

 

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Remainder Theorem

 

The Remainder Theorem: If a polynomial function f(x) is divided by the term x – c, then the remainder is f(c).

 

If the remainder f(c)=0, then c is a zero of f(x). This implies that c is a solution to the polynomial function, the term x – c is a factor of the polynomial function, and c is an x-intercept of the graph of the polynomial function.

 

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Zeros and the Equation

 

Given the degree, zeros, and a point on the graph (x, f(x)) of a polynomial function with real coefficients, use the following guidelines to determine the equation for f(x) in factored form.

 

1.  A polynomial function of degree n has at most n different zeros. (real or complex)

2.  If c is a zero of f, then x – c is a factor of f.

3.  If a + bi is an imaginary zero of f, then the conjugate a – bi is also an imaginary zero of f. This implies (x – (a +bi)) and (x – (a – bi)) are factors of f.

 

 

 

 

 

 

 

 

 

Graphs and Equation

Given the degree and graph of a polynomial function, use the following guidelines to determine the equation for f(x) in factored form.

1.  A polynomial function of degree n has at most n different real complex (real or imaginary) zeros.

2.  Find the x-intercepts of the graph by identifying the points where the graph touches and passes through the x-axis.

3.  If c is an x-intercept that passes through the graph then (x – c)m  where m is an odd whole number is a factor of f. We say c is a zero with multiplicity m.

4.  If c is an x-intercept that touches the graph then (x – c)m  where m is an even whole number is a factor of f. We say c is a zero with multiplicity m.

5.  The degree of f is equal to the total number of real and imaginary zeros, including repeated zeros (zeros of multiplicity).

 

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Descartes Rule of Signs

Descartes’ rule of signs provides information relating to the possible number of positive and negative real zero of f(x).

 

Given a polynomial function f(x) with real coefficients, a nonzero constant term, and terms arranged in order of decreasing powers of x.

1.  The number of positive real zeros of f(x) is either equal to the number of variations of sign (changes from + to – or vise versa) in f(x) or is less than that number by an even integer.

2.  The number of negative real zeros of f(x) is either equal to the number of variations of sign (changes from + to – or vise versa) in f(-x) or is less than that number by an even integer.

   

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Rational Root Theorem

 

The Rational Root Theorem states that if a polynomial function f(x) has integer coefficients, then every rational zero is in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. The Rational Root Theorem provides you with a list of possible rational zeros.

 

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Roots of Polynomial Functions
A polynomial function is given by f(x) = anxn + an-1xn-1 + … a2x2 + a1x + a0. The Fundamental Theorem of Algebra states that the equation f(x) = 0 has at least one real or imaginary root (solution) and at most n solutions. That is, if f(c)= 0 then the c is called a root or solution to the polynomial function. We can also say the c is the x-intercept of the graph of f and (x – c) is a factor of the f.

To find the roots/solutions follow the steps below.

1.  Factor out the greatest common factor, if applicable.

2.  Use special rules or factor by grouping, if needed.

3.  Use the Rational Root Theorem to test possible rational zeros.

4.  Use the Factor Theorem to test if possible rational zeros are actual zeros.

 

The Rational Root Theorem states that if a polynomial function f(x) has integer coefficients, then every rational zero is in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

 

The Factor Theorem states that if x – c is a factor of the polynomial function f(x), then the remainder when using long division will be zero. That is, if a polynomial function f(x) has a factor (xc) if and only if f(c) = 0.

 

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Graphs of Rational Functions

 

A rational function is formed by the quotient of two polynomial functions g(x) and h(x) and is in the form image . The domain of a rational function is all real numbers expect values for which the denominator is equal to zero.

 

A rational function may have vertical, horizontal, or oblique asymptotes. A vertical asymptote describes the behavior of the graph as x approaches some number. At a vertical asymptote the y-values will either approach positive or negative infinity. The graph of a rational function will never intersect its vertical asymptote.

 

 

 

 

 

 

 

 

If the rational function is in lowest terms (no common factors in numerator and denominator other than 1), then the vertical asymptotes are the values were the denominator is equal to zero.

 

If the rational function is not in lowest terms numerator and denominator and they share a common zero, then this zero produces a hole in the graph, not a vertical asymptote.

 

To find the horizontal or oblique asymptote we need to determine how the y-values of rational function behave as x approaches positive or negative infinity. The extreme ends of the graph will either approach a horizontal line or a line with a given slope.

 

The horizontal asymptote, if one exists, can be found by following the rules below:

a.   If the degree of the numerator is less than the degree of the denominator, then the x-axis (the line y = 0) is the horizontal asymptote.

b.  If the degree of the numerator is equal to the degree of the denominator, then the ratio of the leading coefficients is the horizontal asymptote.

c.   If the degree of the numerator is greater than the degree of the denominator, then the graph has no horizontal asymptote.

 

An oblique asymptote for a graph is the line y = ax +b that the graph approaches as x approaches positive or negative infinity. A rational function will have an oblique asymptote only if the degree of the numerator is one greater than the degree of the denominator. It is important to note that if a horizontal asymptote exists, the rational function will not have an oblique asymptote.

 

To find the oblique asymptote, use division algorithm to rewrite the rational function as image  where r(x) is the remainder function.

 

The graphs of a rational function image in lowest terms (numerator and denominator share no common factor other than 1 have the following characteristics to help graph the function:

1.  The x-intercepts of the graph of f are the real zeros of the numerator.

2.  The graph of f has a vertical asymptote at each real zero of the denominator.

3.  The graph of f has at most one horizontal asymptote (h.a.).

a.   If the degree of the numerator is less than the degree of the denominator, then the line y=0 (the x-axis) is the h.a.

b.  If the degrees of the numerator and denominator are equal then the ratio of the leading coefficients is the h.a.

c.   If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote.

4.  The graph of f will have one oblique asymptote if the degree of the numerator is one more than the degree of the denominator.

5.  Using the zeros of both the numerator and denominator, divide the x-axis into intervals and determine if the graph is above or below the x-axis by choosing a number in each interval and evaluating the function at that value.

6.  Analyze the behavior of the graph near each asymptote.

 

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Variation

 

If y varies directly with x, then y = kx If y varies inversely with z, then y = k/z. In both situations, k is a non-zero number and called the constant of variation.

 

The guidelines for finding solving variation problems are outlined below:

1.  Write the general formula that involves variables and the constant of variation, k.

2.  Find the value of k by using the initial data given in the problem.

3.  Rewrite the general formula, using the specified value of k.

4.  Use the new data to solve for the specified variable.

 

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Definition of one-to-one

A function y = f(x) is a relation or rule that creates a correspondence between the domain and range such that each element, x, in the domain corresponds to exactly one element, y, in the range. That is, for every input, x, there is only one output, y.

A function y = f(x) for which every element of the range, y, corresponds to exactly one element of the domain, x, is called a one-to-one function. A function y = f(x) is one-to-one if no two different input elements have the same output element. That is, whenever ab, then f(a) ≠ f(b).

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Verify Inverse Functions

Let f and g be two one-to-one functions. If f(g(x)) = x = g(f(x)), then f and g are inverse functions. The inverse of the function f(x) is denoted as f-1(x).

The graph of an inverse is the reflection of the original graph over the line, y = x.

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Finding an Inverse Function
Let f and g be two one-to-one functions. If f(g(x)) = x = g(f(x)), then f and g are inverse functions. The inverse of the function f(x) is denoted as f-1(x).

To find the inverse of a function f(x), follow the guidelines below.
1. Verify that f is one-to-one throughout its domain. That is, no two different input elements have the same output element.
2. In the equation y = f(x), swap the variables by replacing x with y and vice-versa.
3. Solve the resulting equation for y. This equation is f-1(x).
4. State the domain of f-1(x).

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Evaluating the Inverse Function

The inverse of a function f-1(x) is the set of ordered pairs obtained by interchanging the first and second elements of each pair (x,y) that satisfies the original function f(x). Given the one-to-one function y = f(x) = 2x, we have f(3) = 2(3) = 6. In the inverse function, f-1(6) = 3

To find the composition of two functions, recall the rule:

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Solving Exponential Equations
In an exponential equation the variable appears in the exponent such as 3x = 32x-1. To solve exponential equations without using logarithms, the bases of each exponential expression must be comparable. If the bases are the same on either side of the equal sign, the expressions in the exponents must be equivalent. If am = an, then m = n.

In some cases, you may have to convert one or both bases to another base before you can set the powers equal to each other. For example, (1/3) = 3-1 and 9 = 32.

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Graphing Exponential Equations
The exponential function is given by  y = f(x) = bx where b > 0 and b ≠ 1.

This function has the following characteristics.

1. 
LEAD Technologies Inc. V1.01 The domain is the set of real numbers.
2.  The range is the set of numbers y > 0. This is the case since the exponential equation y =  bx always results in a positive y value for any x value that is plugged into the equation.
3.  The line y = 0 (the x-axis) is the horizontal asymptote for the graph of f. The graph of f has no x-intercepts.
4.  Since b0 = 1 for all values of b, the graph of f has a y-intercept at (0,1).
5.  If b > 1, then the graph of f(x) = bx is an increasing function.
6.  If 0<b<1, then the graph of f(x) = bx is a decreasing function. Note, if b > 1, then the graph of f(x) = b-x is also a decreasing function.

The family of exponential functions in the form y = f(x) = abx + d has similar characteristics. Using transformation of graphs the constant d will shift the graph of y = f(x) = bx vertically d units. This will alter the range, y-intercept, and may or may not produce an x-intercept.

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Compound Interest
Calculating compound interest is one application of exponential functions. If P dollars, the principal, is invested at an interest rate of r% is compounded n times a year, then the amount, A, in the account after t years is given by the formula: image .
The interest period or the number of times interest is compounded per year is given by n.  This value may be measured in years, months, weeks, or days. The chart below provides some values for n.

Interest Period

Yearly

Quarterly

Monthly

Daily

n

1

4

12

365

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Compound Interest

 

Calculating compound interest is one application of exponential functions. If P dollars, the principal, is invested at an interest rate of r% is compounded continuously, then the amount, A, in the account after t years is given by the formula: image .

 

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Finding zeros of functions involving exponentials

 

The typical procedure for finding the zeros (roots, solutions, ­x-intercepts) of a function involving exponentials such as bx is factoring. Recall, that bx > 0 for all values of x.

1. Set the function f(x) = 0.

2. Factor out the greatest common factor.

3. Use the Zero Product Principal and set each term equal to zero.

4. Solve the resulting equations.

 

 

 

 

 

 

 

 

 

 

 

 

 

Exponential Growth and Decay

Exponential growth and decay are applied problems related to the exponential function. Some applied problems include population growth and radioactive decay.

Let A0 be initial amount of a quantity at time t = 0. If the quantity A grows or decays at a continuous rate r, then the value of A after t years is given by: A=Pert where r > 0 is the rate of growth of A and r < 0 is the rate of decay.

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Logarithmic Functions

 

The logarithm of x with base b is defined by y = f(x) = logb(x) if and only if x = by for every x > 0 and every real number y.

 

A logarithmic function is in the form y = logb(x) where b > 0 and b ≠ 1 and x > 0. The logarithm of a negative number or zero is not a real number. Therefore, the domain is the set of real numbers that can be plugged into the logarithmic function so that you will be taking the logarithm of a positive number. The range of a logarithmic function is the set of real numbers.

 

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Definition of Logarithms

 

The logarithm of x with base b is defined by y = f(x) = logb(x) if and only if x = by for every x > 0 and every real number y.

 

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Solving Logarithmic Equations

The logarithmic function with base b is one-to-one.
If logb m = logb n, then m = n.

To solve logarithmic equation that have a single log with the same bases on both sides of the equal sign, you can set the algebraic expressions equal to one another and solve the resulting equation. Extraneous solutions may be introduced when logarithmic equations are solved. Therefore, check the solution(s) to make sure that you are taking the logarithm of positive real numbers.

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Solving Logarithmic Equations

To solve equations involving logarithms and exponentials apply the following laws.
1. logb(mn) = logbm + logbn
2. logb(m/n) = logbm –  logbn
3. logb(mn) = nlogb m
4. logb(b) = 1 or equivalently ln(e) = 1
5.  or equivalently eln m = m

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Logarithmic Functions

 

The logarithm of x with base b is defined by y = f(x) = logb(x) if and only if x = by for every x > 0 and every real number y.

 

A logarithmic function is in the form y = logb(x) where where b > 0 and b ≠ 1 and x > 0. This function has the following characteristics.

1.  The logarithm of a negative number or zero is not a real number. The domain is the set of real numbers that can be plugged into the logarithmic function so that you will be taking the logarithm of a positive number. Because of this domain restriction the graph of f will have a vertical asymptote. 514image

2.  The range is the set of real numbers.

3.  The line x = 0 (the y-axis) is the vertical asymptote for the graph of f. The graph of f has no y-intercepts.

4.  Since b0 = 1 implies logb(1) = 0 for all values of b, the graph of f has a x-intercept at (1,0).

5.  If b > 1, then the graph of f(x) = logb(x)is an increasing function.

6.  If 0<b<1, then the graph of f(x) = logb(x)is a decreasing function.

For more information relating to solving graphing logarithmic functions and exponentials, please visit the links below.

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Simplify Logarithmic Expressions

 

To expand a single logarithmic expression or to condense a logarithmic expressions with logarithms apply the following laws.

 

1. logb(mn) = logbm + logbn

2. logb(m/n) = logbm –  logbn

3. logb(mn) = nlogb m

 

For more information relating to the properties of logarithms, please visit the links below.

 

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  Solving Logarithmic Equations (#525, 528)

 

To solve equations involving multiple logarithms follow the guidelines below:

 

1.   Bring all logarithms to one side of the equal sign and constant terms to the other.

2.   Apply the laws of logarithms to condense into a single logarithm.

a.   logb(mn) = logbm + logbn

b.   logb(m/n) = logbm –  logbn

c.    logb(mn) = nlogb m

3.   Rewrite the single logarithm into exponential form:  
y = logb(x) if and only if x = by

4.   Solve the resulting equation.

5.   Check for extraneous solutions by making sure you are taking the logarithm of positive real numbers.

 

For more information relating to solving equations with logarithms, please visit the links below.

 

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Change of Base

 

To calculate the value of a logarithmic expression that does not use base 10 or base e, use the change of base formula below.

 

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For more information relating to solving equations with logarithms, please visit the links below.

 

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Solving Exponential Equations with Logarithms

 

In an exponential equation the variable appears in the exponent such as 3x = 52x-1. If the bases of each exponential expression are not comparable, then we take the log of both sides to remove the variable from the exponent. That is, logb(mn) = nlogb m.

 

 

 

For more information relating to solving exponential equations with logarithms, please visit the links below.

 

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