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.
For more information relating to the domain and range of functions, visit the links below. Purplemath: Click here to see this information in a popup window.


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. The Regents Exam Prep Center: Click here to see this information in a popup window. Purplemath: Click here to see this information in a popup window.


For more information relating to integer exponents, visit the links below. The Regents Exam Prep Center: Click here to see this information in a popup window. Purplemath: Click here to see this information in a popup window.


The Regents Exam Prep Center has additional information relating to radicals. Click here to see this information in a popup window.
 
For more information relating to rational exponents and radicals, visit the links below. Regents Exam Prep Center: Click here to see this information in a popup window. Purplemath: Click here to see this information in a popup window.
 
For more information relating to rational exponents and radicals, visit the links below. Regents Exam Prep Center: Click here to see this information in a popup window. Purplemath: Click here to see this information in a popup window.
 
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. Regents Exam Prep Center: Click here to see this information in popup window. Purplemath: Click here to see this information in popup window.


For more information relating to simplifying complex fractions, visit the links below. Regents Exam Prep Center: Click here to see this information in popup window. Purplemath: Click here to see this information in popup window.


 
 
The Regents Exam Prep Center has additional information relating to radicals. Click here to see this information in a popup window.


 
 
 
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.
Purplemath has additional information relating to linear
equations. Click here to see this information in popup window.
Solving Equations with Rational Expressions (#202203, #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
Purplemath has additional information relating to solving
rational equations. Click here to see this
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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.
Purplemath has additional information relating to solving
rational literal equations. Click here to see this
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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
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.
Purplemath has additional information relating to solving
applied problems involving geometric concepts. Click here to see this
information in popup window.
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.
Purplemath has additional information relating to solving
applied problems involving geometric concepts. Click here to see this
information in popup window.
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.
Purplemath has additional information relating to solving distance
problems. Click here to see this information in popup window.
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)
Purplemath has additional information relating to solving
work problems. Click here to see this information in popup window.
Solving Quadratic
Equations
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 2^{nd} degree polynomial into the product of two 1^{st} degree (linear) polynomials.
3.
Apply
the Zero Product Principle by setting each factor equal to zero and solving the
resulting linear equation(s).
A quadratic equation can be written in standard form as ax^{2} + bx + c = 0, where a is
not equal to zero. These types of equations are often referred to as 2^{nd} 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:
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An extension of the square root method is the procedure
for rewriting a quadratic equation in the form ax^{2} + 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 ax^{2} + 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 xterm, 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.
If ax^{2} + bx + c = 0, then the solution(s) are given by
.
Purplemath:Click here.
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 =
or equivalently
i^{2} = 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 a – bi
Purplemath has additional information. Click here.
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 =
or equivalently
i^{2} = 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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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.
Regents Exam Prep Center: Click here to see this information in a popup window.
Solving Equations with Rational Exponents
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 QuadraticType Equations
A degreefour trinomial is of quadratictype if it is in
the form Ax^{4} + Bx^{2} + 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 Ax^{4} + Bx^{2} + C.
Example: x^{4} – x^{2} – 12 = (x^{2} – 4)(x^{2} + 3) = (x – 2)(x +
2)(x^{2} + 3)
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).
Purplemath has additional information relating to solving
linear inequalities. Click here to see this
information in popup window.
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 rewrite 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.
Regents Exam Prep Center: Click here Purplemath: Click here
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.
Purplemath has additional information relating to solving
polynomial inequalities. Click here to see this
information in a popup window.
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 popup window.
The distance formula provides a simple method for
computing the distance between two points on the coordinate plane.
The distance, d,
between two points (x_{1}, y_{1}) and (x_{2}, y_{2})
is given by
For more information relating to the distance formula,
visit the links below.
Regents Exam Prep Center: Click here
Purplemath: Click here
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 (x_{1}, y_{1}) and (x_{2}, y_{2}) is
given by
For more information relating to the distance formula,
visit the links below.
Regents Exam Prep Center: Click here
Purplemath: Click here
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 yvalues) to the horizontal change (change in xvalues).
The slope of a line that passes through the points (x_{1},
y_{1}) and (x_{2}, y_{2}) is given by
A vertical line has no run. The slope is undefined since x_{2} – x_{1} = 0.
A horizontal line has no rise. The slope is m = 0 since y_{2} – y_{1} = 0.
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Applied Problems with
Pythagorean’s Theorem
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 setup 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 c^{2} = a^{2} + b^{2}.
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 xintercept(s) and the
yintercept(s).
An xintercept is a point on the coordinate plane where
the graph of the equation/function crosses or touches the xaxis. The xintercept is in the form (a, 0) occurs
when the yvalue is equal to zero. To
find the xintercept, set y = 0 and
solve the resulting equation. An xintercept is sometimes referred to as the
zero of the graph of an equation or as the root of an equation.
A yintercept is a point on the coordinate plane where the
graph of the equation/function crosses the yaxis. The yintercept is in the form (0, b) occurs when the xvalue is equal to zero. To find the yintercept, 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 yaxis –
this means the graph has a mirror image to the left and bright of the yaxis. 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 yaxis.
2.
Symmetric
with respect to the xaxis –
this means the graph has a mirror image above and below the xaxis. 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 xaxis.
3.
Symmetric
with respect to the origin – this means that the graph has been reflected
across the yaxis and then the xaxis. 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.
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 c^{2} = a^{2} + b^{2}.
Let c=d. The
distance, d, between two points (x_{1},
y_{1}) and (x_{2}, y_{2}) is given by
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 r^{2} = (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 Ax^{2} +
By^{2} + 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 (x_{1}, y_{1}) and (x_{2}, y_{2}) can be found using the formula:
There are three useful forms for equations of nonvertical lines.
These include:
1.
SlopeIntercept
Form with slope m and yintercept b
is given by y = mx + b.
2.
PointSlope
Form with slope m and contains the point (x_{1},y_{1}) is given by (y – y_{1}) = m(x – x_{1}).
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.
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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.
For more information relating to parallel and
perpendicualar, visit the links below.
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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) =x^{2} +
2x – 1, f(a + 1) = (a+1)^{2} +
2(a+1) – 1.
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Find the
Domain 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
For more information relating to the domain and range of
functions, visit the links below.
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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 yaxis which means the graph has a mirror image to the left and
right of the yaxis. An odd function
is symmetric with respect to the origin which means that the graph
has been reflected across the yaxis
and then the xaxis. 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.
For more information relating to even and odd functions,
visit the links below.
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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.
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.
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.
Regent Exam Prep Center: Click here Purplemath: Click here
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.
A quadratic function in general form is given by f(x)
= ax^{2} + 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
.
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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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: (f •g)(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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The composition of two function f(x) and g(x) is defined by (f ◦ g)(x) = f(g(x)). The domain of f ◦ g 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)
= x^{2} + 2x – 1 and g(x) = 2x + 1, then (f ◦ g)(x) = f(g(x)) =f(2x + 1).
For more information relating to the composition of functions,
visit the links below.
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1.
If
possible, factor f into the product
of linear or quadratic terms.
2.
Find the yintercept of f by solving f(0). All
polynomial functions have exactly one yintercept.
3.
Find the
xintercept(s) of f. The xintercepts
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 xintercepts.
4.
Between
the xintercepts (zeros of f), the
graph of f is either above the xaxis (f(x)>0) or below the xaxis
(f(x)<0). Use test values within
the intervals between the xintercepts
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 a_{n}.
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.
For more information relating to the graphing polynomial
functions, visit the links below.
We write this relationship as
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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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 xintercept of the graph of
the polynomial function.
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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.
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.
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
xintercepts of the graph by identifying the points where the graph touches and
passes through the xaxis.
3.
If c is
an xintercept 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 xintercept 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 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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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 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
(x − c) if and only if f(c) = 0.
For more information relating to the roots/solutions to
polynomial functions, visit the links below.
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A rational function is formed by the quotient of two
polynomial functions g(x) and h(x) and is in the form
. 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 yvalues 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 yvalues 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 xaxis (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
where r(x) is
the remainder function.
The graphs of a rational function
in lowest terms (numerator and denominator share no common
factor other than 1 have the following characteristics to help graph the
function:
1.
The
xintercepts 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 xaxis) 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 xaxis into
intervals and determine if the graph is above or below the xaxis 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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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 nonzero 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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Verify Inverse Functions
Finding an Inverse Function
Evaluating the Inverse Function
For more
information relating to inverse functions, please visit the links below.
Solving Exponential Equations
Graphing Exponential
Equations
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:
.
For more
information relating to compound interest, please visit the links below.
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Finding zeros of functions
involving exponentials
The
typical procedure for finding the zeros (roots, solutions, xintercepts) of a function involving
exponentials such as b^{x} is
factoring. Recall, that b^{x} > 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
Let A_{0} 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=Pe^{rt} where r > 0 is the rate of growth of A and r < 0 is the rate of decay.
The
logarithm of x with base b is defined
by y = f(x) = log_{b}(x) if and only if x = b^{y} for every x > 0 and every real
number y.
A
logarithmic function is in the form y = log_{b}(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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The
logarithm of x with base b is defined
by y = f(x) = log_{b}(x) if and only if x = b^{y} for every x > 0 and every real
number y.
For more
information relating to logarithms, please visit the links below.
Solving Logarithmic Equations
Solving Logarithmic Equations
The
logarithm of x with base b is defined by y = f(x) =
log_{b}(x) if and only if x = b^{y} for
every x > 0 and every real number y.
A
logarithmic function is in the form y = log_{b}(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.
2.
The range is the set of real
numbers.
3.
The line x = 0 (the yaxis)
is the vertical asymptote for the graph of f. The graph of f has
no yintercepts.
4.
Since b^{0} = 1
implies log_{b}(1) = 0 for all values of b, the graph of f has a xintercept at (1,0).
5.
If b > 1, then the graph
of f(x) = log_{b}(x)is an increasing
function.
6.
If 0<b<1, then the
graph of f(x) = log_{b}(x)is a decreasing
function.
Simplify Logarithmic Expressions
To expand
a single logarithmic expression or to condense a logarithmic expressions with
logarithms apply the following laws.
1. log_{b}(mn)
= log_{b}m + log_{b}n
2. log_{b}(m/n)
= log_{b}m – log_{b}n
3. log_{b}(m^{n})
= nlog_{b} m
For more
information relating to the properties of logarithms, please visit the links
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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.
log_{b}(mn)
= log_{b}m + log_{b}n
b.
log_{b}(m/n)
= log_{b}m – log_{b}n
c.
log_{b}(m^{n}) =
nlog_{b} m
3.
Rewrite
the single logarithm into exponential form:
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
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To
calculate the value of a logarithmic expression that does not use base 10 or
base e, use the change of base formula below.
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 3^{x} = 5^{2x1}. 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, log_{b}(m^{n}) = nlog_{b} m.
For more
information relating to solving exponential equations with logarithms, please
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