College-Level Math Test Review

Solving Systems of equations

Systems of equations are a set of simultaneous equations that all use the same variables.

A solution to a system of equations must be true for each equation in the system. Consistent systems are those with at least one solution. Inconsistent systems are systems of equations that have no solution.

To solve a system of linear equations by substitution, start with the easier equation and solve for one of the variables. Express this variable in terms of the other variable. Substitute this expression in the other equation, and solve for the other variable. The solution should be expressed in the form (x, y). Substitute the values into both of the original equations to check your answer. Consider the following problem.

Solve the system using substitution:

Solving the first equation for x:

Substitute this value in place of x in the second equation, and solve for y:

Plug this value for y back into the first equation to solve for x:

Check both equations if you have time:

Therefore, the solution is (9.6, 0.9).

To solve a system of equations using elimination, begin by rewriting both equations in standard form
.

Check to see if the coefficients of one pair of like variables add to zero. If not, multiply one or both of the equations by a non-zero number to make one set of like variables add to zero. Add the two equations to solve for one of the variables. Substitute this value into one of the original equations to solve for the other variable.

Check your work by substituting into the other equation. Next we will solve the same problem as above, but using the addition method. system using elimination: If we multiply the first equation by 2, we can eliminate the y terms:

Add the equations together and solve for x:

Plug the value for x back in to either of the original equations and solve for y:

Therefore, the solution is (9.6, 0.9).

Equations and Graphing
When algebraic functions and equations are shown graphically, they are usually shown on a Cartesian coordinate plane.

The Cartesian coordinate plane consists of two number lines placed perpendicular to each other, and intersecting at the zero point, also known as the origin. The horizontal number line is known as the x-axis, with positive values to the right of the origin, and negative values to the left of the origin. The vertical number line is known as the y-axis, with positive values above the origin, and negative values below the origin. Any point on the plane can be identified by an ordered pair in the form (x,y), called coordinates. The x-value of the coordinate is called the abscissa, and the y-value of the coordinate is called the ordinate.

The two number lines divide the plane into four quadrants: I, II, III, and IV.

Before learning the different forms equations can be written in, it is important to understand some terminology.

A ratio of the change in the vertical distance to the change in horizontal distance is called the slope.

On a graph with two points,
 and
, the slope is represented by the formula
 ;
.

If the value of the slope is positive, the line slopes upward from left to right. If the value of the slope is negative, the line slopes downward from left to right. If the y-coordinates are the same for both points, the slope is 0 and the line is a horizontal line.

If the x-coordinates are the same for both points, there is no slope and the line is a vertical line.

Two or more lines that have equal slopes are parallel lines.

Perpendicular lines have slopes that are negative reciprocals of each other, such as
 and
. A. mentioned previously, equations can be written many ways. Below is a list of the many forms equations can take. ·     

Standard form:
-intercept is <br><img data-cke-saved-src=" /> · 

 

Slope-intercept form:
is the slope and <i>b </i>is the <i>y</i>-intercept Point-slope form: <br><img src=" />m is the slope and
 is a point on the line Two-point form:
 , where
 and
 are two points on the given line ·     

 

Intercept form:
, where
 is the point at which a line intersects the x-axis, and
 is the point at which the same line intersects the y-axis Equations can also be written as
, where
.

These are referred to as one variable linear equations.

A solution to such an equation is called a root. In the case where we have the equation
, if we solve for  we get a solution of
.

In other words, the root of the equation is –2. This is found by first subtracting 10 from both sides, which gives
.

Next, simply divide both sides by the coefficient of the variable, in this case 5, to get
.

This can be checked by plugging –2 back into the original equation :
.

 

Calculations Using Points

Sometimes you need to perform calculations using only points on a graph as input data.

Using points, you can determine what the midpoint and distance are. If you know the equation for a line you can calculate the distance between the line and the point.

To find the midpoint of two points
-coordinates to get the <i>x</i>-coordinate of the midpoint, and average the <i>y</i>-coordinates to get the <i>y</i>-coordinate of the midpoint. The formula is <br><img data-cke-saved-src=" />. The distance between two points is the same as the length of the hypotenuse of a right triangle with the two given points as endpoints, and the two sides of the right triangle parallel to the x-axis and y-axis, respectively.

The length of the segment parallel to the x-axis is the difference between the x-coordinates of the two points.

The length of the segment parallel to the y-axis is the difference between the y-coordinates of the two points.

Use the Pythagorean Theorem
 or
 to find the distance.

The formula is:
.

When a line is in the format
<i>B</i>, and <i>C</i> are coefficients, you can use a point (<i>x</i>₁, <i>y</i>₁) not on the line and apply the formula <br><img data-cke-saved-src=" /> to find the distance between the line and the point (x1, y1). 

Functions
A function is an equation that has exactly one value of output variable (dependent variable) for each value of the input variable (independent variable).

The set of all values for the input variable (here assumed to be x) is the domain of the function, and the set of all corresponding values of output variable (here assumed to be y) is the range of the function.

When looking at a graph of an equation, the easiest way to determine if the equation is a function or not is to conduct the vertical line test.

If a vertical line drawn through any value of x crosses the graph in more than one place, the equation is not a function.

In functions with the notation f(x), the value substituted for x in the equation is called the argument.

The domain is the set of all values for x in a function. Unless otherwise given, assume the domain is the set of real numbers that will yield real numbers for the range.

This is the domain of definition.

The graph of a function is the set of all ordered pairs (x, y) that satisfy the equation of the function.

The points that have zero as the value for y are called the zeros of the function. These are also the x-intercepts, because that is the point at which the graph crosses, or intercepts, the x-axis.

The points that have zero as the value for x are the y-intercepts because that is where the graph crosses the y-axis.

Any time there are vertical asymptotes or holes in a graph, such that the complete graph cannot be drawn as one continuous line, a graph is said to have discontinuities.

Examples would include the graphs of hyperbolas that are functions, and the function
.

Manipulation of Functions

Horizontal and vertical shift occur when values are added to or subtracted from the x or y values, respectively.

If a constant is added to the y portion of each point, the graph shifts up. If a constant is subtracted from the y portion of each point, the graph shifts down. This is represented by the expression  is a constant. If a constant is added to the x portion of each point, the graph shifts left. If a constant is subtracted from the x portion of each point, the graph shifts right. This is represented by the expression <br><img src=" />k Stretch, compression, and reflection occur when different parts of a function are multiplied by different groups of constants. If the function as a whole is multiplied by a real number constant greater than 1 (stretched vertically. If <i>k</i> in the previous equation is greater than zero but less than 1, the graph is compressed vertically. If <i>k</i> is less than zero, the graph is reflected about the <i>x</i>-axis, in addition to being either stretched or compressed vertically if <i>k</i> is less than or greater than -1, respectively. If instead, just the <i>x</i>-term is multiplied by a constant greater than 1 (<br><img src=" />compressed horizontally. If k in the previous equation is greater than zero but less than 1, the graph is stretched horizontally. If k is less than zero, the graph is reflected about the y-axis, in addition to being either stretched or compressed horizontally if k is greater than or less than -1, respectively.

 

Classification of Functions

There are many different ways to classify functions based on their structure or behavior. Listed here are a few common classifications.

Constant functions are given by the equation y = b or
is a real number. There is no independent variable present in the equation, so the function has a constant value for all <i>x</i>. The graph of a constant function is a horizontal line of slope 0 that is positioned <i>b</i> units from the <i>x</i>-axis. If <i>b</i> is positive, the line is above the <i>x</i>-axis; if <i>b</i> is negative, the line is below the <i>x</i>-axis. Identity functions are identified by the equation <i>y= x </i>or <br><img data-cke-saved-src=" />, where every value of y is equal to its corresponding value of x.

The only zero is the point (0, 0). The graph is a diagonal line with slope 1. In linear functions, the value of the function changes in direct proportion to x.

The rate of change, represented by the slope on its graph, is constant throughout.

The standard form of a linear equation is
<i>b</i>, and <i>c</i> are real numbers. As a function, this equation is commonly written as <br><img src=" />x to get
 , which is the only zero of the function. The domain and range are both the set of all real numbers.

A polynomial function is a function with multiple terms and multiple powers of x, such as
where n is a non-negative integer that is the highest exponent in the polynomial, and
.

The domain of a polynomial function is the set of all real numbers.

If the greatest exponent in the polynomial is even, the polynomial is said to be of even degree and the range is the set of real numbers that satisfy the function.

If the greatest exponent in the polynomial is odd, the polynomial is said to be odd and the range, like the domain, is the set of all real numbers.

A quadratic function is a polynomial function that follows the equation pattern
<i>b</i>, and <i>c</i> are real numbers and <i>a</i> ≠ 0. The domain of a quadratic function is the set of all real numbers. The range is also real numbers, but only those in the subset of the domain that satisfy the equation. To determine the number of roots of a quadratic equation, solve the expression <br><img data-cke-saved-src=" />. If this value is positive, there are two unique real roots.

If this value equals zero, there is one real root, which is a double root. If this value is less than zero, there are no real roots. The root(s) of any quadratic function can be found by plugging the values of a, b, and c into the quadratic formula:

If the expression  is negative, you will instead find complex roots.

A quadratic function has a parabola for its graph.

In the equation
is positive, the parabola will open upward. If <i>a</i> is negative, the parabola will open downward. The axis of symmetry is a vertical line that passes through the vertex. To determine whether or not the parabola will intersect the <i>x</i>-axis, check the number of real roots. An equation with two real roots will cross the <i>x</i>-axis twice. An equation with one real root will have its vertex on the <i>x</i>-axis. An equation with no real roots will not contact the <i>x</i>-axis. A rational function is a function that can be constructed as a ratio of two polynomial expressions: <br><img src=" />domain is the set of all real numbers, except any values for which
range is the set of real numbers that satisfies the function when the domain is applied. When you graph a rational function, you will have vertical asymptotes wherever <br><img src=" />pn and qn-1 are the coefficients of the highest degree terms in their respective polynomials.

A square root function is a function that contains a radical and is in the format
.

The domain is the set of all real numbers that yields a positive radicand or a radicand equal to zero. Because square root values are assumed to be positive unless otherwise identified, the range is all real numbers from zero to infinity. To find the zero of a square root function, set the radicand equal to zero and solve for x.

The graph of a square root function is always to the right of the zero and always above the x-axis.

An absolute value function is in the format
.

Like other functions, the domain is the set of all real numbers.

However, because absolute value indicates positive numbers, the range is limited to positive real numbers.

To find the zero of an absolute value function, set the portion inside the absolute value sign equal to zero and solve for x.

An absolute value function is also known as a piecewise function because it must be solved in pieces – one for if the value inside the absolute value sign is positive, and one for if the value is negative.

The function can be expressed as

This will allow for an accurate statement of the range.

Exponential functions are equations that have the format
> 0 and<i> b</i> ≠ 1. The exponential function can also be written <br><img data-cke-saved-src=" />.

Logarithmic functions are equations that have the format
may be any number except one; however, the most common bases for logarithms are base 10 and base <i>e</i>. The log base <i>e</i> is known the natural logarithm, or <i>ln</i>, expressed by the function <br><img data-cke-saved-src=" />.

Any logarithm that does not have an assigned value of b is assumed to be base 10:
.

Exponential functions and logarithmic functions are related in that one is the inverse of the other.

If
, then
.

This can perhaps be expressed more clearly by the two equations:
 and
.

The following properties apply to logarithmic expressions:

In a one-to-one function, each value of x has exactly one value for y (this is the definition of a function) and each value of y has exactly one value for x.

While the vertical line test will determine if a graph is that of a function, the horizontal line test will determine if a function is a one-to-one function.

If a horizontal line drawn at any value of y intersects the graph in more than one place, the graph is not that of a one-to-one function.

Do not make the mistake of using the horizontal line test exclusively in determining if a graph is that of a one-to-one function.

A one-to-one function must pass both the vertical line test and the horizontal line test. One-to-one functions are also invertible functions.

A monotone function is a function whose graph either constantly increases or constantly decreases. Examples include the functions
,
, or
.

An even function has a graph that is symmetric with respect to the y-axis and satisfies the equation  is any real number and <i>n</i> is a positive even integer. A. odd function has a graph that is symmetric with respect to the origin and satisfies the equation <br><img src=" />a is any real number and n is a positive odd integer. Algebraic functions are those that exclusively use polynomials and roots. These would include polynomial functions, rational functions, square root functions, and all combinations of these functions, such as polynomials as the radicand. These combinations may be joined by addition, subtraction, multiplication, or division, but may not include variables as exponents.

Transcendental functions are all functions that are non-algebraic.

Any function that includes logarithms, trigonometric functions, variables as exponents, or any combination that includes any of these is not algebraic in nature, even if the function includes polynomials or roots.

Related Concepts

According to the Fundamental Theorem of Algebra, every non-constant, single variable polynomial has exactly as many roots as the polynomial's highest exponent.

For example, if  is the largest exponent of a term, the polynomial will have exactly 4 roots.

However, some of these roots may have multiplicity or be non-real numbers. For instance, in the polynomial function , the only real roots are 1 and -1. The root 1 has multiplicity of 2 and there is one non-real root ().

The Remainder Theorem is useful for determining the remainder when a polynomial is divided by a binomial. The Remainder Theorem states that if a polynomial function f(x) is divided by a binomial x – a, where a is a real number, the remainder of the division will be the value of f(a). If f(a) = 0, then a is a root of the polynomial.

The Factor Theorem is related to the Remainder Theorem and states that if f(a) = 0 then (x – a) is a factor of the function.

According to the Rational Root Theorem, any rational root of a polynomial function