Developmental Mathematics Waiver Exam-Definitions Sheet

• Integer Ope rations

- An exponent is shorthand notation for repeated multiplication. Thus

- In general, a negative exponent indicates a reciprocal. Thus

- The absolute value bars make an ex pression positive . Thus

- The proper order of operations is referred to as PEMDAS. Take care of parentheses first,
then exponents, then multiplication and division (from left to right) and finally addition and
subtraction (from left to right).

• Fractions

- Remember that when adding and subtracting fractions , one needs the same number in the de -
nominator (bottom) of the fraction. This is unnecessary when multiplying or dividing.

• Evaluating an Expression

- Evaluating an expression means to plug in the given values for the variables specified. For example
if the expression is 2x + 3y - 7z and x = 1, y = -2 and z = 3. Then

• Simplifying an Expression

- Simplifying an expression involves removing any parentheses (by distributing) and combining like
terms . For example:

• Solving an Equation

- To solve an equation the ultimate goal is to get the variable by itself equal to a number. There
are several ways to manipulate and equation to make this happen.
- You may add or subtract the same term from both sides.
- You may multiply or divide the same non- zero term from both sides.
- You should simplify both sides of the equation first (by combining like terms and or distributing
through parentheses) before you begin applying the above operations.

- Here is a simple example:

• Solving Applications

- There are several types of applications on the waiver exam. One type of problem that is especially
important is working with percents. You should know that in general, the percent refers to the
change divided by the original value. For example, if McDonald's reduces the Big Mac from $2.00
to $1.50, then the price has changed by $.50 and the original price was $2.00. Thus the percent
change is:

• Graphing Linear Equations

- If a line is in slope-intercept (y = mx + b) form, then the number b tells you the y-intercept
which is where the graph crosses the y-axis (the vertical axis).

- The number m tells you the slope of the line. The numerator of the slope tells you how far to
move up or down from the y-intercept (the rise) and the denominator tells you how far to move
to the left or right (the run) from that point.

- For example to graph , start by placing the point (0, 1) on the graph. This is your
y-intercept. Then move down 2 and to the right 3 (which takes you to the point (3,-1) on the
graph. Connect the dots and you have the graph of the line.

- If a line is not in slope-intercept form, you can solve for y (get y by itself) to put it in slope-intercept
form.

- A vertical line is of the form x = 5 and a horizontal line is of the form y = 6.

• Equations of Lines

- To find the x-intercept (where the graph crosses the x-axis), set y = 0 and solve for x. To find
the y-intercept, set x = 0 and solve for y. For example, for 12x - 30y = 60.

* When we set y = 0, we get x = 5.
* When we set x = 0, we get y = -2.
* We write the final answers as points: (5, 0) and (0,-2).

- To find the equation of a line using a point and the slope it is helpful to know the point-slope
version of the equation of a line, . So to find the equation of the line with
slope 11 which passes through (-1, 2). We have y - 2 = 11(x - (-1)). To get the final answer in
slope-intercept form, we solve for y and the solution is y = 11x + 13.

- To find the slope between two points and we use the formula

• Functions

- We evaluate a function at a given value by plugging in the given value for the independent variable.
For example, if f(x) = x³ - 4x² and we want to find f(2). We get:

• Polynomials

- When subtracting polynomials remember to distribute the minus sign to each term in the second
polynomial. For example.

- To multiply two polynomials, you can multiply vertically (as depicted on the sample exam solution
guide) or you can use the FOIL method. If both polynomials have two terms you can use the
FOIL shortcut: First, Outer Inner Last. So for example:

- Factoring a polynomial is essentially in our case to find what binomials multiply together to give
our polynomial. So for example, 2x² - 9x - 5 factors as (2x + 1)(x - 5).

- If every term of the polynomial is divisible by some common factor , factor this out before beginning
to factor into binomials. For example:

- To solve a quadratic equation, the first method to try is to

* Get the polynomial = 0.
* Factor the polynomial.
* Set each factor - 0.
* Solve the resulting equations.