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# Page 1 Mathematics Study Guide for the TASC

## How to prepare for the TASC Math Test

### General Information

Like all TASC sections, the Math section is heavily aligned with Common Core State Standards but does not specifically address Common Core objectives that are labeled “advanced.” The TASC also presents items that assess college and career readiness. Be sure you are thoroughly familiar with all of the concepts that follow and seek more practice with any area in which you experience difficulty.

Math concepts are grouped in sections here by order of their emphasis on the test.

### Most Emphasized Concepts

You will find quite a few questions about algebra and functions on this test, as well as items relating to geometry and numeration.

#### Operations with Polynomials

Questions involving polynomials in the TASC math section deal mainly with arithmetic operations, factoring, and finding the zeroes of polynomials.

Polynomials are algebraic expressions consisting of constants, variables and exponents combined by addition, subtraction, multiplication and division. By definition, a polynomial will not have a term divided by a variable.

Addition and subtraction of polynomials are done by grouping like terms together. “Like terms” are terms with the same variable and exponent; their coefficients (or the constants attached to the variables) can be different. In either operation, the result will still be a polynomial.

Example: Subtract $x^2 + 6x + 5$ from $x^3 + 4x^2 - 15x + 8$

$(x^3 + 4x^2 - 15x + 8) - (x^2 + 6x + 5)$
$x^3 + 4x^2 - x^2 - 15x - 6x + 8 - 5$
$x^3 + 3x^2 - 21x + 3$

To multiply polynomials, each term of a polynomial is multiplied by each term of another polynomial. The product of polynomials is a polynomial.

Example: Multiply $(x^2 + 3y)$ by $(x + 1)$

$(x^2 + 3y) (x + 1)$
$x^3 + 3xy + x^2 + 3y$
$x^3 + x^2 + 3xy + 3y$

Polynomials can be divided, but the result may or may not be a polynomial. Each term of the first polynomial must be divided by each term of the second polynomial. Factoring can also be used to simplify division.

Example: Divide $-5x^3 - 15x$ by $x^2 + 3$

“Solving a polynomial”, or finding its roots or zeroes means finding the x-intercepts of the polynomial when f(x) equals zero. This can be done by graphing or solving numerically.

#### Equations and Inequalities

Some questions may involve equations and require finding an unknown value. To solve equations with two unknowns (represented by variables), at least two equations must be given or derived. Practice solving equations by the elimination, substitution and graphing methods.

Example:

Find the values of x and y in these equations: $2x = 3y$ and $x – y + 3 = 5$

Solution by the substitution method:

Use the first equation and express x in terms of y:

Use this expression and substitute to the x terms in the second equation:

Plug in the value of y to either of the two equations (TIP: choose the equation that looks simpler):

$2x = 3y$
$2x = 3 \cdot 4$
$x = 6$

Remember, your answers can always be verified by plugging both your x and y solution into either equation and confirming that the values yield a true statement.

Solution by the elimination method:

Write second equation below the first (the same way you do vertical subtraction):

$x – y + 3 = 5$
- $2x = 3y$

Examine the equations and see how you can modify either of the equations so that one variable can be eliminated by performing vertical subtraction of the two equations. In this particular question, we can multiply the first equation by 2 and eliminate 2x by subtraction:

Multiply by 2: $2x – 2y + 6 = 10$

Rearrange it: $2x = 2y + 4$

Subtract the second equation from it:

$2x = 2y + 4$
- ($2x = 3y$)
———————-
$0 = -y + 4$
$y = 4$

Substitute this value to either of the two equations to find x = 6.

Solution by the graphing method:

Plot/Graph the two lines. The coordinates of the point where the two lines meet are the values for x and y.

Inequalities show the relative size of one algebraic expression compared to another and use inequality symbols, such as $\gt$, $\lt$, $\ge$ and $\le$, between the left and right expressions.

The inequality below states that the expression $x^2 - 9x + 20$ is greater than or equal to $10x - 6$:

When solving inequalities, it is important to remember that a symbol of inequality changes direction when a negative (-) value is introduced by multiplication or division, or when the two sides of the equation are switched or interchanged.

#### Equations and Expressions

Review algebraic expressions and equations, polynomials, degrees of polynomials, factoring polynomials, linear equations and quadratic equations. Develop the skill of properly interpreting expressions.

The degree of a polynomial in one variable refers to the highest exponent of the variable.

This polynomial has a degree of 2

The degree of a polynomial with 2 or more variables is the highest sum of the exponents of a term.

This polynomial has a degree of 4 (3 + 1)

Linear equations are examples of polynomials with degree 1. The linear equation $y = 3x + 5$ can be interpreted as a straight line that has a slope of 3 and crosses the y-axis at $y = 5$. A positive slope shows a straight line that increases from left to right.

Quadratic equations are examples of polynomials with degree 2. Below is a quadratic equation:

It follows the standard form of quadratic equations which is:

The sample quadratic equation can be factored and yields the following values of x where the curve crosses the x-axis:

$(x - 2)(x - 1) = 0$
$x = 2$
$x = 1$

A positive a coefficient shows a parabola opening upwards. To find its vertex, we get a clue from the expression below, which is the value of x at the vertex.

Solving for y when x is 3/2 or 1.5, we get

We then interpret $x^2 - 3x + 2$ as a parabola facing upward that crosses the x-axis at points (2, 0) and (1, 0), and has a vertex at (1.5, -0.25).

The skill to interpret and visualize equations is especially useful when solving word problems.

#### Making Sense of Functions

A function, usually denoted by f(x) (but can also be denoted by g(x), m() or other variables and notations), describes a relationship that takes an input, or an element of a set (called the domain), to produce an output, or exactly one element of a set (called the range). Consider the function below:

For every input of x, one output of f(x) comes out; for instance:

Conversely, the equation $(m) = \pm\sqrt{m}$ is not a function because for every value of m, two values come out from $(m)$.

It is important to know how to perform operations with functions. Functions can be added, subtracted, multiplied, divided, and composed.

Addition: Simply add the two functions. To add f(x) and g(x), the operation is performed this way:

(f + g)(x) = f(x) + g(x)

Example: What is $(f + g)(x)$ if $f(x) = x^2 + 1$ and $g(x) = 3x$?

Subtraction: The operation is written as $(f - g)(x)$ and, using the example above, subtraction is performed as:

Take note that for $(g - f)(x)$, the result will be different.

Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

Let’s illustrate using the same sample functions.

Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$

So: $(\frac{f}{g})(x) = \frac{x^2+1}{3x}$

Composition: Apply the function to the result of another function. The symbol for composition is a tiny circle, i.e., (f g)(x). Using the same functions given as example, the following are two different function compositions:

#### Linear, Quadratic, and Exponential Functions

Below are some common functions:

Linear Function

The standard form for linear functions is given as $f(x) = mx + b$.

If we plot and observe the input-output values in a linear function, the change will be uniformly increasing or decreasing.

Example: $f(x) = 3x + 1$
The input-output values are:

There is a uniform increase of 3, which is actually the slope of the line.

The standard form for quadratic functions is given as $f(x) = ax^2 + bx + c$

Take note of the change in input-output values in the example below.

Example: $f(x) = x^2 + 7x - 12$
The input-output values are:

The pattern of change in quadratic functions is definitely different from our previous linear model. On the graph, this will show a parabola pointing upward and increasing faster vertically than it increases horizontally.

Exponential Function

Exponential functions have this standard form: $f(x) = a^x$ where $a \gt 0$

Example: $f(t) = 3e^t$

The input-output values are:

The rate of change in an exponential function significantly exceeds the rates of change in other functions modeled; hence, we say that the change in this model is exponential. Exponential functions are seen in math problems involving compound interest and growth and decay formulas.

#### Measuring Geometric Shapes and Solids

Common formulas for geometric shapes are provided in the TASC math reference sheets, so there’s no need to memorize them. Practice solving for the surface area of closed flat shapes and for the volume or capacity of solid geometric shapes.

Questions may not directly state the shape of the solid, and you need to be able to visualize situations given. For instance, a question may ask for the volume of a solid generated after rotating a rectangle 3-cm wide and 5-cm length rectangle, if the flat surface is rotated along its length.

#### Rational and Irrational Numbers

Rational numbers are real numbers that can be written as ratios or fractions, or in the form:

$\frac{m}{n}$ where $n \ne 0$

These are rational numbers: 25, 0.5, -0.25, ½, $\sqrt{4}$.

Irrational numbers, on the other hand, are real numbers that cannot be written as ratios or fractions. There are examples of irrational numbers: $\pi$, $\sqrt{5}$, Euler’s number e.

Rational exponents are exponents that are fractions; hence, they are also called fractional exponents, and the whole expression is called a radical.

A radical expression in the form $x^{\frac{m}{n}}$ is the same as $\sqrt[n]{x^m}$ or $(\sqrt[n]{x})^m$.

The laws of exponents that apply to whole-number exponents work in the same manner for rational exponents as shown below:

• $x^m \cdot x^n = x^{m+n}$

example: $x^{\frac{2}{3}} \cdot x^{\frac{1}{2}} = x^{\frac{2}{3} + \frac{1}{2}} = x^{\frac{7}{6}}$

• $\frac{x^m}{x^n} = x^{m-n}$

example: $\frac{x^{\frac{2}{3}}}{x^{\frac{1}{2}}} = x^{\frac{2}{3}-\frac{1}{2}} = x^{\frac{1}{6}}$

• $(x^m)^n = x^{m \cdot n}$

example: $(x^{\frac{2}{3}})^{\frac{1}{2}} = x^{\frac{2}{3} \cdot \frac{1}{2}} = x^{\frac{1}{3}}$

• $(xy)^m = x^m \cdot y^m$

example: $(xy)^{\frac{1}{2}} = x^{\frac{1}{2}} \cdot y^{\frac{1}{2}}$

• $(\frac{x}{y})^m = \frac{x^m}{y^m}$

example: $(\frac{x}{y})^{\frac{1}{2}} = \frac{x^\frac{1}{2}}{y^\frac{1}{2}}$

• $x^{-m} = \frac{1}{x^m}$

example: $x^{-\frac{1}{2}} = \frac{1}{x^\frac{1}{2}}$