# Test Course

## Course Outline

## Unit 1: Foundations

#### Terms to Know

- absolute value
- The absolute value of a number is its distance from $0$on the number line.

- additive identity
- The number 0 is the additive identity because adding 0 to any number does not change its value.

- additive inverse
- The opposite of a number is its additive inverse.

- coefficient
- The coefficient of a term is the constant that multiplies the variable in a term.

- complex fraction
- A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

- composite number
- A composite number is a counting number that is not prime. It has factors other than 1 and the number itself.

- constant
- A constant is a number whose value always stays the same.

- denominator
- In a fraction, written
$\frac{a}{b},$where
$b\ne 0,$the denominator
*b*is the number of equal parts the whole has been divided into.

- divisible by a number
- If a number
*m*is a multiple of*n*, then*m*is divisible by*n*.

- equation
- An equation is two expressions connected by an equal sign.

- equivalent fractions
- Equivalent fractions are fractions that have the same value.

- evaluate an expression
- To evaluate an expression means to find the value of the expression when the variables are replaced by given numbers.

- expression
- An expression is a number, a variable, or a combination of numbers and variables using operation symbols.

- factors
- If
$a\xb7b=m,$, then
*a*and*b*are factors of*m*.

- fraction
- A fraction is written
$\frac{a}{b},$, where
$b\ne 0,$, and
*a*is the numerator and*b*is the denominator. A fraction represents parts of a whole.

- integers
- The whole numbers and their opposites are called the integers.

- irrational number
- An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

- least common denominator
- The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

- least common multiple
- The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

- like terms
- Terms that are either constants or have the same variables raised to the same powers are called like terms.

- multiple of a number
- A number is a multiple of
*n*if it is the product of a counting number and*n.*

- multiplicative identity
- The number 1 is the multiplicative identity because multiplying 1 by any number does not change its value.

- multiplicative inverse
- The reciprocal of a number is its multiplicative inverse.

- negative numbers
- Numbers less than $0$are negative numbers.

- numerator
- In a fraction, written
$\frac{a}{b},$, where
$b\ne 0,$, the numerator
*a*indicates how many parts are included.

- opposite
- The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

- order of operations
- The order of operations are established guidelines for simplifying an expression.

- percent
- A percent is a ratio whose denominator is 100.

- prime factorization
- The prime factorization of a number is the product of prime numbers that equals the number.

- prime number
- A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.

- principal square root
- The positive square root is called the principal square root.

- rational number
- A rational number is a number of the form
$\frac{p}{q},$, where
*p*and*q*are integers and $q\ne 0.$Its decimal form stops or repeats.

- real number
- A real number is a number that is either rational or irrational.

- reciprocal
- The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.

- simplify an expression
- To simplify an expression means to do all the math possible.

- square of a number
- If
${n}^{2}=m,$, then
*m*is the square of*n*.

- square root of a number
- If
${n}^{2}=m,$, then
*n*is a square root of*m*.

- term
- A term is a constant, or the product of a constant and one or more variables.

- variable
- A variable is a letter that represents a number whose value may change.

### Key Concepts

#### 1.1

Use the Language of Algebra**Divisibility Tests**A number is divisible by: 2 if the last digit is 0, 2, 4, 6, or 8. 3 if the sum of the digits is divisible by 3. 5 if the last digit is 5 or 0. 6 if it is divisible by both 2 and 3. 10 if it ends with 0.**How to find the prime factorization of a composite number.**- Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
- Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
- Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
- Step 4. Write the composite number as the product of all the circled primes.

**How To Find the least common multiple using the prime factors method.**- Step 1. Write each number as a product of primes.
- Step 2. List the primes of each number. Match primes vertically when possible.
- Step 3. Bring down the columns.
- Step 4. Multiply the factors.

**Equality Symbol**$a=b$is read “*a*is equal to*b*.” The symbol “=” is called the equal sign.**Inequality****Inequality Symbols**Inequality Symbols Words $a\ne b$ *a*is*not equal to b.*$a<b$ *a*is*less than b.*$a\le b$ *a*is*less than or equal to b.*$a>b$ *a*is*greater than b.*$a\ge b$ *a*is*greater than or equal to b.***Grouping Symbols**$\begin{array}{cccccc}\text{Parentheses}\hfill & & & & & \left(\phantom{\rule{0.2em}{0ex}}\right)\hfill \\ \text{Brackets}\hfill & & & & & \left[\phantom{\rule{0.2em}{0ex}}\right]\hfill \\ \text{Braces}\hfill & & & & & \left\{\phantom{\rule{0.2em}{0ex}}\right\}\hfill \end{array}$**Exponential Notation**${a}^{n}$means multiply*a*by itself,*n*times. The expression ${a}^{n}$is read*a*to the ${n}^{th}$power.**Simplify an Expression**To simplify an expression, do all operations in the expression.**How to use the order of operations.**- Step 1.
Parentheses and Other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

- Step 2.
Exponents
- Simplify all expressions with exponents.

- Step 3.
Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.

- Step 4.
Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.

- Step 1.
**How to combine like terms.**- Step 1. Identify like terms.
- Step 2. Rearrange the expression so like terms are together.
- Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.

Operation Phrase Expression **Addition***a*plus*b*the sum of $a$and*b**a*increased by*b**b*more than*a*the total of*a*and*b**b*added to*a*$a+b$ **Subtraction***a*minus $b$ the difference of*a*and*b**a*decreased by*b**b*less than*a**b*subtracted from*a*$a-b$ **Multiplication***a*times*b*the product of $a$and $b$ twice*a*$a\xb7b,ab,a(b),(a)(b)$ $2a$ **Division***a*divided by*b*the quotient of*a*and*b*the ratio of*a*and*b**b*divided into*a*$a\xf7b,a\text{/}b,\frac{a}{b},b\overline{)a}$

#### 1.2

Integers**Opposite Notation**$$\begin{array}{c}\text{\u2212}a\phantom{\rule{0.2em}{0ex}}\text{means the opposite of the number}\phantom{\rule{0.2em}{0ex}}a\hfill \\ \text{The notation}\phantom{\rule{0.2em}{0ex}}\text{\u2212}a\phantom{\rule{0.2em}{0ex}}\text{is read as \u201cthe opposite of}\phantom{\rule{0.2em}{0ex}}a\text{.\u201d}\hfill \end{array}$$**Absolute Value**The absolute value of a number is its distance from 0 on the number line. The absolute value of a number*n*is written as $\left|n\right|$and $\left|n\right|\ge 0$for all numbers. Absolute values are always greater than or equal to zero.**Grouping Symbols**$$\begin{array}{cccccccccc}\text{Parentheses}\hfill & & & \left(\phantom{\rule{0.2em}{0ex}}\right)\hfill & & & \text{Braces}\hfill & & & \left\{\phantom{\rule{0.2em}{0ex}}\right\}\hfill \\ \text{Brackets}\hfill & & & \left[\phantom{\rule{0.2em}{0ex}}\right]\hfill & & & \text{Absolute value}\hfill & & & \phantom{\rule{0.2em}{0ex}}\left|\phantom{\rule{0.2em}{0ex}}\right|\hfill \end{array}$$**Subtraction Property**$\phantom{\rule{2em}{0ex}}a-b=a+\left(\text{\u2212}b\right)$ Subtracting a number is the same as adding its opposite.**Multiplication and Division of Signed Numbers**For multiplication and division of two signed numbers:__Same signs____Result__• Two positives Positive • Two negatives Positive __Different signs____Result__• Positive and negative Negative • Negative and positive Negative **Multiplication by**$\mathrm{-1}$ $\phantom{\rule{2em}{0ex}}\mathrm{-1}a=\text{\u2212}a$ Multiplying a number by $\mathrm{-1}$gives its opposite.**How to Use Integers in Applications.**- Step 1.
**Read**the problem. Make sure all the words and ideas are understood - Step 2.
**Identify**what we are asked to find. - Step 3.
**Write a phrase**that gives the information to find it. - Step 4.
**Translate**the phrase to an expression. - Step 5.
**Simplify**the expression. - Step 6.
**Answer**the question with a complete sentence.

- Step 1.

#### 1.3

Fractions**Equivalent Fractions Property**If*a*,*b*, and*c*are numbers where $b\ne 0,c\ne 0,$then $\phantom{\rule{2em}{0ex}}\frac{a}{b}=\frac{a\xb7c}{b\xb7c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{a\xb7c}{b\xb7c}=\frac{a}{b}.$**How to simplify a fraction.**- Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
- Step 2. Simplify using the Equivalent Fractions Property by dividing out common factors.
- Step 3. Multiply any remaining factors.

**Fraction Multiplication**If*a*,*b*,*c*, and*d*are numbers where $b\ne 0,$and $d\ne 0,$then $\phantom{\rule{2em}{0ex}}\frac{a}{b}\xb7\frac{c}{d}=\frac{ac}{bd}.$ To multiply fractions, multiply the numerators and multiply the denominators.**Fraction Division**If*a*,*b*,*c*, and*d*are numbers where $b\ne 0,c\ne 0,$and $d\ne 0,$then $\phantom{\rule{2em}{0ex}}\frac{a}{b}\xf7\frac{c}{d}=\frac{a}{b}\xb7\frac{d}{c}.$ To divide fractions, we multiply the first fraction by the reciprocal of the second.**Fraction Addition and Subtraction**If*a*,*b*, and*c*are numbers where $c\ne 0,$then $\phantom{\rule{2em}{0ex}}\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}.$ To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.**How to add or subtract fractions.**- Step 1.
Do they have a common denominator?
- Yes—go to step 2.
- No—rewrite each fraction with the LCD (least common denominator).
- Find the LCD.
- Change each fraction into an equivalent fraction with the LCD as its denominator.

- Step 2. Add or subtract the fractions.
- Step 3. Simplify, if possible.

- Step 1.
**How to simplify an expression with a fraction bar.**- Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
- Step 2. Simplify the fraction.

**Placement of Negative Sign in a Fraction**For any positive numbers*a*and*b*, $\phantom{\rule{2em}{0ex}}\frac{\text{\u2212}a}{b}=\frac{a}{\text{\u2212}b}=-\frac{a}{b}.$**How to simplify complex fractions.**- Step 1. Simplify the numerator.
- Step 2. Simplify the denominator.
- Step 3. Divide the numerator by the denominator. Simplify if possible.

#### 1.4

Decimals**How to round decimals.**- Step 1. Locate the given place value and mark it with an arrow.
- Step 2. Underline the digit to the right of the place value.
- Step 3.
Is the underlined digit greater than or equal to $5?$
- Yes: add 1 to the digit in the given place value.
- No: do
__not__change the digit in the given place value

- Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.

**How to add or subtract decimals.**- Step 1. Determine the sign of the sum or difference.
- Step 2. Write the numbers so the decimal points line up vertically.
- Step 3. Use zeros as placeholders, as needed.
- Step 4. Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.
- Step 5. Write the sum or difference with the appropriate sign

**How to multiply decimals.**- Step 1. Determine the sign of the product.
- Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
- Step 3. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
- Step 4. Write the product with the appropriate sign.

**How to multiply a decimal by a power of ten.**- Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
- Step 2. Add zeros at the end of the number as needed.

**How to divide decimals.**- Step 1. Determine the sign of the quotient.
- Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places—adding zeros as needed.
- Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
- Step 4. Write the quotient with the appropriate sign.

**How to convert a decimal to a proper fraction and a fraction to a decimal.**- Step 1. To convert a decimal to a proper fraction, determine the place value of the final digit.
- Step 2.
Write the fraction.
- numerator—the “numbers” to the right of the decimal point
- denominator—the place value corresponding to the final digit

- Step 3. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

**How to convert a percent to a decimal and a decimal to a percent.**- Step 1. To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
- Step 2. To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.

**Square Root Notation**$\sqrt{m}$is read “the square root of*m*.” If $m={n}^{2},$then $\sqrt{m}=n,$for $n\ge 0.$ The square root of*m*, $\sqrt{m},$is the positive number whose square is*m*.**Rational or Irrational**If the decimal form of a number*repeats or stops*, the number is a rational number.*does not repeat and does not stop*, the number is an irrational number.

**Real Numbers**

#### 1.5

Properties of Real NumbersCommutative Property
When adding or multiplying, changing the order gives the same result
$\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{9.1em}{0ex}}a+b& =\hfill & b+a\hfill \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{9.1em}{0ex}}a\xb7b& =\hfill & b\xb7a\hfill \end{array}$ |

Associative Property
When adding or multiplying, changing the grouping gives the same result.
$\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\left(a+b\right)+c& =\hfill & a+\left(b+c\right)\hfill \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\left(a\xb7b\right)\xb7c& =\hfill & a\xb7\left(b\xb7c\right)\hfill \end{array}$ |

Distributive Property
$\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill \phantom{\rule{12.4em}{0ex}}a\left(b+c\right)& =\hfill & ab+ac\hfill \\ \\ & & & \hfill \phantom{\rule{12.4em}{0ex}}\left(b+c\right)a& =\hfill & ba+ca\hfill \\ \\ & & & \hfill \phantom{\rule{12.4em}{0ex}}a\left(b-c\right)& =\hfill & ab-ac\hfill \\ \\ & & & \hfill \phantom{\rule{12.4em}{0ex}}\left(b-c\right)a& =\hfill & ba-ca\hfill \end{array}$ |

Identity Property
$\begin{array}{cccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a\text{:}\hfill & & & \phantom{\rule{12.1em}{0ex}}a+0=a\hfill \\ \phantom{\rule{4em}{0ex}}0\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{additive identity}\hfill & & & \phantom{\rule{12.1em}{0ex}}0+a=a\hfill \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a\text{:}\hfill & & & \phantom{\rule{12.65em}{0ex}}a\xb71=a\hfill \\ \phantom{\rule{4em}{0ex}}1\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{multiplicative identity}\hfill & & & \phantom{\rule{12.65em}{0ex}}1\xb7a=a\hfill \end{array}$ |

Inverse Property
$\begin{array}{cccc}\phantom{\rule{2em}{0ex}}\text{of addition}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,\hfill & & & \hfill \phantom{\rule{7.1em}{0ex}}a+\left(\text{\u2212}a\right)=0\\ \phantom{\rule{4em}{0ex}}\text{\u2212}a\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{additive inverse}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}a\hfill & & & \\ \phantom{\rule{4em}{0ex}}\text{A number and its}\phantom{\rule{0.2em}{0ex}}opposite\phantom{\rule{0.2em}{0ex}}\text{add to zero.}\hfill & & & \\ \phantom{\rule{2em}{0ex}}\text{of multiplication}\phantom{\rule{0.2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,a\ne 0\hfill & & & \hfill \phantom{\rule{7.4em}{0ex}}a\xb7\frac{1}{a}=1\\ \\ \phantom{\rule{4em}{0ex}}\frac{1}{a}\phantom{\rule{0.2em}{0ex}}\text{is the}\phantom{\rule{0.2em}{0ex}}\text{multiplicative inverse}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}a\hfill & & & \\ \phantom{\rule{4em}{0ex}}\text{A number and its}\phantom{\rule{0.2em}{0ex}}reciprocal\phantom{\rule{0.2em}{0ex}}\text{multiply to one.}\hfill & & & \end{array}$ |

Properties of Zero
$\begin{array}{cccc}\phantom{\rule{2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,\hfill & & & \phantom{\rule{16.7em}{0ex}}a\xb70=0\hfill \\ & & & \phantom{\rule{16.7em}{0ex}}0\xb7a=0\hfill \\ \phantom{\rule{2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,a\ne 0,\hfill & & & \phantom{\rule{17.7em}{0ex}}\frac{0}{a}=0\hfill \\ \phantom{\rule{2em}{0ex}}\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,\hfill & & & \phantom{\rule{15.7em}{0ex}}\frac{a}{0}\phantom{\rule{0.2em}{0ex}}\text{is undefined}\hfill \end{array}$ |

### Key Concepts

#### 3.1

Graph Linear Equations in Two Variables**Points on the Axes**- Points with a
*y*-coordinate equal to 0 are on the*x*-axis, and have coordinates $\left(a,0\right).$ - Points with an
*x*-coordinate equal to $0$are on the*y*-axis, and have coordinates $\left(0,b\right).$

- Points with a
**Quadrant**$$\begin{array}{cccc}\hfill \mathbf{\text{Quadrant I}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant II}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant III}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant IV}}\hfill \\ \hfill (x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill \\ \hfill (+,+)\hfill & \hfill \phantom{\rule{2em}{0ex}}(-,+)\hfill & \hfill \phantom{\rule{2em}{0ex}}(-,-)\hfill & \hfill \phantom{\rule{2em}{0ex}}(+,-)\hfill \end{array}$$**Graph of a Linear Equation:**The graph of a linear equation $Ax+By=C$is a straight line. Every point on the line is a solution of the equation. Every solution of this equation is a point on this line.**How to graph a linear equation by plotting points.**- Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

*x*-intercept and*y*-intercept of a Line- The
*x*-intercept is the point $\left(a,0\right)$where the line crosses the*x*-axis. - The
*y*-intercept is the point $\left(0,b\right)$where the line crosses the*y*-axis.

- The
**Find the***x*– and*y*-intercepts from the Equation of a Line- Use the equation of the line. To find:
the
*x*-intercept of the line, let $y=0$and solve for*x*. the*y*-intercept of the line, let $x=0$and solve for*y*.

- Use the equation of the line. To find:
the
**How to graph a linear equation using the intercepts.**- Step 1.
Find the
*x*– and*y*-intercepts of the line. Let $y=0$ and solve for*x.*Let $x=0$and solve for*y*. - Step 2. Find a third solution to the equation.
- Step 3. Plot the three points and check that they line up.
- Step 4. Draw the line

- Step 1.
Find the

#### 3.2

Slope of a Line**Slope of a Line**- The slope of a line is $m=\frac{\text{rise}}{\text{run}}.$
- The rise measures the vertical change and the run measures the horizontal change.

**How to find the slope of a line from its graph using**$m=\frac{\text{rise}}{\text{run}}.$- Step 1. Locate two points on the line whose coordinates are integers.
- Step 2. Starting with one point, sketch a right triangle, going from the first point to the second point.
- Step 3. Count the rise and the run on the legs of the triangle.
- Step 4. Take the ratio of rise to run to find the slope: $m=\frac{\text{rise}}{\text{run}}.$

**Slope of a line between two points.**- The slope of the line between two points
$({x}_{1},{y}_{1})$and
$({x}_{2},{y}_{2})$is:
$$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}.$$

- The slope of the line between two points
$({x}_{1},{y}_{1})$and
$({x}_{2},{y}_{2})$is:
**How to graph a line given a point and the slope.**- Step 1. Plot the given point.
- Step 2. Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
- Step 3. Starting at the given point, count out the rise and run to mark the second point.
- Step 4. Connect the points with a line.

**Slope Intercept Form of an Equation of a Line**- The slope–intercept form of an equation of a line with slope
*m*and*y*-intercept, $\left(0,b\right)$is $y=mx+b$

- The slope–intercept form of an equation of a line with slope
**Parallel Lines**- Parallel lines are lines in the same plane that do not intersect.
Parallel lines have the same slope and different
*y*-intercepts. If ${m}_{1}$and ${m}_{2}$are the slopes of two parallel lines then ${m}_{1}={m}_{2}.$ Parallel vertical lines have different*x*-intercepts.

- Parallel lines are lines in the same plane that do not intersect.
Parallel lines have the same slope and different
**Perpendicular Lines**- Perpendicular lines are lines in the same plane that form a right angle.
- If ${m}_{1}$and ${m}_{2}$are the slopes of two perpendicular lines, then: their slopes are negative reciprocals of each other, ${m}_{1}=-\frac{1}{{m}_{2}}.$ the product of their slopes is $\mathrm{-1},$ ${m}_{1}\xb7{m}_{2}=\mathrm{-1}.$
- A vertical line and a horizontal line are always perpendicular to each other.

#### 3.3

Find the Equation of a Line**How to find an equation of a line given the slope and a point.**- Step 1. Identify the slope.
- Step 2. Identify the point.
- Step 3. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 4. Write the equation in slope-intercept form.

**How to find an equation of a line given two points.**- Step 1. Find the slope using the given points. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$
- Step 2. Choose one point.
- Step 3. Substitute the values into the point-slope form: $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 4.
Write the equation in slope-intercept form.
To Write an Equation of a Line **If given:****Use:****Form:**Slope and *y*-intercept**slope-intercept**$y=mx+b$ Slope and a point **point-slope**$y-{y}_{1}=m\left(x-{x}_{1}\right)$ Two points **point-slope**$y-{y}_{1}=m\left(x-{x}_{1}\right)$

**How to find an equation of a line parallel to a given line.**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the parallel line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form: $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 5. Write the equation in slope-intercept form

**How to find an equation of a line perpendicular to a given line.**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the perpendicular line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$
- Step 5. Write the equation in slope-intercept form.

#### 3.4

Graph Linear Inequalities in Two Variables**How to graph a linear inequality in two variables.**- Step 1. Identify and graph the boundary line. If the inequality is $\le \text{or}\ge ,$ the boundary line is solid. If the inequality is $<\phantom{\rule{0.2em}{0ex}}\text{or}>,$the boundary line is dashed.
- Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
- Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side.

#### 3.5

Relations and Functions**Function Notation:**For the function $y=f(x)$*f*is the name of the function*x*is the domain value- $f(x)$is the range value
*y*corresponding to the value*x*We read $f(x)$as*f*of*x*or the value of*f*at*x*.

**Independent and Dependent Variables:**For the function $y=f(x),$*x*is the independent variable as it can be any value in the domain*y*is the dependent variable as its value depends on*x*

#### 3.6

Graphs of Functions**Vertical Line Test**- A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
- If any vertical line intersects the graph in more than one point, the graph does not represent a function.

**Graph of a Function**- The graph of a function is the graph of all its ordered pairs,
$\left(x,y\right)$or using function notation,
$\left(x,f\left(x\right)\right)$where
$y=f\left(x\right).$
$$\begin{array}{cccc}\hfill f& & & \text{name of function}\hfill \\ \hfill x& & & x\text{-coordinate of the ordered pair}\hfill \\ \hfill f\left(x\right)& & & y\text{-coordinate of the ordered pair}\hfill \end{array}$$

- The graph of a function is the graph of all its ordered pairs,
$\left(x,y\right)$or using function notation,
$\left(x,f\left(x\right)\right)$where
$y=f\left(x\right).$
**Linear Function****Constant Function****Identity Function****Square Function****Cube Function****Square Root Function****Absolute Value Function**

### Key Concepts

#### 3.1 Graph Linear Equations in Two Variables

**Points on the Axes**- Points with a
*y*-coordinate equal to 0 are on the*x*-axis, and have coordinates $\left(a,0\right).$ - Points with an
*x*-coordinate equal to $0$ are on the*y*-axis, and have coordinates $\left(0,b\right).$

- Points with a
**Quadrant**$$\begin{array}{cccc}\hfill \mathbf{\text{Quadrant I}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant II}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant III}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant IV}}\hfill \\ \hfill (x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill \\ \hfill (+,+)\hfill & \hfill \phantom{\rule{2em}{0ex}}(-,+)\hfill & \hfill \phantom{\rule{2em}{0ex}}(-,-)\hfill & \hfill \phantom{\rule{2em}{0ex}}(+,-)\hfill \end{array}$$**Graph of a Linear Equation:**The graph of a linear equation $Ax+By=C$ is a straight line.

Every point on the line is a solution of the equation.

Every solution of this equation is a point on this line.**How to graph a linear equation by plotting points.**- Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

*x*-intercept and*y*-intercept of a Line- The
*x*-intercept is the point $\left(a,0\right)$ where the line crosses the*x*-axis. - The
*y*-intercept is the point $\left(0,b\right)$ where the line crosses the*y*-axis.

- The
**Find the***x*– and*y*-intercepts from the Equation of a Line- Use the equation of the line. To find:

the*x*-intercept of the line, let $y=0$ and solve for*x*.

the*y*-intercept of the line, let $x=0$ and solve for*y*.

- Use the equation of the line. To find:
**How to graph a linear equation using the intercepts.**- Step 1.
Find the
*x*– and*y*-intercepts of the line.

Let $y=0$ and solve for*x.*

Let $x=0$ and solve for*y*. - Step 2. Find a third solution to the equation.
- Step 3. Plot the three points and check that they line up.
- Step 4. Draw the line

- Step 1.
Find the

#### 3.2 Slope of a Line

**Slope of a Line**- The slope of a line is $m=\frac{\text{rise}}{\text{run}}.$
- The rise measures the vertical change and the run measures the horizontal change.

**How to find the slope of a line from its graph using**$m=\frac{\text{rise}}{\text{run}}.$- Step 1. Locate two points on the line whose coordinates are integers.
- Step 2. Starting with one point, sketch a right triangle, going from the first point to the second point.
- Step 3. Count the rise and the run on the legs of the triangle.
- Step 4. Take the ratio of rise to run to find the slope: $m=\frac{\text{rise}}{\text{run}}.$

**Slope of a line between two points.**- The slope of the line between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is:

$$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}.$$

- The slope of the line between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is:
**How to graph a line given a point and the slope.**- Step 1. Plot the given point.
- Step 2. Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
- Step 3. Starting at the given point, count out the rise and run to mark the second point.
- Step 4. Connect the points with a line.

**Slope Intercept Form of an Equation of a Line**- The slope–intercept form of an equation of a line with slope
*m*and*y*-intercept, $\left(0,b\right)$ is $y=mx+b$

- The slope–intercept form of an equation of a line with slope
**Parallel Lines**- Parallel lines are lines in the same plane that do not intersect.

Parallel lines have the same slope and different*y*-intercepts.

If ${m}_{1}$ and ${m}_{2}$ are the slopes of two parallel lines then ${m}_{1}={m}_{2}.$

Parallel vertical lines have different*x*-intercepts.

- Parallel lines are lines in the same plane that do not intersect.
**Perpendicular Lines**- Perpendicular lines are lines in the same plane that form a right angle.
- If ${m}_{1}$ and ${m}_{2}$ are the slopes of two perpendicular lines, then:

their slopes are negative reciprocals of each other, ${m}_{1}=-\frac{1}{{m}_{2}}.$

the product of their slopes is $\mathrm{-1},$${m}_{1}\xb7{m}_{2}=\mathrm{-1}.$ - A vertical line and a horizontal line are always perpendicular to each other.

#### 3.3 Find the Equation of a Line

**How to find an equation of a line given the slope and a point.**- Step 1. Identify the slope.
- Step 2. Identify the point.
- Step 3. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 4.
Write the equation in slope-intercept form.

**How to find an equation of a line given two points.**- Step 1. Find the slope using the given points. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$
- Step 2. Choose one point.
- Step 3. Substitute the values into the point-slope form: $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 4.
Write the equation in slope-intercept form.

To Write an Equation of a Line **If given:****Use:****Form:**Slope and *y*-intercept**slope-intercept**$y=mx+b$ Slope and a point **point-slope**$y-{y}_{1}=m\left(x-{x}_{1}\right)$ Two points **point-slope**$y-{y}_{1}=m\left(x-{x}_{1}\right)$

**How to find an equation of a line parallel to a given line.**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the parallel line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form: $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 5. Write the equation in slope-intercept form

**How to find an equation of a line perpendicular to a given line.**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the perpendicular line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$
- Step 5. Write the equation in slope-intercept form.

#### 3.4 Graph Linear Inequalities in Two Variables

**How to graph a linear inequality in two variables.**- Step 1.
Identify and graph the boundary line.

If the inequality is $\le \text{or}\ge ,$ the boundary line is solid.

If the inequality is $<\phantom{\rule{0.2em}{0ex}}\text{or}>,$ the boundary line is dashed. - Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
- Step 3.
Shade in one side of the boundary line.

If the test point is a solution, shade in the side that includes the point.

If the test point is not a solution, shade in the opposite side.

- Step 1.
Identify and graph the boundary line.

#### 3.5 Relations and Functions

**Function Notation:**For the function $y=f(x)$*f*is the name of the function*x*is the domain value- $f(x)$ is the range value
*y*corresponding to the value*x*

We read $f(x)$ as*f*of*x*or the value of*f*at*x*.

**Independent and Dependent Variables:**For the function $y=f(x),$*x*is the independent variable as it can be any value in the domain*y*is the dependent variable as its value depends on*x*

#### 3.6 Graphs of Functions

**Vertical Line Test**- A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
- If any vertical line intersects the graph in more than one point, the graph does not represent a function.

**Graph of a Function**- The graph of a function is the graph of all its ordered pairs, $\left(x,y\right)$ or using function notation, $\left(x,f\left(x\right)\right)$ where $y=f\left(x\right).$

$$\begin{array}{cccc}\hfill f& & & \text{name of function}\hfill \\ \hfill x& & & x\text{-coordinate of the ordered pair}\hfill \\ \hfill f\left(x\right)& & & y\text{-coordinate of the ordered pair}\hfill \end{array}$$

- The graph of a function is the graph of all its ordered pairs, $\left(x,y\right)$ or using function notation, $\left(x,f\left(x\right)\right)$ where $y=f\left(x\right).$
**Linear Function**

**Constant Function**

**Identity Function**

**Square Function**

**Cube Function**

**Square Root Function**

**Absolute Value Function**