Test Course

1.1

• 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.
  1. Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
  2. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
  3. Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
  4. 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.
  1. Step 1. Write each number as a product of primes.
  2. Step 2. List the primes of each number. Match primes vertically when possible.
  3. Step 3. Bring down the columns.
  4. 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.

  Table 1.4
• Grouping Symbols $\begin{array}{cccccc}\text{Parentheses}\hfill & & & & & \left(\right)\hfill \\ \text{Brackets}\hfill & & & & & \left[\right]\hfill \\ \text{Braces}\hfill & & & & & \left\{\right\}\hfill \end{array}$
• 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.
  1. Step 1.
     Parentheses and Other Grouping Symbols

     • Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
  2. Step 2.
     Exponents

     • Simplify all expressions with exponents.
  3. Step 3.
     Multiplication and Division

     • Perform all multiplication and division in order from left to right. These operations have equal priority.
  4. Step 4.
     Addition and Subtraction

     • Perform all addition and subtraction in order from left to right. These operations have equal priority.
• How to combine like terms.
  1. Step 1. Identify like terms.
  2. Step 2. Rearrange the expression so like terms are together.
  3. 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	$a·b,ab,a(b),(a)(b)$ $2a$
  Division	a divided by b the quotient of a and b the ratio of a and b b divided into a	$a÷b,a\text{/}b,\frac{a}{b},b\overline{)a}$

  Table 1.5

1.2

• Opposite Notation
  $$\begin{array}{c}\text{−}a\text{means the opposite of the number}a\hfill \\ \text{The notation}\text{−}a\text{is read as “the opposite of}a\text{.”}\hfill \end{array}$$
• 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
  $$\begin{array}{cccccccccc}\text{Parentheses}\hfill & & & \left(\right)\hfill & & & \text{Braces}\hfill & & & \left\{\right\}\hfill \\ \text{Brackets}\hfill & & & \left[\right]\hfill & & & \text{Absolute value}\hfill & & & \left|\right|\hfill \end{array}$$
• Subtraction Property $a-b=a+\left(\text{−}b\right)$ 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

    $$If the signs are the same, the result is positive.

  Different signs	Result
  •  Positive and negative	Negative
  •  Negative and positive	Negative

    $$If the signs are different, the result is negative.
• Multiplication by $\mathrm{-1}$  $\mathrm{-1}a=\text{−}a$ Multiplying a number by $\mathrm{-1}$gives its opposite.
• How to Use Integers in Applications.
  1. Step 1. Read the problem. Make sure all the words and ideas are understood
  2. Step 2. Identify what we are asked to find.
  3. Step 3. Write a phrase that gives the information to find it.
  4. Step 4. Translate the phrase to an expression.
  5. Step 5. Simplify the expression.
  6. Step 6. Answer the question with a complete sentence.

1.3

• Equivalent Fractions Property If a, b, and c are numbers where $b\ne 0,c\ne 0,$then $\frac{a}{b}=\frac{a·c}{b·c}\text{and}\frac{a·c}{b·c}=\frac{a}{b}.$
• How to simplify a fraction.
  1. Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
  2. Step 2. Simplify using the Equivalent Fractions Property by dividing out common factors.
  3. 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 $\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}.$ 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 $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}.$ 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 $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\text{and}\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}.$ To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
• How to add or subtract fractions.
  1. 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.
  2. Step 2. Add or subtract the fractions.
  3. Step 3. Simplify, if possible.
• How to simplify an expression with a fraction bar.
  1. Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Step 2. Simplify the fraction.
• Placement of Negative Sign in a Fraction For any positive numbers a and b, $\frac{\text{−}a}{b}=\frac{a}{\text{−}b}=-\frac{a}{b}.$
• How to simplify complex fractions.
  1. Step 1. Simplify the numerator.
  2. Step 2. Simplify the denominator.
  3. Step 3. Divide the numerator by the denominator. Simplify if possible.

1.4

• How to round decimals.
  1. Step 1. Locate the given place value and mark it with an arrow.
  2. Step 2. Underline the digit to the right of the place value.
  3. 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
  4. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.
• How to add or subtract decimals.
  1. Step 1. Determine the sign of the sum or difference.
  2. Step 2. Write the numbers so the decimal points line up vertically.
  3. Step 3. Use zeros as placeholders, as needed.
  4. 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.
  5. Step 5. Write the sum or difference with the appropriate sign
• How to multiply decimals.
  1. Step 1. Determine the sign of the product.
  2. 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.
  3. 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.
  4. Step 4. Write the product with the appropriate sign.
• How to multiply a decimal by a power of ten.
  1. Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
  2. Step 2. Add zeros at the end of the number as needed.
• How to divide decimals.
  1. Step 1. Determine the sign of the quotient.
  2. 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.
  3. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
  4. Step 4. Write the quotient with the appropriate sign.
• How to convert a decimal to a proper fraction and a fraction to a decimal.
  1. Step 1. To convert a decimal to a proper fraction, determine the place value of the final digit.
  2. Step 2.
     Write the fraction.

     • numerator—the “numbers” to the right of the decimal point
     • denominator—the place value corresponding to the final digit
  3. 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.
  1. Step 1. To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
  2. 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

  

  Figure 1.9

1.5

3.1

• 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 $\left(0,b\right).$
• Quadrant
  $$\begin{array}{cccc}\hfill \mathbf{\text{Quadrant I}}\hfill & \hfill \mathbf{\text{Quadrant II}}\hfill & \hfill \mathbf{\text{Quadrant III}}\hfill & \hfill \mathbf{\text{Quadrant IV}}\hfill \\ \hfill (x,y)\hfill & \hfill (x,y)\hfill & \hfill (x,y)\hfill & \hfill (x,y)\hfill \\ \hfill (+,+)\hfill & \hfill (-,+)\hfill & \hfill (-,-)\hfill & \hfill (+,-)\hfill \end{array}$$
  
• 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.
  1. Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
  2. Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
  3. 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.
• 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.
• How to graph a linear equation using the intercepts.
  1. 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.
  2. Step 2. Find a third solution to the equation.
  3. Step 3. Plot the three points and check that they line up.
  4. Step 4. Draw the line

3.2

• 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 $m=\frac{\text{rise}}{\text{run}}.$
  1. Step 1. Locate two points on the line whose coordinates are integers.
  2. Step 2. Starting with one point, sketch a right triangle, going from the first point to the second point.
  3. Step 3. Count the rise and the run on the legs of the triangle.
  4. 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:
    $$m=\frac{y_2-y_1}{x_2-x_1}.$$
• How to graph a line given a point and the slope.
  1. Step 1. Plot the given point.
  2. Step 2. Use the slope formula $m=\frac{\text{rise}}{\text{run}}$  to identify the rise and the run.
  3. Step 3. Starting at the given point, count out the rise and run to mark the second point.
  4. 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 $y=mx+b$
• 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.
• 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 $\mathrm{-1},$ $m_1·m_2=\mathrm{-1}.$
  • A vertical line and a horizontal line are always perpendicular to each other.

3.3

• How to find an equation of a line given the slope and a point.
  1. Step 1. Identify the slope.
  2. Step 2. Identify the point.
  3. Step 3. Substitute the values into the point-slope form, $y-y_1=m\left(x-x_1\right).$ 
  4. Step 4. Write the equation in slope-intercept form.
• How to find an equation of a line given two points.
  1. Step 1. Find the slope using the given points. $m=\frac{y_2-y_1}{x_2-x_1}$ 
  2. Step 2. Choose one point.
  3. Step 3. Substitute the values into the point-slope form: $y-y_1=m\left(x-x_1\right).$ 
  4. 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.
  1. Step 1. Find the slope of the given line.
  2. Step 2. Find the slope of the parallel line.
  3. Step 3. Identify the point.
  4. Step 4. Substitute the values into the point-slope form: $y-y_1=m\left(x-x_1\right).$ 
  5. Step 5. Write the equation in slope-intercept form
• How to find an equation of a line perpendicular to a given line.
  1. Step 1. Find the slope of the given line.
  2. Step 2. Find the slope of the perpendicular line.
  3. Step 3. Identify the point.
  4. Step 4. Substitute the values into the point-slope form, $y-y_1=m\left(x-x_1\right)$ 
  5. Step 5. Write the equation in slope-intercept form.

3.4

• How to graph a linear inequality in two variables.
  1. 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 $<\text{or}>,$the boundary line is dashed.
  2. Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. 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

• 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

• 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 $y=f\left(x\right).$ 
    $$\begin{array}{cccc}\hfill f& & & \text{name of function}\hfill \\ \hfill x& & & x\text{-coordinate of the ordered pair}\hfill \\ \hfill f\left(x\right)& & & y\text{-coordinate of the ordered pair}\hfill \end{array}$$
• Linear Function
• Constant Function
• Identity Function
• Square Function
• Cube Function
• Square Root Function
• Absolute Value Function

• 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 $\left(0,b\right).$
• Quadrant
  $$\begin{array}{cccc}\hfill \mathbf{\text{Quadrant I}}\hfill & \hfill \mathbf{\text{Quadrant II}}\hfill & \hfill \mathbf{\text{Quadrant III}}\hfill & \hfill \mathbf{\text{Quadrant IV}}\hfill \\ \hfill (x,y)\hfill & \hfill (x,y)\hfill & \hfill (x,y)\hfill & \hfill (x,y)\hfill \\ \hfill (+,+)\hfill & \hfill (-,+)\hfill & \hfill (-,-)\hfill & \hfill (+,-)\hfill \end{array}$$
  
• 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.
  1. Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
  2. Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
  3. 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.
• 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.
• How to graph a linear equation using the intercepts.
  1. 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.
  2. Step 2. Find a third solution to the equation.
  3. Step 3. Plot the three points and check that they line up.
  4. Step 4. Draw the 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 $m=\frac{\text{rise}}{\text{run}}.$
  1. Step 1. Locate two points on the line whose coordinates are integers.
  2. Step 2. Starting with one point, sketch a right triangle, going from the first point to the second point.
  3. Step 3. Count the rise and the run on the legs of the triangle.
  4. 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:
    
    $$m=\frac{y_2-y_1}{x_2-x_1}.$$
• How to graph a line given a point and the slope.
  1. Step 1. Plot the given point.
  2. Step 2. Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
  3. Step 3. Starting at the given point, count out the rise and run to mark the second point.
  4. 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 $y=mx+b$
• 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.
• 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·m_2=\mathrm{-1}.$
  • A vertical line and a horizontal line are always perpendicular to each other.

• How to find an equation of a line given the slope and a point.
  1. Step 1. Identify the slope.
  2. Step 2. Identify the point.
  3. Step 3. Substitute the values into the point-slope form, $y-y_1=m\left(x-x_1\right).$
  4. Step 4. Write the equation in slope-intercept form.
     
     
     
     
• How to find an equation of a line given two points.
  1. Step 1. Find the slope using the given points. $m=\frac{y_2-y_1}{x_2-x_1}$
  2. Step 2. Choose one point.
  3. Step 3. Substitute the values into the point-slope form: $y-y_1=m\left(x-x_1\right).$
  4. 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.
  1. Step 1. Find the slope of the given line.
  2. Step 2. Find the slope of the parallel line.
  3. Step 3. Identify the point.
  4. Step 4. Substitute the values into the point-slope form: $y-y_1=m\left(x-x_1\right).$
  5. Step 5. Write the equation in slope-intercept form
• How to find an equation of a line perpendicular to a given line.
  1. Step 1. Find the slope of the given line.
  2. Step 2. Find the slope of the perpendicular line.
  3. Step 3. Identify the point.
  4. Step 4. Substitute the values into the point-slope form, $y-y_1=m\left(x-x_1\right)$
  5. Step 5. Write the equation in slope-intercept form.

• How to graph a linear inequality in two variables.
  1. 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 $<\text{or}>,$ the boundary line is dashed.
  2. Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. 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.

• 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

• 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 $y=f\left(x\right).$
    
    $$\begin{array}{cccc}\hfill f& & & \text{name of function}\hfill \\ \hfill x& & & x\text{-coordinate of the ordered pair}\hfill \\ \hfill f\left(x\right)& & & y\text{-coordinate of the ordered pair}\hfill \end{array}$$
• Linear Function
  
• Constant Function
  
• Identity Function
  
• Square Function
  
• Cube Function
  
• Square Root Function
  
• Absolute Value Function