Benchmark Review

​Benchmark Review

By Dr. Collier

Combining Like Terms!

Slope

Topics in this quizizz:

- Identifying slope (direction)

- Finding slope of a line (steepness)

- Slope from a Table

- Slope from an Equation

Standard Form to Slope Intercept Form

Slope from Two Points

I can calculate slope using the slope formula and two points.

Functions

Objective: I can determine if a relation is a function

Linear Functions

Write an equation in slope intercept form for each representation.OBJEC

Solving Systems of Equations

Scatterplots and Trend Lines

8th Grade

What is a Trend Line?

When data is collected, points do not form a straight line, but they may approximate to a linear relationship

A trend line is a line that is

VERY CLOSE to most of the data points.

It is used to make conjectures (predictions)

Multiple Choice

What is a trend line used for?

• To make predictions

• To make the graph look pretty

• To connect the points

First, Lets make a trend line!

- Count the number of points, make a line that best fits the data points.

TIP: Count the number of total points. Try to have the same number of points on both sides of the trend line

Make a Prediction!

How many visitors would the beach have at 90 degrees?

Make a prediction!

What will the weight of the shipment be for 7 books?

Multiple Choice

Which graph shows a line of best fit (trend line) for the scatter plot?

• Graph A

• Graph B

• Graph C

• Graph D

Multiple Choice

Which of the following lines represents the line of best fit for the scatter plot?

• A
• B
• C
• D

Multiple Choice

Based on the scatterplot, what is the best prediction of the average number of hours a person spends at work every week, if that person spends an average of 10 hours on recreational activities every week?

• 33 hours
• 85 hours
• 50 hours
• 65 hours

Multiple Choice

Based on the scatterplot, what is the best prediction of the average amount of money spent on groceries for a household of 7 people?

• $240
• $190
• $210
• $300

The following questions are for you to do independently...

Multiple Choice

Based on the trend in the data, approximately how many free throws would a player be expected to make if he attempted 60 free throws?

• 50
• 35
• 25
• 60

Multiple Choice

Based on this scatterplot, approximately what score would a student with 6 absences expect to receive on the final exam?

• 65

• 92

• 67

• 76

Multiple Choice

Based on the scatterplot, about how much money would a group of 10 people be expected to spend on food and beverages at this restaurant?

• $135
• $115
• $105
• $150

Methods We Have Used to Solve Systems of Equations

• Graphing

• Elimination

• Substitution

Graphing Method

• Use the Desmos Graphing Calculator

• One Solution: Intersection of the two lines

• No Solution: Parallel lines, lines with the same slopes

• Infinitely Many Solutions: One line (Both of the lines are the same)

Multiple Choice

Solve:
y = 2x -11
-3y = -6x -15

• (-1.5,-4)
• (1.5,4)
• No solution (parallel lines)
• Infinitely many solutions

Multiple Choice

Solve:
y = 7x + 9
2y + 2x = -18

• (5, 0)
• (9, 18)
• (-2, -5)
• (1, 16)

Multiple Choice

Solve the following system by graphing.  What is the solution?

• Infinite number of solutions
• (3, 3)
• (3, -3)
• (-3, 3)

Multiple Choice

^{What is the solution?} 

• 1
• -2
• (1, 2)
• (1, -1)

Multiple Choice

When you graph the exact same equation twice,

• you will have no solution. 
• you will have one solution.
• you will have infinite solutions. 
• you will graph a giraffe. 

Multiple Choice

How many solutions will this system have? 

• No solution
• One Solution
• I Don't Know
• Infinitely Many Solutions

Elimination

• Best when the equation is in Standard Form

• Look for matching coefficients and variables that OPPOSITE signs

• If needed you can multiply by a constant to make variables match and be opposite

• 
  

Substitution

• Used when we we have a single variable isolated.

• The coefficient of the isolated variable should be 1.

• Replace the variable in 1 equation, with the isolated variable from the other equation.

• Find the other solution by replacing the unknown variable with the newly found variable.

Multiple Choice

What is the best method to use?

2x − 3y = −1

y =x − 1

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

What is the best method to use?

−4x − 2y = −12

4x + 8y = −24

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

What variable do you eliminate?

4x + 8y = 20

−4x + 2y = −30

• X because they have opposite signs

• Y because they have opposite signs

Multiple Choice

What is the proper way to set up the equation to begin solving for x?

y = 6x − 11

−2x − 3y = −7

• -2(-11) -3y= -7

• -2(6x-11) -3y = -7

• -2x -3(11) = -7

• -2x - 3(6x-11)= -7

Multiple Choice

What is the best method to use?

−4x − 2y = −12

4x + 8y = −24

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

Which variable do you eliminate and why?

7x + 2y = 24

8x + 2y = 30

• You pick x because it's LCM is 56

• You pick y because it's LCM is 2.

Multiple Choice

What variable do you eliminate, and what do you multiply the equation(s) by?

5x + y = 9

10x − 7y = −18

• You eliminate x, and multiply the top equation by -2

• You eliminate y, and multiply the top equation by 7

Multiple Choice

What is the best method do you use?

y = 6x − 11

−2x − 3y = −7

• Graphing

• Substitution because a variable is defined

• Elimination because both equations are in standard form.

Writing Systems of Equations

• Identify the variables

• Look for slopes and starting (y-intercepts)

• Write the equation

• Solve with Desmos

Multiple Choice

Some students want to order shirts with their school logo. One company charges $9.65 per shirt plus a setup fee of $43. Another company charges $8.40 per shirt plus a $58 fee. Which equation represents the number of shirts when both companies charge the same amount? 

• y = 9.65 + x
  y = 8.40 + x
• y = 9.65x + 43
  y = 8.40x + 58
• y =9.65x
  y = 8.40x
• y = 9.65x - 43
  y = 8.40x - 58

Multiple Choice

A large pizza at Palanzio’s Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese pizza at Guido’s Pizza is $7.30 plus $0.65 for each topping. Which system of equations could be used to find the number of toppings when both companies cost the same amount? 

• y = 6.80 + .65x
  y=7.30+.90x
• x + y = 6.80
  x + y = 7.30
• y = 6.80+.90x
  y = 7.30 + .65x
• y + .90x = 6.80
  y + .65x = 7.30

Multiple Choice

Two brothers went shopping at a back-to-school sale where all shirts and shorts were the same price. The younger brother spent $175 on 7 new shirts and 7 pairs of shorts. The older brother purchased 6 new shirts and 7 pairs of shorts and paid a total of $165. How much did one shirt cost?

• $5
• $10
• $15
• $20

Remember...

1.) Find the slope

2.) Find the y-intercept (b)

3.) Plug both values into y = mx + b

Multiple Choice

Write an equation in slope-intercept form:

$$slope\ =\ -\frac{2}{3}$$   $$y-intercept\ =\ \left(0,\ -9\right)$$  

• $$y\ =\ -\frac{2}{3}x\ +\ 9$$  

• $$y\ =\ -\frac{2}{3}x\ -\ 9$$  

• $$y\ =\ -9x\ +\ \frac{2}{3}$$  

• $$y\ =\ -9x\ -\ \frac{2}{3}$$  

Multiple Choice

Write an equation in slope-intercept form:

$$Slope\ =\ 10$$  

$$y-intercept\ =\ \left(0,\ 8\right)$$  

• $$y\ =\ 8x\ -\ 10$$  

• $$y\ =\ 8x\ +\ 10$$  

• $$y\ =\ 10x\ -\ 8$$  

• $$y\ =\ 10x\ +\ 8$$  

Multiple Choice

Write an equation in slope-intercept form:

$$Slope\ =\ \frac{3}{8}$$  

$$y-intercept\ =\ \left(0,\ -1\right)$$  

• $$y\ =\ \frac{3}{8}x\ -\ 1$$  

• $$y\ =\ \frac{3}{8}x\ +\ 1$$  

• $$y\ =\ 1x\ +\ \frac{3}{8}$$  

• $$y\ =\ 1x\ -\ \frac{3}{8}$$  

From a Graph

Find the Slope

Find the y-intercept

Plug in the values to y = mx + b

Multiple Choice

Write the equation for the graph shown.

• $$y\ =\ \frac{2}{1}x\ +\ 9$$  

• $$y\ =\ -\frac{1}{2}x\ -\ 9$$  

• $$y\ =\ -\frac{2}{1}x\ +\ 9$$  

• $$y\ =\ \frac{1}{2}x\ +\ 9$$  

Multiple Choice

Write the equation for the graph shown.

• $$y\ =\ -\frac{2}{4}x\ +\ 8$$  

• $$y\ =\ \frac{2}{4}x\ +\ 8$$  

• $$y\ =\ 8x\ +\ \frac{2}{4}$$  

• $$y\ =\ -\frac{4}{2}x\ +\ 8$$  

Multiple Choice

Write an equation for the graph shown.

• $$y\ =\ -\frac{3}{1}x\ +\ 2$$  

• $$y\ =\ \frac{1}{3}x\ +\ 2$$  

• $$y\ =\ \frac{3}{1}x\ +\ 2$$  

• $$y\ =\ 2x\ +\ \frac{3}{1}$$  

From a Scenario

Find the slope (Per, Each, Every)

Find the y-intercept (initial value)

Multiple Choice

A photographer has already taken 10 pictures. She takes an additional 3 pictures per hour.

• $$y\ =\ 10x\ +\ 3$$

• $$y\ =\ 3x\ +\ 10$$

• $$y\ =\ 3x\ -\ 10$$

Multiple Choice

Gil has $200 saved for snacks. He will spend $5 on snack per week until he runs out.

• $$y\ =\ 200-5x\$$

• $$y\ =\ 5x\ +\ 200$$

• $$y\ =\ -200x\ +\ 5$$

Multiple Choice

Bad Bunny currently has $5 to go to Mexico. He is selling his famous cupcakes for $25 per dozen to raise more money.

• $$y\ =\ 25x\ -\ 5$$

• $$y\ =\ 25x\ +\ 5$$

• $$y\ =\ 5x\ +\ 25$$

• $$y\ =\ -5x\ +\ 25$$

From a Table

Find the slope

Find the y-intercept

Plug the values into y = mx + b

Multiple Choice

Write an equation for the table shown.

• $$y\ =\ \frac{1}{2}x\ +\ 4$$

• $$y\ =\ \frac{1}{2}x\ -\ 2$$

• $$y\ =\ \frac{2}{1}x\ -\ 2$$

• $$y\ =\ \frac{1}{2}x\ -\ 3$$  

Multiple Choice

Write an equation for the table shown.

• $$y\ =\ 3x\ +\ 4$$

• $$y\ =\ 3x\ -\ 2$$

• $$y\ =\ \frac{1}{3}x\ +\ 4$$

• $$y\ =\ -3x\ +\ 4$$  

Multiple Choice

Write an equation for the table shown.

• $$y\ =\ \frac{1}{3}x\ +\ 6$$

• $$y\ =\ \frac{1}{3}x\ -\ 2$$

• $$y\ =\ -2x\ +\ \frac{1}{3}$$

• $$y\ =\ \frac{3}{1}x\ -\ 2$$

Functions

• When the x that you try gives you only one y THAT'S A FUNCTION

• A function is a special relation where each x-value only has one y-value

When you put an input into a machine you can only get out one input.

THAT'S A FUNCTION!

(Each input has only one output)

These are all functions

EACH X VALUE

HAS ONLY ONE Y VALUE

These are NOT functions

In each example

there is an X VALUE that has more than one Y VALUE

Multiple Choice

Is the relation a function? Why.

• Yes, because the x-value 11 has two y-values paired with it.

• Yes, because each x-value has only one y-value paired with it.

• No, because the x-value 11 has two y-values paired with it.

• No, because each x-value has only one y-value paired with it.

Multiple Choice

Is this table a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this mapping a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this set of ordered pairs a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this set of ordered pairs a function or not a function?

• Function

• Not a Function

The vertical line test

• To tell if a graph is a function, draw a vertical line anywhere on the graph.

• If any vertical line touches the graph in more than one location it is NOT a function

Multiple Choice

Which graph does NOT pass the vertical line test?

• Graph 1

• Graph 2

• Graph 3

• Graph 4

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Multiple Choice

Is this graph a function or not a function?

• Function

• Not a Function

Use the formula to find the slope between the two points.

Multiple Choice

Use the formula to find the slope between the two points.

(5, 10) and (8, 12)

• $$\frac{2}{3}$$

• $$-\frac{2}{3}$$

• $$\frac{3}{2}$$

• $$-\frac{3}{2}$$

Multiple Choice

Use the formula to find the slope between the two points.

(-2, 7) and (3, 8)

• $$-\frac{1}{1}$$  

• $$\frac{1}{5}$$  

• $$-\frac{1}{5}$$

• $$\frac{1}{1}$$  

Multiple Choice

Use the formula to find the slope between the two points.

(-1, -9) and (-4, -7)

• $$\frac{2}{3}$$  
  

• $$-\frac{2}{3}$$  
  

• $$\frac{3}{2}$$  

• $$-\frac{3}{2}$$  
  

Multiple Choice

Use the formula to find the slope between the two points.

(5, -2) and (5, -7)

• 1

• 0

• undefined

• $$\frac{7}{5}$$  
  

Multiple Choice

Use the formula to find the slope between the two points.

(4, 9) and (7, 5)

• $$-\frac{4}{5}$$  
  

• $$-\frac{4}{3}$$  
  

• $$\frac{4}{3}$$  

• $$-\frac{3}{4}$$  
  

Multiple Choice

Use the formula to find the slope between the two points.

(4, -3) and (7, -3)

•  0
  

• undefined

• $$\frac{6}{3}$$  
  

• $$-\frac{3}{6}$$  
  

Multiple Choice

Use the formula to find the slope between the two points.

(-1, -2) and (8, -3)

•   $$\frac{9}{1}$$  
  

• $$-\frac{9}{1}$$  

• $$\frac{1}{9}$$
  

• $$-\frac{1}{9}$$  

Multiple Choice

Which equation is in standard form?

• y = mx + b

• $$y-y_1=m\left(x-x_1\right)$$

• Ax + By = C

• $$m=\frac{y_2-y_1}{x_2-x_1}$$

To change from Standard Form to Slope Intercept Form, you solve the equation for y.

• First, add or subtract the "x" term.

• Second, divide by the coefficient (number in front of) y.

• Finally, simplify if you can.

Write 3x + 2y = 6 in slope intercept form

Multiple Choice

What is the first thing you would do to write 2x - y = 7 in slope intercept form?

• Add y to both sides

• Subtract 2x from both sides

• Divide both sides by 2

• Subtract 7 from both sides

You would subtract the 2x first

So, -y = -2x + 7 (Don't forget the negative in front of the y needs to come with it!)

Next, you would need to divide by the negative 1.

Multiple Choice

What is the resulting equation? (Remember you had -y = -2x + 7)

• y = 2x + 7

• y = -2x - 7

• y = 2x - 7

• y = 2x + 8

Multiple Choice

Which equation represents 4x - 2y = 6 in slope intercept form.

• y = 2x - 3

• y = -2x + 3

• 
  

  $$y\ =\ \frac{1}{2}x\ -\ 3$$

• y = 2x + 8

What would happen with x + 3y = -9

Multiple Choice

Write x + 5y = 10 in slope intercept form.

• $$y=-5x\ +\ 2$$

• $$y\ =\ -\frac{1}{5}x\ +\ 5$$

• $$y\ =\ 5x\ +\ 2$$

• $$y\ =\ -\frac{1}{5}x\ +\ 2$$

Multiple Choice

Write x + y = 3 in slope intercept form.

• y = x + 3

• y = -3x

• y = -x + 3

• y = -x - 3

Let's try:                 - 3x + 2y = 8

Multiple Choice

Which represents  -2x - 3y = 12 in slope intercept form.

• $$y\ =\ -\frac{2}{3}x\ -\ 4$$

• $$y\ =\ \frac{2}{3}x\ -\ 4$$

• $$y\ =\ -\frac{2}{3}x\ +\ 4$$

• $$y\ =\ \frac{2}{3}x\ +\ 4$$

Multiple Choice

Last one, which represents -x - y = 0 in slope intercept form?

• y = x

• y = -x

• y = 0

• y = x + 1

Identifying Slope of a Line

• Slope can tell us the direction of the line!

• Refresh yourself of the 4 types of slope using the picture to the right:

Multiple Choice

What type of slope?

• positive

• negative

• zero

• undefined

Multiple Choice

What type of slope?

• negative
• steep
• undefined
• zero slope

Multiple Choice

What type of slope?

• zero slope
• undefined
• negative
• positive

Multiple Choice

What type of slope?

• undefined
• positive slope
• negative slope
• zero slope

Finding the Slope of a Line

• Slope can tell us how steep a line is!

• Each line can be assigned a value (a #) that tells us how steep the line is.

• Slope is written as a ratio (rise/run)

• Always make sure to simplify your answer!

Multiple Choice

Find the slope of the line below using rise/run.

• $$\frac{2}{3}$$

• $$\frac{3}{2}$$

• 3

• $$\frac{1}{3}$$

Multiple Choice

Find the slope of the line below using rise/run.

• $$\frac{1}{2}$$  

• $$\frac{2}{1}$$  

• 2

• 1

Multiple Choice

Find the slope of the line below using rise/run.

• 3

• $$\frac{1}{3}$$  

• 2

• 1

Multiple Choice

Find the slope of the line below using rise/run.

• $$\frac{5}{3}$$  
  

• $$\frac{3}{5}$$  
  

• 5

• $$\frac{5}{4}$$  
  

Multiple Choice

Find the slope of the line below using rise/run.

• $$\frac{1}{2}$$  
  

• $$\frac{4}{2}$$  
  

• 2

• 3

• $$\frac{2}{3}$$  

Finding slope from a Table

• Slope = change in y/change in x

• Use the table to find the value the y is changing by. That is the numerator!

• Use the table to find the value the x is changing by. That is the denominator!

• ANOTHER OPTION: Graph the points and count rise/run

Multiple Choice

Find the slope. Simplify, if you can.

• $$\frac{2}{5}$$

• $$\frac{5}{2}$$

• $$2\ \frac{1}{2}$$

• 2

Multiple Choice

Find the slope. Simplify, if you can.

• $$\frac{1}{2}$$  
  

• $$\frac{4}{8}$$  

• 4

• 2

Multiple Choice

Find the slope. Simplify, if you can.

• 3

•   $$\frac{1}{3}$$  

• $$-\frac{1}{3}$$  

• -3

Slope from an equation

• The number in front of the x in the equation is the slope!

Multiple Choice

Identify the slope in the equation:

$$y=8x$$  

• 8

• 1/8

• -8

• 0

Multiple Choice

Identify the slope in the equation:

$$y=\frac{5}{6}x$$  

• $$\frac{5}{6}$$  

• $$\frac{6}{5}$$  
  

• 5

• 6

Fill in the Blanks

$y=x+5$  

Identify the slope in the equation:

How many bottles of water do you have?

How many burgers did you order?

How many bags of doritos did you get?

w= water

b=burger

d=doritos

Therefore we have 3w+2b+1d

Multiple Choice

Simplify by combining like terms:

$$7b-3b+4f$$  

• 
  

  $$10b+4f$$  

• $$8b$$  

• $$4b-4f$$
  

• $$4b+4f$$   

Multiple Choice

Simplify by combining like terms:

$$4t-t+3$$  

• $$4t-4$$  

• $$5t+3$$  

• $$3t+3$$  

• $$5t-3$$  

Multiple Choice

Simplify by combining like terms:

$$8x+4x-3-3$$  

• 6

• $$6x$$  

• $$4x-6$$  

• $$12x-6$$  

Multiple Choice

Simplify by combining like terms:

$$7x+2y+9x+8$$  

• $$18xy+8$$  

• $$16x+2y+8$$  

• $$9xy+9x+8$$  

• $$16x+10y$$  

Multiple Choice

Simplify by combining like terms:

$$7f+8g+4f+3f+6f+2g$$  

• $$20f+11g$$  

• $$21f+10g$$  

• $$20f+10g$$  

• $$22f+10g$$  

Multiple Choice

Simplify by combining like terms:

$$9x+7y+4x-11y+10x-8x$$  

• $$15x-3y$$  

• $$15x-4y$$  

• $$16x-4y$$  

• $$15x-5y$$  

Multiple Choice

Simplify by combining like terms:

$$12x-15y-18x+17y+20x$$  

• $$14x+y$$  

• $$14x+2y$$  

• $$14x+3y$$  

• $$15x+2y$$  

Multiple Choice

Simplify by combining like terms:

$$3g+3h+3+2g+8$$  

• $$5g+3h+11$$  

• $$8gh+11$$  

• $$6g+5h+8$$  

• $$6g+2h+11$$  

Review of solving two step equations:

Special Case of two step equations:

If the entire variable side is multiplied or divided undo multiplication or division first and then adding or subtracting.

Example 2: 3(2x – 7) = 3

  

Multiple Choice

Solve for v.

• 60
• -3
• 15
• 3

Multiple Choice

$$96=6\left(k+8\right)$$  

• $$12$$  

• $$8$$  

• $$-4$$  

• $$9$$  

Multistep Equations

Day 1

Multipstep Equations

1) Combine like terms

2) complete two step equation steps

Example 4: 7x + 2x + 6 = -12

Example 5: 27 = -3(5x – 2 + 2x)

Multiple Choice

7 + 3x + 4 =23

• x = 20/3
• x = 4
• x = 12
• x = 34/3

Multiple Choice

3x + 8x - 8 = 3

• 4

• 3

• 2

• 1

Multiple Choice

8v - 4v - 32 = 8

• 2

• 10

• 4

• -4