8.F.B.4: Modeling Linear Functions with Rate of Change and Initial Value
In plain English: 8.F.B.4 is the Common Core grade 8 math standard that asks students to write a linear function that models a real situation. Students find the rate of change and the initial value from a description, from two (x, y) pairs, from a table or from a graph, and explain what each number means in the situation. It is a core modeling standard of Grade 8 Math.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Common Core State Standards for Mathematics · Domain: Functions (F) · Cluster: Use functions to model relationships between quantities. Also written as 8.F.4 · Official standard
Students learn to build a linear function (a function, a rule that gives each input exactly one output, whose graph is a straight line) that models a real situation, and to write it as an equation y = mx + b. Two numbers describe every such model. The rate of change is how much y changes for each increase of 1 in x; on a graph it is the slope (the steepness of the line, rise ÷ run). The initial value is the value of y when x = 0, the starting amount; on a graph it is where the line crosses the y-axis. Students find both numbers from four kinds of information: a description in words, two (x, y) pairs, a table, and a graph.
The second half of the standard is interpretation. Students explain what the two numbers mean in the situation (dollars per class, centimeters per day, a one-time fee, a starting height) and where they show up in a table or a graph. Every rule on this page is written as an equation such as y = -6x + 50 or in words. Function notation such as f(x) is not needed in grade 8 and is not used in the quiz or homework.
Learning Objectives
By the end of this lesson, students will be able to:
Write an equation y = mx + b that models a linear situation described in words
Find the rate of change and the initial value from two (x, y) pairs, including pairs read from a table or a graph
Work backward in a table that does not start at x = 0 to find the initial value
Explain what the rate of change and the initial value mean in the situation, with units
Point to the rate of change and the initial value on a graph and in a table of values
Prior Knowledge Required
Students should already be comfortable with:
Knowing that a function gives each input exactly one output 8.F.A.1
Reading the unit rate as the slope of a proportional graph 8.EE.B.5
Finding slope as rise ÷ run and knowing that y = mx + b crosses the y-axis at b 8.EE.B.6
Adding, subtracting, multiplying and dividing positive and negative numbers 7.NS.A.1
Write the prompt on the board and give pairs three minutes. Ask them to make a small table first.
Warm-Up Prompt
"The climbing wall at a rec center charges $6 to rent a harness, plus $2 for each climb. Make a table of the total cost for 0, 1, 2 and 3 climbs. Which number do you start with? Which number do you keep adding?"
Collect the table on the board: 0 climbs, $6; 1 climb, $8; 2 climbs, $10; 3 climbs, $12. Students usually say "you start with 6 and add 2 each time." Tell them that these two numbers have names. The $6 is the initial value: the cost when x = 0. The $2 per climb is the rate of change: the change in cost for each extra climb. Write the model y = 2x + 6 under the table and point out that the rate multiplies x and the initial value is added on.
Direct Instruction20 minutes
Put these words on an anchor chart, one at a time, each with a small sketch:
Linear function: a function whose graph is a straight line. In grade 8 it is written y = mx + b.
Model: an equation (or table or graph) that describes how two quantities in a real situation are related. Here, x is the input quantity and y is the output quantity.
Rate of change: the change in y for each increase of 1 in x. From two points, rate of change = (change in y) ÷ (change in x). It is the slope m of the line. Recall from 8.EE.B.6 that slope = rise ÷ run, where the rise is the vertical change and the run is the horizontal change between two points. Its unit is "y-units per x-unit," such as dollars per class.
Initial value: the value of y when x = 0. It is b in y = mx + b. On a graph it is the y-intercept, the point (0, b) where the line crosses the y-axis. In a table it is the row where x = 0, even if you have to work backward to find it.
Sign of the rate: a positive rate means y goes up as x goes up. A negative rate means y goes down, as when a candle burns or money is spent.
Work through the five examples. For each one, ask two questions before any arithmetic: "What are x and y, with units?" and "What is y when x = 0?"
From a description
A pottery class charges a $40 materials fee plus $15 for each class you attend. Write a model for the total cost y after x classes.
Equation: The rate of change is $15 per class and the initial value is $40, so y = 15x + 40. After 8 classes: 15(8) + 40 = $160.
From two (x, y) values
A candle is 20 cm tall after burning for 2 hours and 14 cm tall after 5 hours. It burns at a steady rate. Model its height y after x hours.
Equation: Rate = (14 - 20) ÷ (5 - 2) = -6 ÷ 3 = -2 cm per hour. Going back 2 hours from (2, 20) adds 4 cm, so the initial value is 24 cm and y = -2x + 24. The candle burns out when 24 - 2x = 0, after 12 hours.
From a table that does not start at 0 (Diagram 2)
A gift card is used only for smoothies that all cost the same. The table shows the balance y after x smoothies: 2 smoothies, $38; 4 smoothies, $26; 6 smoothies, $14. Find the price, the starting balance and the model.
Equation: Each time x goes up 2, y goes down 12, so the rate is -12 ÷ 2 = -$6 per smoothie. Going back from x = 2 to x = 0 adds $12, so the initial value is $50 and y = -6x + 50.
From a graph (Diagram 1)
A graph shows the height of a bean seedling, in centimeters, x days after a student starts measuring it. The line crosses the y-axis at (0, 4) and passes through (2, 7) and (4, 10).
Equation: The initial value is 4 cm, the height on the first day of measuring. From (2, 7) to (4, 10): rise 3, run 2, so the rate is 3 ÷ 2 = 1.5 cm per day. Model: y = 1.5x + 4.
Interpreting a model in the situation, the graph and the table
The cost in dollars to rent a moving truck for one day is y = 0.75x + 30, where x is the number of miles driven. What do 0.75 and 30 mean?
Equation: 0.75 is the rate: each mile adds $0.75. 30 is the initial value: a $30 daily fee paid even before driving. On the graph, the line crosses the y-axis at (0, 30) and rises $75 for every 100 miles to the right. In a table with rows for 0, 100 and 200 miles, the costs are $30, $105 and $180.
Use Diagram 1 with Example 4: students read the initial value where the line meets the y-axis, then draw their own slope triangle (a right triangle under the line whose sides are the run and the rise) between any two marked points and get 1.5 again. Use Diagram 2 with Example 3 to show working backward in a table, and point out that the graph is separate dots because you cannot buy half a smoothie. Stress the order in y = mx + b: the rate multiplies x, and the initial value stands alone. Swapping them is a frequent mistake.
Guided Practice15 minutes
Pairs work through four problems, one for each kind of information. One partner finds the rate of change, the other finds the initial value, and they write the model together.
Guided practice problems with answers
Problem
Answer
Description: a bowling alley charges $4 for shoe rental plus $5.50 per game. Model the cost y of x games.
Rate $5.50 per game, initial value $4: y = 5.5x + 4
Two values: a hiker's water bottle holds 800 mL after 1 hour of hiking and 400 mL after 3 hours. Model the water y after x hours and say when it is empty.
Rate -400 ÷ 2 = -200 mL per hour, initial value 1,000 mL: y = -200x + 1000. Empty after 5 hours
Table: a bakery charges $8.50 for 12 cookies, $14.50 for 24 cookies and $20.50 for 36 cookies, in one box. Find the model and explain the initial value.
Rate 6 ÷ 12 = $0.50 per cookie. Initial value 8.50 - 6 = $2.50, a charge for the box: y = 0.5x + 2.5
Graph: the water depth in a pool being drained is a line through (0, 48) and (6, 30), with hours on the x-axis and inches on the y-axis. Find and interpret both numbers.
Initial value 48 inches deep at the start. Rate (30 - 48) ÷ 6 = -3 inches per hour: y = -3x + 48
Listen for students who use the first table row ($8.50) as the initial value, and for students who divide the change in x by the change in y.
Independent Practice15 minutes
Students work alone, then compare with a partner.
Independent practice problems with answers
Problem
Answer
A class already has $80 for a trip and earns $7 profit on each T-shirt it sells. Model the money y after selling x T-shirts.
y = 7x + 80
A linear function passes through (3, 4) and (7, 24). Find the rate of change, the initial value and the equation.
Rate 20 ÷ 4 = 5; initial value 4 - 15 = -11; y = 5x - 11
A table has x = 0, 4, 8, 12 and y = 25, 22, 19, 16. Write the model.
Rate -3 ÷ 4 = -0.75; initial value 25; y = -0.75x + 25
The pages y left to read in a novel after x days are y = 320 - 25x. What do 320 and -25 mean? After how many days is the book finished?
320 pages at the start; 25 fewer pages left each day. 320 ÷ 25 = 12.8, so the book is finished on day 13
In one sentence, explain why the initial value of a model is where its graph crosses the y-axis.
Every point on the y-axis has x = 0, and the initial value is the y-value when x = 0
Closure5-10 minutes
Exit ticket: (1) An electrician charges a $75 visit fee plus $90 per hour. Write the model for the cost y of an x-hour job. (y = 90x + 75.) (2) A linear function passes through (4, 10) and (8, 4). Find its rate of change and initial value. (Rate -6 ÷ 4 = -1.5; initial value 10 + 6 = 16.) (3) In Diagram 1, what does the 4 in y = 1.5x + 4 mean for the seedling? (The seedling was 4 cm tall when the measuring started.)
Differentiation Strategies
For Struggling Students
Give a table template whose first row is always x = 0, and have students fill that row in first by working backward
Use a sentence frame: "y starts at ___ and changes by ___ for each 1 ___ of x"
Color-code: circle the rate of change in blue and the initial value in black, in the words, the table, the graph and the equation
For Advanced Students
Give two models for the same situation, such as two phone plans, and ask when they cost the same (a preview of 8.EE.C.8, beyond this standard)
Ask students to write a situation where the initial value makes no sense in real life, and to explain why the model still needs it
Give three (x, y) pairs where one pair does not fit a line and ask students to find it and explain how
Assessment Guidance
What to Look For
A strong answer names both numbers with units and says what they mean, for example "$15 per class" and "a $40 fee paid once." Watch for four errors: swapping the rate and the initial value in y = mx + b, using the first row of a table as the initial value when that row is not x = 0, dividing the change in x by the change in y, and dropping the negative sign when y goes down.
02
Classroom Activities
3 Activities
1
Four Ways to Show a Model: Card Match
20 minPairs
Each pair gets 16 cards that show four linear functions. Every function appears once as words, once as a table, once as a graph (given by two points) and once as an equation. Pairs build four groups of four cards and write the rate of change and the initial value on each group.
The 16 Cards
Card 1 (Words): A water tank starts with 10 liters, and a hose adds 2 liters each minute.
Card 2 (Table): x: 1, 2, 4; y: 12, 22, 42
Card 3 (Graph): a line through (0, 30) and (10, 10)
Card 4 (Equation): y = 0.5x + 30
Card 5 (Words): A 30-meter roll of ribbon; each gift bow uses 2 meters.
Card 6 (Table): x: 4, 10, 20; y: 32, 35, 40
Card 7 (Graph): a line through (0, 10) and (4, 18)
Card 8 (Equation): y = 10x + 2
Card 9 (Words): Maya has $2 and earns $10 for each dog walk.
Card 10 (Table): x: 1, 3, 5; y: 12, 16, 20
Card 11 (Graph): a line through (0, 30) and (20, 40)
Card 12 (Equation): y = -2x + 30
Card 13 (Words): A class jar has $30, and students add 50 cents each day.
Card 14 (Table): x: 2, 5, 8; y: 26, 20, 14
Card 15 (Graph): a line through (0, 2) and (3, 32)
Card 16 (Equation): y = 2x + 10
Answer Key
Tank: Cards 1, 7, 10 and 16. Rate 2 liters per minute, initial value 10 liters
Ribbon: Cards 3, 5, 12 and 14. Rate -2 meters per bow, initial value 30 meters
Dog walks: Cards 2, 8, 9 and 15. Rate $10 per walk, initial value $2
Class jar: Cards 4, 6, 11 and 13. Rate $0.50 per day, initial value $30
Procedure
Start with the four Words cards and write the rate of change and the initial value on a sticky note for each
For every Table and Graph card, find the rate from two pairs and the initial value from x = 0 before matching
When all four groups are built, check with another pair and settle any disagreement with a calculation
Discussion Questions
Cards 2 and 10 both contain the pair (1, 12). Why do they belong to different functions?
Cards 3 and 11 both cross the y-axis at 30. What is different about them?
Which two functions have rates with the same size but opposite signs? What does the sign tell you in each story?
Modification for Distance Learning
Put the 16 cards on a shared slide. Pairs drag them into four boxes and type the rate and the initial value in each box.
2
Stacking Cups: Build a Model from Measurements
20 minGroups of 3-4
Groups measure the height of stacks of 1 to 6 plastic cups of the same size with a centimeter ruler, record (number of cups, height) pairs, and build a linear model. The activity shows an initial value that is useful in the model but does not match a real stack.
Sample Data (invented)
An invented group measured stacks of 1, 2, 3, 4, 5 and 6 cups at 12.0, 13.1, 14.2, 15.3, 16.4 and 17.5 cm.
Procedure
Stand the stack upside down on the table and measure from the table to the top, to the nearest millimeter
Record each (cups, height) pair in a table and plot the pairs on grid paper
Find the rate of change from two rows: in the sample, (17.5 - 12.0) ÷ (6 - 1) = 1.1 cm per cup
Work back from 1 cup to 0 cups to find the initial value: 12.0 - 1.1 = 10.9 cm. Write the model: y = 1.1x + 10.9 in the sample
Use the model to predict the height of 20 cups (32.9 cm in the sample), then test it if you have enough cups
Discussion Questions
What does the rate 1.1 cm per cup measure on a real cup? (The part of each cup that sticks out above the one below it.)
A stack of 0 cups has no height, but the model says 10.9 cm. What does 10.9 cm measure on one cup?
In the sample, a shelf has 50 cm of space. What is the largest number of cups that fits? (35 cups: 49.4 cm; 36 cups would need 50.5 cm.)
Challenge Variation
Give each group a second cup size. Groups build a second model and explain which number, the rate or the initial value, changed more and why.
3
Model Stations Gallery Walk
20 minGroups of 4, rotating
Four posters around the room each show one situation in a different form. Groups spend 4 minutes at each station and write, on a sticky note, the model, the rate of change with units, what the initial value means, and one prediction. The next group checks the note before adding its own.
The Four Stations
Station 1 (graph): a hiker walks down a mountain. The graph of elevation (height above sea level) in meters against hours of walking is a line through (0, 2400) and (3, 1500). Predict the elevation after 5 hours
Station 2 (table): a phone plan costs $31 for 2 GB (gigabytes) of data, $40 for 5 GB and $52 for 9 GB. Predict the cost of 12 GB
Station 3 (words): a school pays a $150 bus fee plus $4 per student for a museum trip. Predict the cost for 60 students
Station 4 (two values): 4 weeks after a family adopts a puppy, it weighs 9.4 kg; 10 weeks after, it weighs 13.0 kg. Assume steady growth and predict its weight 16 weeks after adoption
Answer Key
Station 1: y = -300x + 2400. The hiker descends 300 m per hour from a starting elevation of 2,400 m; after 5 hours, 900 m
Station 2: y = 3x + 25. Each GB adds $3, and the base charge is $25; 12 GB cost $61
Station 3: y = 4x + 150. Each student adds $4, and the bus costs $150; 60 students cost $390
Station 4: y = 0.6x + 7. The puppy gains 0.6 kg per week and weighed 7 kg when adopted; after 16 weeks, 16.6 kg
Discussion Questions
Only one station has a negative rate of change. Which one, and what does the sign mean there?
At Station 4, the initial value is not "a fee" or "a start of a trip." What is it?
Should the Station 4 model be trusted a year after adoption? Why might steady growth stop?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Reading a Model from a Graph
The height of a bean seedling, drawn to scale. The line crosses the y-axis at (0, 4), so the initial value is 4 cm. The slope triangle from (2, 7) to (4, 10) has a run of 2 and a rise of 3, so the rate of change is 1.5 cm per day and the model is y = 1.5x + 4.
Diagram 2: Working Backward in a Table
The gift card table starts at x = 2. Each step of 2 smoothies lowers the balance by $12, so going back one step to x = 0 adds $12 and gives the initial value, $50. On the graph, the dots fall on a line that crosses the y-axis at (0, 50). The dots are not joined because only whole smoothies can be bought.
04
Homework Assignment
~30 min
8.F.B.4 Homework: Building and Reading Linear Models
Directions: For every model, name what x and y stand for, with units. Show how you found the rate of change and the initial value, and use grid paper for graphs.
Part 1: Build the Model (Problems 1-3)
A car's gas tank holds 14 gallons when full. On a highway trip, the car uses 0.04 gallons per mile. (a) Write a model for the gas y left after x miles. (b) Give the rate of change and the initial value with units, and say what each means. (c) How much gas is left after 150 miles? (d) How many miles can the car go before the tank is empty?
An online shop prints custom T-shirts. An order of 10 shirts costs $135 and an order of 25 shirts costs $285. The cost is linear. (a) Find the rate of change. (b) Find the initial value and explain what it could stand for. (c) Write the model. (d) What does an order of 40 shirts cost?
A bathtub drains at a steady rate. After 1 minute it holds 32 gallons, after 3 minutes 24 gallons, after 4 minutes 20 gallons and after 6 minutes 12 gallons. (a) Find the rate of change, and explain why you cannot just subtract one row from the next. (b) How much water was in the tub when it started draining? (c) Write the model and find when the tub is empty.
Part 2: Read and Interpret Models (Problems 4-6)
A graph shows the height of a young maple tree, in feet, x years after it was planted. The line passes through (0, 7) and (5, 17). (a) What are the initial value and the rate of change? Say what each means for the tree. (b) Write the model. (c) If the growth stays steady, how tall is the tree after 8 years?
An electric scooter ride costs y = 0.35x + 1 dollars for x minutes. (a) What do 0.35 and 1 mean? (b) Where does the graph cross the y-axis, and what does that point mean? (c) What does a 20-minute ride cost? (d) Explain what the point (10, 4.50) on the graph tells you.
Kai looks at this table: x = 3, 6, 9 and y = 26, 38, 50. He says, "The initial value is 26, because it is the first number, and the rate of change is 12." (a) Explain both of Kai's mistakes. (b) Find the correct rate of change and initial value. (c) Write the model and graph it for x = 0 to 9.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Rate of Change
Correct value and sign, with units, found from two pairs
Correct value but wrong sign, or no units
Missing or wrong
Initial Value
Correct value at x = 0, found by working backward when needed
Uses a table row that is not x = 0, or one arithmetic error
Missing or wrong
Model
Equation y = mx + b with m and b in the right places
Rate and initial value swapped
No equation
Interpretation
Explains both numbers in the situation and on the graph or table
Explains one number
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A gym charges a $25 joining fee plus $30 per month. For the total cost y after x months, what are the rate of change and the initial value?
Answer: B
The cost goes up $30 for each month, so the rate is $30 per month. The $25 fee is paid once, at x = 0, so it is the initial value: y = 30x + 25. Choice A swaps the two numbers. Choice C adds them into one monthly charge, which would make 2 months cost $110 instead of $85. Choice D forgets the joining fee.
Question 2 of 20 · Multiple Choice
A linear function passes through (2, 5) and (6, 17). What is its rate of change?
Answer: A
Rate = (17 - 5) ÷ (6 - 2) = 12 ÷ 4 = 3. Choice B divides the change in x by the change in y (4 ÷ 12). Choice C stops at the change in y. Choice D divides one y-value by its x-value (17 ÷ 6), which works only for a line through (0, 0).
Question 3 of 20 · Multiple Choice
The points (4, 11) and (10, 26) are on the graph of a linear function. What is its initial value?
Answer: C
The rate is (26 - 11) ÷ (10 - 4) = 15 ÷ 6 = 2.5. Going back 4 steps from x = 4 to x = 0 subtracts 4 × 2.5 = 10, so the initial value is 11 - 10 = 1. Choice A is the rate, not the initial value. Choice B is the y-value at x = 4, not at x = 0. Choice D adds the 10 instead of subtracting it.
Question 4 of 20 · Multiple Choice
A table shows x = 0, 1, 2, 3 and y = 40, 36, 32, 28. Which equation models the table?
Answer: D
When x = 0, y = 40, so the initial value is 40. Each time x goes up 1, y goes down 4, so the rate is -4. Choice A loses the negative sign. Choice B swaps the rate and the initial value. Choice C uses the last y-value, 28, as the initial value.
Question 5 of 20 · Multiple Choice
A table for a linear function shows x = 3, 5, 8 and y = 20, 26, 35. What is the initial value?
Answer: A
From x = 3 to x = 5, y goes up 6, so the rate is 6 ÷ 2 = 3 (check: from 5 to 8, 9 ÷ 3 = 3). Going back 3 steps from x = 3 to x = 0 subtracts 3 × 3 = 9, so the initial value is 20 - 9 = 11. Choice B is the first row, where x = 3, not 0. Choice C is the rate. Choice D goes back only one step: 20 - 3.
Question 6 of 20 · Multiple Choice
A graph shows a line that crosses the y-axis at (0, -2) and passes through (4, 6). Which equation models the line?
Answer: C
The initial value is -2. From (0, -2) to (4, 6): rise 8, run 4, so the rate is 2 and y = 2x - 2. Choice A uses the y-value 6 of the second point as the rate. Choice B divides run by rise (4 ÷ 8). Choice D swaps the rate and the initial value.
Question 7 of 20 · Multiple Choice
A graph shows the money y, in dollars, in a class account x weeks after the class starts spending it. The line crosses the y-axis at (0, 150) and goes down $20 for each week to the right. What does the point (0, 150) mean?
Answer: B
The point (0, 150) is on the y-axis, where x = 0 weeks, so it is the initial value: $150 at the start. Choices A and D read the initial value as a rate. Choice C reads 150 as an x-value; the account is actually empty after 150 ÷ 20 = 7.5 weeks.
Question 8 of 20 · Multiple Choice
The water y, in liters, in a tank x minutes after a valve opens is y = 120 - 8x. What does the -8 mean?
Answer: A
The -8 multiplies x, so it is the rate of change: 8 fewer liters for each minute. Choice B confuses the rate with the initial value, which is 120 liters. Choice C is also wrong: the tank is empty after 120 ÷ 8 = 15 minutes. Choice D ignores the negative sign.
Question 9 of 20 · Multiple Choice
The cost of a sports camp is y = 45x + 60 dollars for x days. What does the 60 represent?
Answer: D
The 60 is the initial value: the cost when x = 0, a fee that does not depend on the number of days. Choice A describes the 45. Choice B would be 45 + 60 = $105. Choice C confuses the output with the input.
Question 10 of 20 · Multiple Choice
A classroom starts the year with a 500-sheet pack of printer paper and uses 40 sheets each school day. Which equation gives the sheets y left after x school days?
Answer: C
The pack starts with 500 sheets (initial value) and loses 40 each day (rate -40), so y = 500 - 40x, which is the same as y = -40x + 500. Choice A adds paper instead of using it. Choice B swaps the two numbers. Choice D subtracts once and then treats the result as a rate.
Question 11 of 20 · Multiple Choice
In a table for a linear function, each time x goes up by 2, y goes up by 7. What is the rate of change?
Answer: B
Rate = change in y ÷ change in x = 7 ÷ 2 = 3.5. Choice A forgets that x changed by 2, not 1. Choice C divides the change in x by the change in y. Choice D multiplies instead of dividing.
Question 12 of 20 · Multiple Choice
Which description gives a linear function with a rate of change of 4 and an initial value of 9?
Answer: D
Starting with 9 gives the initial value 9, and getting 4 more each day gives the rate 4: y = 4x + 9. Choice A swaps the numbers. Choice B has a rate of -4. Choice C starts at 0 and has no initial value of 9; it only describes 9 days.
Question 13 of 20 · Multiple Choice
A rain barrel is drained to water a garden. A table shows 0, 1, 2 and 3 hours with 95, 83, 71 and 59 liters left. Which statement interprets the table correctly?
Answer: A
The row for 0 hours shows 95 liters, the initial value, and each hour the amount goes down 12 liters: y = -12x + 95. Choice B reads the table backward. Choice C uses the row for 1 hour as the start. Choice D uses the total change over 3 hours as the change for one hour.
Question 14 of 20 · Multiple Choice
A graph of a linear function passes through (-4, -1) and (4, 17). What is its initial value?
Answer: B
The rate is (17 - (-1)) ÷ (4 - (-4)) = 18 ÷ 8 = 2.25. Going from x = 4 back to x = 0 subtracts 4 × 2.25 = 9, so the initial value is 17 - 9 = 8: y = 2.25x + 8. Choices A and D are the y-values at x = -4 and x = 4, not at x = 0. Choice C is the rate.
Question 15 of 20 · Short Answer
A water park charges $18 per person plus $12 to park one car. Write a model for the total cost y for a group of x people who come in one car. Then find the cost for 5 people.
y = 18x + 12. The rate is $18 per person and the initial value is the $12 parking fee. For 5 people: 18(5) + 12 = $102.
Question 16 of 20 · Short Answer
A linear function passes through (3, 50) and (8, 30). Find its rate of change and its initial value, and write the equation.
Rate = (30 - 50) ÷ (8 - 3) = -20 ÷ 5 = -4. Going back 3 steps from x = 3 adds 3 × 4 = 12, so the initial value is 62. Equation: y = -4x + 62. Check: -4(8) + 62 = 30.
Question 17 of 20 · Short Answer
Dani saves money at a steady rate. Her savings table shows $110 after 2 months, $185 after 5 months and $285 after 9 months. Find the rate of change and the initial value, and explain what each one means.
Rate = (185 - 110) ÷ (5 - 2) = 75 ÷ 3 = $25 per month (check: 100 ÷ 4 = 25): she saves $25 each month. Initial value = 110 - 2(25) = $60: she had $60 when she started. Model: y = 25x + 60.
Question 18 of 20 · Short Answer
A pickup truck towing a trailer starts a trip with 26 gallons of gas. A graph of the gas y left after x miles is a line through (0, 26) and (100, 18). Find the rate of change and explain what it means. How much gas is left after 250 miles?
Rate = (18 - 26) ÷ 100 = -0.08 gallons per mile: the truck uses 0.08 gallons for each mile. Model: y = -0.08x + 26. After 250 miles: 26 - 0.08(250) = 6 gallons.
Question 19 of 20 · Short Answer
A taxi charges a $3 pickup fee plus $2.25 per mile. Leah writes the model y = 3x + 2.25. Explain her mistake, write the correct model, and find the cost of an 8-mile ride.
Leah swapped the rate and the initial value. The $2.25 per mile multiplies the miles, and the $3 fee is added once. Correct model: y = 2.25x + 3. An 8-mile ride costs 2.25(8) + 3 = $21. (Leah's model would give $26.25.)
Question 20 of 20 · Short Answer
The graph of a linear function crosses the y-axis at (0, -3) and rises 2 units for every 5 units to the right. Write its equation and find y when x = 10.
The initial value is -3 and the rate is 2 ÷ 5 = 0.4, so y = 0.4x - 3. When x = 10: 0.4(10) - 3 = 1.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.F.B.4 mean?
8.F.B.4 means students can write a linear function that models a real situation and explain its two key numbers. The rate of change tells how much y changes for each 1 unit of x, and the initial value is y when x = 0. Students find both from words, two (x, y) pairs, a table or a graph, and say what they mean with units.
Is 8.F.B.4 taught in grade 8 or in Algebra I?
8.F.B.4 is a grade 8 standard, usually taught in the functions unit of Grade 8 Math after slope (8.EE.B.6). Algebra I revisits the same skill with more function types in HSF.LE.A.2, and schools that teach Algebra I in grade 8 cover it early in that course.
What is the difference between rate of change and slope?
For a linear function they are the same number. "Slope" describes the steepness of the line on the graph, found as rise ÷ run. "Rate of change" describes the situation, with units, such as 1.5 cm per day. The standard asks students to connect the two.
How do you find the initial value when a table does not start at x = 0?
Find the rate of change first, then work backward to x = 0. In the gift card table on this page, x goes up 2 while the balance goes down $12, so going back from x = 2 to x = 0 adds $12. You can also substitute one pair into y = mx + b and solve for b.
What if the initial value does not make sense in the situation?
The model still needs it, but it may not describe a real object. In the stacking cups activity, a stack of 0 cups cannot be 10.9 cm tall; the number is the height of one cup without its lip. Ask students to say whether x = 0 is possible before they interpret b.
How is 8.F.B.4 different from 8.EE.B.5 and 8.EE.B.6?
8.EE.B.5 is about proportional relationships, whose graphs pass through (0, 0), and 8.EE.B.6 explains slope and derives y = mx + b. 8.F.B.4 uses those tools to model situations with a starting amount, where the initial value is usually not 0, and to interpret both numbers in context.
Do students need f(x) notation for 8.F.B.4?
No. The grade 8 function standards carry a footnote (at 8.F.A.1) that says function notation is not required in grade 8. Students write models as equations such as y = 15x + 40. Function notation comes in high school, in HSF.IF.A.2.
What mistakes do students make with rate of change and initial value?
A frequent one is swapping the two numbers, writing the fee as the rate. Others are using the first table row as the initial value when that row is not x = 0, dividing the change in x by the change in y, and losing the negative sign when an amount goes down.
Why are some models graphed as dots and others as lines?
It depends on which inputs make sense. If x counts whole things, such as smoothies or T-shirts, the graph is separate dots on a line. If x can be any number in a range, such as hours or miles, the dots fill in to make a solid line segment.
How can parents help with 8.F.B.4 at home?
Look for costs with a fixed part and a per-unit part: a ride-share fare, a delivery fee plus price per item, a membership plus a price per visit. Ask your child which number is the starting amount, which is the amount per unit, and to write the equation and use it for one prediction.
07
Related Standards
6 standards
These standards connect to 8.F.B.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.EE.B.5Prerequisite
Graph proportional relationships and read the unit rate as the slope
