8.F.A.2: Comparing Functions Shown in Different Ways
In plain English: 8.F.A.2 is the Common Core grade 8 math standard that asks students to compare two functions when each one is shown in a different way: an equation, a graph, a table or a description in words. Students find and compare rates of change, starting values and outputs for the same input. It follows 8.F.A.1 in the functions unit of Grade 8 Math.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Common Core State Standards for Mathematics · Domain: Functions (F) · Cluster: Define, evaluate, and compare functions. Also written as 8.F.2 · Official standard
In 8.F.A.1 students learned that a function gives each input exactly one output. Now they compare two functions when each one is shown in a different representation: an equation, a graph, a table or a verbal description (a sentence in words). To compare, students first find the same property in both, then compare the two numbers. The main properties are the rate of change (how much the output changes when the input goes up by 1), the initial value (the output when the input is 0), whether the function is increasing or decreasing, and the output for a chosen input.
The official example asks students to compare a linear function given by a table with one given by an equation and decide which has the greater rate of change; Example 1 and Diagram 2 work through it. Most functions here are linear (their graphs are straight lines), and one example compares a linear function with a function whose rate of change is not constant. The grade 8 functions standards carry the note "Function notation is not required in Grade 8," so every function on this page is named Function A or Function B and written in words or as an equation such as y = 3x + 5.
Learning Objectives
By the end of this lesson, students will be able to:
Find the rate of change and the initial value of a linear function given by an equation, a graph, a table or a verbal description
Decide which of two functions, shown in different ways, has the greater rate of change, and explain how they know
Compare the initial values of two functions and their outputs for the same input
State a comparison in the words of the situation, with units, for example "Tank A drains 20 more gallons per minute than Tank B"
Prior Knowledge Required
Students should already be comfortable with:
Knowing that a function gives each input exactly one output 8.F.A.1
Reading a unit rate as the slope of a graph 8.EE.B.5
Using the slope m and the equation y = mx + b of a line 8.EE.B.6
Computing unit rates, including rates with fractions 7.RP.A.1
Show the table and read the prompt aloud. Pairs have three minutes to answer all three questions.
Maya's savings (invented)
Week
0
1
2
3
Savings ($)
25
45
65
85
Warm-Up Prompt
"Leo says: I started with $40, and I add $15 every week. Who saves money faster, Maya or Leo? Who had more money at week 0? Who will have more money after 4 weeks?"
Collect answers. Maya saves faster: her savings go up $20 each week, and Leo's go up $15. Leo started with more, $40 against $25. After 4 weeks Maya has 85 + 20 = $105 and Leo has 40 + 4(15) = $100, so Maya has more. Point out that Maya's plan is a table and Leo's is a sentence, yet the class compared them. Today students name the two numbers they used: the rate of change and the initial value.
Direct Instruction20 minutes
Define each term and write it on an anchor chart:
Representation: one way of showing a function. The four ways are an equation, a graph, a table and a verbal description (a sentence in words).
Rate of change: how much the output changes when the input goes up by 1. For a linear function (a function whose graph is a straight line), it is the same between any two points, and it equals the slope (the steepness of the line: rise over run, as in 8.EE.B.6).
Initial value: the output when the input is 0. On a graph it is where the line crosses the y-axis.
Increasing or decreasing: a function is increasing if its outputs go up as the inputs go up (positive rate), and decreasing if they go down (negative rate).
Finding the rate in each representation: in y = mx + b, the rate is m, the coefficient of x (the number multiplied by x), and b is the initial value. In a table or on a graph, pick two points and divide the change in output by the change in input. In words, look for "per", "each" or "every".
Work through the four examples. For each function, students first write "rate = ..." and "initial value = ..." and only then compare.
Table and equation: greater rate of change (official example)
Function A is given by a table: x = 0, 2, 4, 6 gives y = 3, 10, 17, 24. Function B is y = 3x + 5. Which function has the greater rate of change?
Equation: Function A: (10 - 3) ÷ (2 - 0) = 7 ÷ 2 = 3.5. Function B: 3. Function A has the greater rate of change. Careful: the outputs go up by 7 each row, but the inputs go up by 2, so the rate is not 7.
Graph and words: two draining tanks
Two hot tubs are drained by pumps. Tank A: the graph of water (gallons) against time (minutes) is a line through (0, 600) and (8, 200). Tank B: "It holds 450 gallons and loses 30 gallons each minute." Which drains faster, and which started with more water?
Equation: Tank A: (200 - 600) ÷ (8 - 0) = -50 gallons per minute. Tank B: -30 gallons per minute. Tank A drains faster, 50 gallons a minute against 30, and it also started with more water, 600 gallons against 450.
Equation and graph: starting value and later outputs
Plant A's height in centimeters after w weeks is h = 1.5w + 4. Plant B's graph is a line through (0, 2) and (4, 12). Compare the starting heights, the growth rates and the heights after 6 weeks.
Equation: Starting heights: A 4 cm, B 2 cm. Rates: A 1.5 cm per week, B (12 - 2) ÷ 4 = 2.5 cm per week. After 6 weeks: A 1.5(6) + 4 = 13 cm, B 2 + 2.5(6) = 17 cm. B started shorter but is taller by week 6.
Table and words: a rate that is not constant
Function A: x = 1, 2, 3, 4, 5 gives y = 2, 4, 8, 16, 32. Function B: "The output is 6 times the input." Compare the outputs at x = 2 and x = 5, and decide which function has a constant rate of change.
Equation: At x = 2: A gives 4 and B gives 12, so B is greater. At x = 5: A gives 32 and B gives 30, so A is greater. A's outputs go up by 2, 4, 8, 16, so its rate is not constant and A is not linear. B's rate is always 6.
Use Diagram 1 with Example 2 to show the rate on a graph as a slope triangle (a right triangle drawn between two points of the line: its horizontal side is the change in input and its vertical side is the change in output). Draw Tank B's line from its words: start at 450 and fall 30 each minute. Use Diagram 2 with Example 1. The lines of A and B meet at (4, 17): B starts higher, but A grows faster, so A is greater after x = 4. Stress that a comparison names a property. "A is bigger" is not a full answer; "A has the greater rate of change" is.
Guided Practice15 minutes
Pairs work on four items, one at a time. After each one, a pair explains its answer at the board.
Function A: y = -2x + 9. Function B: x = 0, 1, 2, 3 gives y = 4, 7, 10, 13. Which function is increasing? Which has the greater initial value? (B is increasing, with rate 3; A is decreasing, with rate -2. A has the greater initial value, 9 against 4.)
Car A: the graph of miles against hours is a line through (0, 0) and (3, 180). Car B: "It drives 65 miles each hour." Which car is faster? (Car A goes 180 ÷ 3 = 60 miles per hour, so Car B is faster.)
Plan P: "$20 a month plus $5 for each gigabyte of data." Plan Q: c = 8g + 12, where c is the cost in dollars and g is gigabytes. Which plan has the greater rate of change? Which costs less for 3 gigabytes? (Q has the greater rate, $8 per gigabyte. For 3 gigabytes P costs $35 and Q costs $36, so P costs less.)
Elevator A: at 0, 5, 10 and 15 seconds its height is 60, 45, 30 and 15 meters. Elevator B: the graph of height against time is a line through (0, 40) and (10, 0). Which elevator goes down faster? (A: -15 ÷ 5 = -3 meters per second. B: -40 ÷ 10 = -4 meters per second. B goes down faster.)
Listen for students who read the initial value as the rate, and for students who say -2 is a "bigger drop" than -4.
Independent Practice15 minutes
Students work alone on four items, then compare with a partner:
Function A: y = 4x - 1. Function B: "Start at 10 and add 3 each time the input goes up by 1." Which has the greater rate? The greater initial value? Which output is greater at x = 5? (A has the greater rate, 4. B has the greater initial value, 10. At x = 5, A gives 19 and B gives 25, so B is greater.)
Function A: x = 1, 3, 5, 7 gives y = 11, 17, 23, 29. Function B: y = 2.5x + 9. Which has the greater rate, and what is A's initial value? (A's rate is 6 ÷ 2 = 3, greater than 2.5. Step back one unit from x = 1: 11 - 3 = 8, so A's initial value is 8.)
Function A: a line through (0, 15) and (5, 5). Function B: y = -3x + 20. Which decreases faster? (A's rate is -10 ÷ 5 = -2; B's rate is -3. B decreases faster.)
Write a verbal description of a function that has a greater rate of change than y = 2x + 7 but a smaller initial value. Trade with a partner and check each other's.
Closure5-10 minutes
Exit ticket: (1) Function A is y = 5x + 2. Function B: x = 0, 4, 8 gives y = 1, 23, 45. Which has the greater rate of change? Show both rates. (B: 22 ÷ 4 = 5.5, and A's rate is 5.) (2) Name one property, other than the rate of change, that you can compare, and say where you find it in a table. (For example, the initial value: the output in the column where x = 0.)
Differentiation Strategies
For Struggling Students
Give a comparison organizer with two columns (Function A, Function B) and rows for rate of change, initial value, and increasing or decreasing. Students fill in both columns before they compare
Start with tables whose inputs go up by 1, then move to tables that skip inputs, such as 0, 2, 4
For graphs, have students draw the slope triangle and label its run and its change in output before dividing
For Advanced Students
Give two functions in different representations and ask for the input where their outputs are equal, found from a table of values (a preview of 8.EE.C.8, beyond this standard)
Ask students to write the same function in all four representations, then give a partner two of them and ask which property is easier to read from each
Compare a linear function with a function whose outputs double each step, and find the first whole-number input where the doubling function is greater
Assessment Guidance
What to Look For
A full answer names the property, gives both values and states the comparison, for example "B has the greater rate of change: 5.5 against 5." Watch for four errors: reading the initial value b as the rate, dividing only the change in output and forgetting the change in input when a table skips inputs, reading the first row of a table as the initial value when its input is not 0, and saying that -2 decreases faster than -4. Ask students to state each comparison with units in context problems.
02
Classroom Activities
3 Activities
1
Rate Race Card Sort
15 minPairs
Each of the 8 cards shows how a student's pay (in dollars) depends on the hours worked at a summer job. The cards use all four representations. Pairs find the hourly rate on every card and line the cards up from the smallest rate to the greatest.
The 8 Cards (with answers)
Card 1, equation: y = 15x (rate $15 per hour, initial value $0)
Card 2, table: 0, 2, 4, 6 hours pay $10, $36, $62, $88 (rate $13 per hour, initial value $10)
Card 3, graph: a line through (0, 0) and (5, 80) (rate $16 per hour, initial value $0)
Card 4, words: "$14 per hour plus a $25 bonus on the first day" (rate $14 per hour, initial value $25)
Card 5, equation: y = 12.5x + 30 (rate $12.50 per hour, initial value $30)
Card 6, table: 1, 3, 5, 7 hours pay $18, $54, $90, $126 (rate $18 per hour, initial value $0)
Card 7, graph: a line through (0, 20) and (4, 72) (rate $13 per hour, initial value $20)
Card 8, words: "$120 for an 8-hour day, paid by the hour, with no bonus" (rate $15 per hour, initial value $0)
Procedure
Write "rate = ..." and "initial value = ..." on a sticky note for each card before ordering
Line up the cards from the smallest rate to the greatest. Cards with the same rate go in the same spot
Check the order with another pair and settle any difference by showing the division
Discussion Questions
The order is Card 5, then Cards 2 and 7 (tied), Card 4, Cards 1 and 8 (tied), Card 3 and Card 6. Which two pairs of cards tie, and how could two cards that look so different tie?
Card 5 has the greatest initial value but the smallest rate. After how many whole hours does Card 6 pay more than Card 5?
Card 6 starts at 1 hour. How did you find its initial value?
Modification for Distance Learning
Put the cards on a shared slide with a number line from 12 to 19 dollars per hour. Pairs drag each card to its rate and type the division next to it.
2
Cup Stack Lab
20 minGroups of 3
Groups measure the height of a stack of identical plastic cups as they add cups one at a time. Their measurements make a table. They compare their stack with a rival stack that is described by an equation.
Procedure
Stand one cup upright and measure its height to the nearest tenth of a centimeter. Add cups one at a time, up to 5 cups, and measure after each one
Record a table: input = number of cups, output = height of the stack in cm
Find the rate of change: how many centimeters each added cup adds
The rival stack follows h = 1.2n + 8, where n is the number of cups and h is the height in cm. Compare the rival's rate with yours, and compare the heights of a 10-cup stack
Sample Data (invented)
One group measured 9.5 cm for 1 cup, then 10.2, 11.0, 11.7 and 12.4 cm for 2 to 5 cups. From 1 cup to 5 cups the height rose 2.9 cm over 4 added cups, a rate of about 0.7 cm per cup. A 10-cup stack is about 12.4 + 5(0.725) ≈ 16 cm tall. The rival rate is 1.2 cm per cup, and its 10-cup stack is 1.2(10) + 8 = 20 cm tall.
Discussion Questions
In the sample data, one rival cup (9.2 cm) is shorter than one of our cups (9.5 cm), but the rival stack of 10 is taller. How is that possible?
Why are the measured steps (0.7, 0.8, 0.7, 0.7 cm) not all exactly the same? Is the stack still close to linear?
What does the 8 in h = 1.2n + 8 describe? Does a stack of 0 cups make sense?
Challenge Variation
Groups measure a second kind of cup, write its rule as an equation, and draw both stacks on one graph. Which property can they see directly on the graph, and which one do they need to compute?
3
Design a Rival
20 minPairs
Each pair gets 4 challenge cards. Each card shows a champion function and asks for a rival in a different representation that beats or matches it in a named way. Partners trade rivals and check them.
Challenge Cards (with one possible answer)
Card A: the champion is y = 2x + 10. Make a table rival with a greater rate of change and a smaller initial value. (x = 0, 1, 2 gives y = 4, 7, 10: rate 3, initial value 4.)
Card B: the champion is a table, x = 0, 5, 10 gives y = 50, 40, 30. Make a graph rival that decreases faster and starts lower. (A line through (0, 40) and (10, 0): rate -4 against -2, initial value 40 against 50.)
Card C: the champion is a line through (0, 6) and (3, 0). Write a verbal rival with the same initial value that decreases half as fast. ("Start at 6 and go down 1 each time the input goes up by 1": rate -1 against -2.)
Card D: the champion is "A taxi charges $3 plus $2 per mile." Write an equation rival that costs the same for a 4-mile ride but has a smaller rate. (c = 1.5m + 5: both cost $11 for 4 miles, and 1.5 < 2.)
Procedure
Write the champion's rate of change and initial value first
Choose the rival's rate and initial value, then write the rival in the representation the card asks for
Trade with your partner. The partner computes the rival's rate and initial value from the new representation and signs off only if the card's condition is met
Discussion Questions
For Card B, why does "decreases faster" mean a rate farther from 0, such as -4 compared with -2?
For Card D, is there more than one correct rival? What do all correct rivals have in common?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Reading and Comparing Rates on a Graph
Tank A is the graph from Example 2, drawn to scale: its slope triangle has a run of 8 minutes and a change of -400 gallons, so its rate is -50 gallons per minute. Tank B (dashed) is drawn from its words: it starts at 450 gallons and loses 30 gallons each minute. Tank A's line is steeper, so it drains faster.
Diagram 2: The Official Example, Table Against Equation
The table points of Function A (dots) and the line y = 3x + 5 of Function B (dashed), drawn to scale. B starts higher (5 against 3), but A has the greater rate of change (3.5 against 3), so the two lines meet at (4, 17) and A is greater for inputs above 4.
04
Homework Assignment
~30 min
8.F.A.2 Homework: Comparing Functions
Directions: For every comparison, first write the rate of change and the initial value of each function, then answer the question. Use units when there is a context.
Part 1: Rates and Starting Values (Problems 1-3)
Function A: x = 0, 3, 6, 9 gives y = 2, 14, 26, 38. Function B: y = 5x - 3. (a) Which function has the greater rate of change? (b) Which has the greater initial value?
Kayak shop X charges "a $12 launch fee plus $9 for each hour." Kayak shop Y's graph of cost against hours is a line through (0, 5) and (2, 27). (a) Which shop has the greater hourly rate? (b) What does each shop charge for 3 hours? (c) Which shop is cheaper for 4 hours?
Phone A's battery charge, in percent, after h hours of video is b = 100 - 8h. Phone B: after 0, 2, 4 and 6 hours of video its charge is 90%, 78%, 66% and 54%. (a) Which battery drains faster? (b) Which phone has more charge after 5 hours?
Part 2: Compare and Explain (Problems 4-6)
Hiker A's elevation graph, in meters against hours, is a line through (0, 1200) and (3, 2100). Hiker B "starts at 1,500 meters and climbs 250 meters each hour." (a) Who climbs faster? (b) Who is higher after 2 hours, and by how many meters?
Function A: x = 0, 1, 2, 3, 4 gives y = 1, 3, 9, 27, 81. Function B: y = 10x + 1. (a) Compare the outputs at x = 2 and at x = 4. (b) Which function has a constant rate of change? Use the table to explain.
Jin says Function A, y = 2x + 30, is "bigger" than Function B, shown by the table x = 0, 10, 20 and y = 0, 45, 90, because 30 is greater than 0. (a) Which property is Jin comparing? (b) Which function has the greater rate of change? (c) Find each function's output at x = 20. (d) Write one sentence that compares the two functions using two properties.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Rates of Change
Every rate is correct, including tables that skip inputs and decreasing functions
One or two rates are wrong
Most rates are wrong or missing
Initial Values and Outputs
All initial values and outputs are correct
One or two errors
Many errors
Comparisons
Each comparison names the property and both values
Comparisons given without the property or the values
No comparisons
Context and Units
Answers use the situation's words and units
Units missing in some answers
No context or units
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 linear function is shown in a table: x = 0, 1, 2, 3 gives y = 7, 11, 15, 19. A second linear function is y = 3x + 10. Which function has the greater rate of change?
Answer: A
The table's outputs go up by 4 for each increase of 1 in x, so its rate is 4, and 4 is greater than the equation's rate, 3. Choice B takes the initial value 10 as the rate. Choice D takes the table's first output, 7, as the rate. Choice C has the right rate for the equation, but 3 is less than 4.
Question 2 of 20 · Multiple Choice
Function A: x = 0, 5, 10, 15 gives y = 2, 12, 22, 32. Function B: y = 2.5x + 1. Which statement is true?
Answer: B
A's outputs go up by 10 while the inputs go up by 5, so A's rate is 10 ÷ 5 = 2. B's rate is 2.5, which is greater. Choice A forgets to divide by the change in input. Choice C has A's rate right but compares it the wrong way.
Question 3 of 20 · Multiple Choice
The graph of Function A is a line through (0, 8) and (6, 20). Function B: "Start at 3 and add 3 each time the input goes up by 1." Which statement is true?
Answer: D
A's rate is (20 - 8) ÷ 6 = 2 and its initial value is 8. B's rate is 3 and its initial value is 3. So B has the greater rate and A has the greater initial value. Choice A comes from dividing 20 by 6 without subtracting the starting value 8, which gives about 3.3, more than 3.
Question 4 of 20 · Multiple Choice
Candle 1 is 24 cm tall and burns down 1.5 cm each hour. Candle 2's height in centimeters after t hours is h = 30 - 2t. Which candle burns down faster?
Answer: B
The rate of change of Candle 2 is -2 cm per hour and that of Candle 1 is -1.5 cm per hour. A drop of 2 cm each hour is faster than 1.5 cm. Choices A and D compare the starting heights (the initial values), which say nothing about how fast a candle burns. Choice C names the smaller drop.
Question 5 of 20 · Multiple Choice
Function A: x = 0, 2, 4 gives y = 1, 9, 17. Function B: the graph is a line through (0, 5) and (2, 11). Which function has the greater output at x = 6?
Answer: C
A's rate is 8 ÷ 2 = 4, so at x = 6 it gives 1 + 4(6) = 25. B's rate is 6 ÷ 2 = 3, so it gives 5 + 3(6) = 23. A is greater. Choice D uses B's rise, 6, as its rate without dividing by the run of 2: 5 + 6(6) = 41. Choice A is B's correct output, but 23 is less than 25.
Question 6 of 20 · Multiple Choice
Gym A charges "a $50 joining fee plus $20 each month." Gym B's total cost after 1, 2, 3 and 4 months is $45, $70, $95 and $120. Which statement is true?
Answer: C
Gym B's cost goes up $25 each month, more than Gym A's $20. Its joining fee is the cost at 0 months: $45 - $25 = $20, less than $50. Choice B reads the cost at 1 month, $45, as the joining fee, but the table starts at month 1, not month 0. Choice D gets the monthly comparison backward.
Question 7 of 20 · Multiple Choice
Function A is y = -4x + 12. Function B's graph is a line through (0, 18) and (6, 0). Which function's outputs decrease faster?
Answer: D
B's rate is (0 - 18) ÷ 6 = -3, so B drops 3 for each 1 unit and A drops 4. A drop of 4 is faster. Choice C compares the signed numbers: -3 is greater than -4, but a greater number here means a slower drop. Choices A and B compare initial values, not rates.
Question 8 of 20 · Multiple Choice
Function A: x = 1, 2, 3, 4 gives y = 3, 6, 12, 24. Function B: "The output is 5 times the input." Which statement is true?
Answer: A
At x = 4, A gives 24 and B gives 5(4) = 20, so A is greater. Choice B is false: A's outputs go up by 3, 6 and 12, so its rate is not constant. Choice D uses A's first step, 3, as if it were A's rate everywhere, and B's rate is 5, not 3.
Question 9 of 20 · Multiple Choice
Bowling alley A charges c = 4.5g + 3 dollars for g games, including shoe rental. Alley B's graph of cost against games is a line through (0, 5) and (2, 13). Which statement is true?
Answer: C
B's rate is (13 - 5) ÷ 2 = $4 per game, with $5 for shoes. For 6 games, A costs 4.5(6) + 3 = $30 and B costs 5 + 4(6) = $29, so B is cheaper. For 2 games, A costs $12 and B costs $13, so choice B is false. Choice D would need equal rates, but $4.50 is not $4.
Question 10 of 20 · Multiple Choice
The graph of Function A is a line that goes down from left to right. Function B: x = 0, 1, 2 gives y = 9, 9, 9. Which statement is true?
Answer: A
A line that goes down from left to right has a negative rate of change: A is decreasing. B's output stays 9, so the change in output is 0 and B's rate is 0. Choice D takes the output 9 as the rate. Choice C also reads the direction of A backward.
Question 11 of 20 · Multiple Choice
In Town A, snow is 10 cm deep and gets 2.5 cm deeper each hour. In Town B, the snow depth in centimeters after h hours is d = 3h + 4. After 4 hours, which town has deeper snow, and by how much?
Answer: D
After 4 hours Town A has 10 + 2.5(4) = 20 cm and Town B has 3(4) + 4 = 16 cm, so Town A is deeper by 4 cm. Choice B is the difference at the start, 10 - 4 = 6 cm. Choice C multiplies the difference in rates, 0.5 cm per hour, by 4 hours and credits the wrong town. Choice A has the right gap but the wrong town.
Question 12 of 20 · Multiple Choice
Function A has a rate of change of 3 and an initial value of -2. Function B's graph is a line through (1, 4) and (3, 10). Which statement is true?
Answer: B
B's rate is (10 - 4) ÷ (3 - 1) = 3, the same as A's. B's initial value is one step back from (1, 4): 4 - 3 = 1, which is greater than -2. Choice D comes from dividing 10 by 3 (about 3.3) instead of using the changes. Choice C compares -2 and 1 the wrong way.
Question 13 of 20 · Multiple Choice
Tank A: after 0, 4 and 8 minutes it holds 20, 44 and 68 liters. Tank B "starts with 20 liters and gains 5 liters each minute." Which property is the same for both tanks?
Answer: C
Both tanks hold 20 liters at 0 minutes, so the initial values match. Tank A's rate is 24 ÷ 4 = 6 liters per minute, not 5, so choice A is wrong. After 4 minutes A has 44 liters and B has 40; after 8 minutes A has 68 and B has 60.
Question 14 of 20 · Multiple Choice
Which function has a greater rate of change than y = 1.5x + 2?
Answer: D
Choice D's rate is 5 ÷ 3, about 1.67, which is greater than 1.5. Choice A's rate is 3 ÷ 2 = 1.5, equal but not greater. Choice B's rate is 5 ÷ 4 = 1.25. Choice C has a greater initial value, 10, but its rate is only 1.
Question 15 of 20 · Short Answer
Function A: x = 0, 4, 8, 12 gives y = 4, 10, 16, 22. Function B: y = 1.4x + 6. Which function has the greater rate of change? Show both rates.
Function A. A's outputs go up by 6 while the inputs go up by 4, so its rate is 6 ÷ 4 = 1.5. B's rate is 1.4, and 1.5 > 1.4.
Question 16 of 20 · Short Answer
Club A's graph of money raised (dollars) against water bottles sold is a line through (0, 0) and (40, 60). Club B earns $2 for each bottle and started with $0. Which club earns more per bottle, and how much more money does it raise when each club sells 100 bottles?
Club B, $2 per bottle against Club A's 60 ÷ 40 = $1.50. For 100 bottles, B raises $200 and A raises $150, so B raises $50 more.
Question 17 of 20 · Short Answer
Function A: y = -2x + 15. Function B: x = 0, 1, 2, 3 gives y = 20, 17, 14, 11. Which function decreases faster? Which has the greater output at x = 5?
B decreases faster: its rate is -3, and A's is -2. At x = 5, A gives -2(5) + 15 = 5 and B gives 20 - 3(5) = 5, so the outputs are equal at x = 5.
Question 18 of 20 · Short Answer
Aiden says, "y = 6x + 1 has a greater rate of change than the line through (0, 2) and (1, 9), because 6 is greater than 2." Explain his mistake and give the correct comparison.
Aiden used the line's initial value, 2, as if it were its rate. The line's rate is (9 - 2) ÷ (1 - 0) = 7. The line has the greater rate of change, 7 against 6.
Question 19 of 20 · Short Answer
In Town A the temperature was 48°F at 6 am and rose 3°F each hour. In Town B, the graph of temperature against hours after 6 am is a line through (0, 42) and (4, 58). Which town warmed faster, and what was the temperature in each town at 11 am?
Town B warmed faster: (58 - 42) ÷ 4 = 4°F per hour, against 3°F. At 11 am, 5 hours after 6 am, Town A was 48 + 3(5) = 63°F and Town B was 42 + 4(5) = 62°F.
Question 20 of 20 · Short Answer
Function A: x = 0, 1, 2, 3, 4 gives y = 0, 2, 6, 12, 20. Function B: "The output is 4 times the input." (a) Which function has a constant rate of change? (b) Which output is greater at x = 1, and which at x = 4?
(a) Only B: its outputs go up by 4 each time. A's outputs go up by 2, 4, 6 and 8, so A's rate is not constant. (b) At x = 1, B is greater (4 against 2). At x = 4, A is greater (20 against 16).
0 of 20 answered · 0 correct
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Frequently Asked Questions
10 Questions
What does 8.F.A.2 mean?
8.F.A.2 means students can compare two functions when each one is shown in a different way. One might be an equation and the other a graph, a table or a sentence. Students find the same property in both, such as the rate of change or the initial value, and say which function has more of it.
Is 8.F.A.2 only about linear functions?
No, but most grade 8 comparisons use linear functions. The official example compares two linear functions, and linear functions are the main topic of the grade 8 functions unit. Students can also compare the outputs of a function whose rate is not constant, such as one whose outputs double, with a linear one, as in Example 4.
How do you find the rate of change from a table?
Divide the change in the outputs by the change in the inputs between two rows. If the inputs go 0, 10, 20 and the outputs go 3, 8, 13, the rate is 5 ÷ 10 = 0.5, not 5. Checking the input column first prevents the error of reading the output step as the rate.
What properties of two functions can students compare?
In grade 8, the usual ones are the rate of change, the initial value, whether each function is increasing or decreasing, the output for a given input, and whether the rate of change is constant. A strong answer always names the property it compares.
Do students need f(x) notation for 8.F.A.2?
No. The grade 8 functions standards carry the footnote "Function notation is not required in Grade 8." Students name functions in words (Function A, Plan B) and write equations such as y = 3x + 5. Function notation comes in high school, in HSF.IF.A.2.
What mistakes do students make when comparing functions?
Four errors come up often. Students read the number b in y = mx + b as the rate, forget to divide by the change in input when a table skips inputs, use the first row of a table as the initial value when the first input is not 0, and say that a rate of -2 falls faster than a rate of -4 because -2 is the greater number.
How does 8.F.A.2 connect to 8.EE.B.5?
8.EE.B.5 compares two proportional relationships given in different ways, for example two speeds. 8.F.A.2 does the same for any functions, including ones that do not start at 0. So the unit rate students compared in 8.EE.B.5 becomes the rate of change, and a new property, the initial value, joins it.
What comes after 8.F.A.2 in high school?
The high school standard HSF.IF.C.9 asks the same question for more kinds of functions, such as quadratic and exponential ones, in Algebra I and later courses. Students also meet the average rate of change over an interval in HSF.IF.B.6, which extends the rate of change to functions whose graphs are curves.
How can parents practice comparing functions at home?
Use real choices. Compare two phone or streaming plans, one from an ad (words) and one from a price table, and ask which costs more per month and which costs more to start. Ask your child which plan is cheaper for 6 months and to show how they know.
Why do the four representations matter for 8.F.A.2?
Because the same property looks different in each one. The rate of change is the coefficient of x in an equation, the slope of a graph, the change in output over the change in input in a table, and a "per" or "each" amount in words. Students who can find it in all four can compare any two functions.
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Related Standards
6 standards
These standards connect to 8.F.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.A.1Prerequisite
Understand that a function assigns to each input exactly one output
