HSF.IF.B.6: Calculating and Interpreting Average Rate of Change
In plain English: HSF.IF.B.6 is the Common Core functions standard that asks students to calculate the average rate of change of a function over a given interval, from an equation or a table, and to explain what it means with units, such as degrees per minute. Students also estimate the rate of change by reading points from a graph. It is usually taught in Algebra I and used again in Algebra II.
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Common Core State Standards for Mathematics · Domain: Interpreting Functions (IF) · Cluster: Interpret functions that arise in applications in terms of the context Also written as HSF-IF.B.6 or F-IF.6 · Official standard
The average rate of change of a function f over an interval from x = a to x = b is (f(b) - f(a)) / (b - a): the change in output divided by the change in input. Students already know this number as the slope of a line. In this lesson they apply it to functions that are not linear, where the rate is different on different intervals, so the interval always has to be named.
Students calculate average rates of change from equations and from tables, and they interpret every answer as a rate with units, such as "the stone fell 19.6 meters per second on average between 1 and 3 seconds". They then estimate rates from graphs by reading two points and see the average rate of change as the slope of the secant line through those points.
Learning Objectives
By the end of this lesson, students will be able to:
Calculate the average rate of change of a function given by an equation over a specified interval
Calculate the average rate of change from a table of values over a specified interval
Interpret an average rate of change in context, with units and the correct sign, and explain what it does not tell
Estimate a rate of change from a graph by reading points and connect it to the slope of a secant line
Prior Knowledge Required
Students should already be comfortable with:
Unit rates and slope as a rate 8.EE.B.5
Finding the rate of change of a linear function from a table or graph 8.F.B.4
Evaluating functions written in function notation HSF.IF.A.2
Working with negative numbers and fractions in division
Pose a question students can answer without a formula:
Warm-Up Prompt
"A family leaves home at 9:00 a.m. and has driven 150 miles by 11:30 a.m. What was their average speed? Did the car travel at that speed the whole time? How could two cars with very different trips have the same average speed?"
The average speed is 150 miles ÷ 2.5 hours = 60 miles per hour. The car stopped at lights and sped up on the highway, so it did not go 60 mph the whole time. Tell students that average speed is an average rate of change: the change in distance divided by the change in time. Today they will compute the same kind of rate for any function and any interval.
Direct Instruction20-25 minutes
Write the definition and a procedure:
Name the interval: a ≤ x ≤ b. The average rate of change always belongs to an interval.
Find the outputs: evaluate f(a) and f(b) from the equation, or read them from the table.
Divide the changes: (f(b) - f(a)) / (b - a), output change over input change, in that order.
Attach units: output units per input unit, such as meters per second or dollars per year.
Interpret: "On average, between ___ and ___, the (output) increased/decreased by ___ per (input unit)."
From an equation
A stone is dropped from a cliff. The distance it has fallen after t seconds is d(t) = 4.9t² meters. Find the average rate of change on 1 ≤ t ≤ 3 (Diagram 2).
Equation: (d(3) - d(1)) / (3 - 1) = (44.1 - 4.9)/2 = 19.6 m/s: the stone's average speed from 1 s to 3 s. On 0 ≤ t ≤ 1 it is only 4.9 m/s.
From a table
A town's population in thousands (invented data): 2010: 84.0, 2014: 91.2, 2018: 95.6, 2022: 97.2. Compare 2010-2018 with 2018-2022.
Equation: (95.6 - 84.0)/8 = 1.45 thousand per year; (97.2 - 95.6)/4 = 0.4 thousand per year. Growth slowed from about 1,450 to 400 people per year.
Estimated from a graph
Diagram 1 shows the temperature of a cup of tea. Read the points at m = 0, 10 and 20 and estimate the average rates of change.
Equation: ≈ (46 - 90)/10 = -4.4°C per minute, then ≈ (30 - 46)/10 = -1.6°C per minute. The tea cools more slowly as it nears room temperature.
Exponential model, equal intervals
A savings account balance is A(t) = 500(1.04)t dollars after t years. Compare 0 ≤ t ≤ 5 with 5 ≤ t ≤ 10.
Equation: A(5) ≈ 608.33 and A(10) ≈ 740.12. Rates: ≈ $21.67 per year, then ≈ $26.36 per year. The balance grows faster later.
Negative rate from a table
An airplane descends. Altitude in feet at minutes 0, 4, 8, 12: 33,000; 26,600; 17,800; 9,000. Find the rate on 0 ≤ t ≤ 12 and on each 4-minute interval.
Equation: (9,000 - 33,000)/12 = -2,000 ft per minute. Intervals: -1,600, -2,200, -2,200 ft per minute. The negative sign means losing altitude.
After the table example, ask: "Did the population grow by exactly 1,450 people every year from 2010 to 2018?" No: the average hides year-to-year changes, which is why the interval matters. After the graph example, stress that readings from a graph are estimates: the exact values are T(10) ≈ 45.7 and T(20) ≈ 30.3, which give -4.43 and -1.54°C per minute, close to the estimates.
Guided Practice15 minutes
Pairs work two problems on mini whiteboards, one representation at a time. (1) Equation: a bacteria culture has B(t) = 300 · 2t cells after t hours. Find the average rate of change on 0 ≤ t ≤ 3 (700 cells per hour) and on 3 ≤ t ≤ 4 (2,400 cells per hour), and explain why the second rate is larger. (2) Table: a phone battery reads 100% at 0 minutes, 91% at 20, 80% at 40 and 70% at 60 minutes of video streaming. Find the rate on 0 ≤ t ≤ 60 (-0.5 percentage points per minute, or -30 per hour) and on 20 ≤ t ≤ 40 (-0.55 per minute). Circulate and listen for students who divide the input change by the output change, or who drop the negative sign. Ask every pair to say each answer aloud as a sentence with units.
Independent Practice15 minutes
Students work alone: (1) Use Diagram 1 to estimate the tea's average rate of change from m = 20 to m = 30 by reading both points from the graph (about (25 - 30)/10 ≈ -0.5°C per minute). (2) For g(x) = 2x, find the average rate of change on 0 ≤ x ≤ 3 and on 3 ≤ x ≤ 6 (7/3 and 56/3) and explain why they differ. (3) Use the airplane table from the examples: on which 4-minute interval did the plane descend fastest, and how do you know? Students then compare their graph estimate with a partner's and discuss why two careful readings can differ slightly.
Closure5-10 minutes
Exit ticket: (1) A bike rental shop earned $2,400 in month 1 and $3,300 in month 4. Find the average rate of change of its monthly earnings from month 1 to month 4 and interpret it. (Answer: $300 per month.) (2) For f(x) = x², the average rate of change on -2 ≤ x ≤ 2 is 0. Does that mean f is constant on that interval? Explain. (No: f(-2) = f(2) = 4, but f decreases and then increases in between.)
Differentiation Strategies
For Struggling Students
Give a template with four boxes: interval, f(a) and f(b), the fraction (change in output)/(change in input), and the sentence with units
Start with tables where the input changes by 1, so the rate is just the change in output, then move to wider intervals
Have students write the units of the answer before calculating it
For Advanced Students
For f(x) = x², find the average rate of change on 3 ≤ x ≤ 3 + h for h = 1, 0.5 and 0.1, and describe the pattern (going further: this leads to the idea of an instantaneous rate)
Show that for any linear function the average rate of change is the same on every interval, and explain why this is not true for f(x) = x²
Find two different intervals on which h(t) = -16t² + 96t has an average rate of change of 0, and explain what this means for a ball's height
Assessment Guidance
What to Look For
Every answer should name the interval, carry units and have the right sign. Watch for the reversed fraction (input change over output change) and for students who subtract the outputs in one order and the inputs in the other. In interpretations, look for the word "average" or "on average": a rate of $300 per month does not mean earnings grew by exactly $300 each month. For graph estimates, accept a range of reasonable readings and ask students to explain which points they read.
02
Classroom Activities
3 Activities
1
Rate of Change Stations
25 minGroups of 3-4
Groups rotate through four stations, about 6 minutes each. Each station presents a function in a different way, and each group records the rate, units and a one-sentence interpretation on a shared recording sheet.
The Four Stations
Station A, equation: a balloon being pumped up holds V(t) = 0.5t² + 2t liters of air after t seconds. Rate on 2 ≤ t ≤ 6: (30 - 6)/4 = 6 liters per second
Station B, table: a runner's distance in a 10 km race at minutes 0, 10, 20, 30, 40 and 48 is 0, 2.1, 4.3, 6.3, 8.2 and 10.0 km. Rates: 0.215 km per minute on 0 ≤ t ≤ 20, 0.225 km per minute on 40 ≤ t ≤ 48 (a faster finish)
Station C, graph: a printed graph of a bean plant's height through the points (0, 3), (2, 8), (4, 17), (6, 24) and (8, 27), in weeks and centimeters. Estimate the rate on 2 ≤ w ≤ 6: about (24 - 8)/4 = 4 cm per week
Station D, interpretation: from 2019 to 2023, the average rate of change of a town's median home price was $18,500 per year. Groups decide which of three statements must be true: the price rose $74,000 in total (true); the price rose $18,500 every year (not necessarily); the price never fell (not necessarily)
Procedure
At each station, one student computes, one checks, and one writes the interpretation sentence; roles rotate
At the end, groups compare Station B answers and discuss what the runner's rates show about her race
Discussion Questions
At Station C, why might two groups give slightly different answers? Are both acceptable?
At Station D, what information would you need to know if the price ever went down?
2
When Did It Change Fastest?
15 minPairs
Pairs compute the average rate of change of a podcast episode's total downloads over several intervals and tell the story of the episode's popularity from the rates alone.
Data (invented)
Total downloads after 0, 1, 2, 3, 5 and 10 days: 0; 1,200; 1,900; 2,300; 2,700; 3,100.
Tasks
Compute the rates on 0 ≤ d ≤ 1, 1 ≤ d ≤ 3, 3 ≤ d ≤ 5 and 5 ≤ d ≤ 10 (1,200; 550; 200; 80 downloads per day)
Write a two-sentence story of the episode using the rates
Explain why the rate on 0 ≤ d ≤ 10 (310 downloads per day) describes no single day well
Modification for Distance Learning
Share the data in a spreadsheet. Pairs write a formula for the average rate of change between two rows and fill it down, then post their story in the class discussion board.
3
Secant Line Lab
20 minPairs
Pairs graph f(x) = x² carefully on graph paper, draw secant lines, and connect the slope they measure on the graph with the average rate of change they compute from the equation.
Procedure
Graph f(x) = x² for -1 ≤ x ≤ 4 with 1 unit = 2 grid squares
Draw the secant lines from (1, 1) to (3, 9), to (2, 4) and to (1.5, 2.25). Estimate each slope with a ruler by counting rise and run
Compute each average rate of change from the equation: 4, 3 and 2.5, and compare with the estimates
Draw the secant from (-1, 1) to (3, 9) and compute its slope (2). Explain why it is smaller than the slope from 1 to 3
Discussion Questions
How close were your graph estimates to the computed rates? What limits the accuracy of a reading?
What happens to the slope as the right endpoint moves closer to x = 1?
Challenge Variation
Going further: repeat with the right endpoint at 1.1 and 1.01. Predict the slope the secant lines approach, and describe what that number would mean if f(x) gave distance and x gave time.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Estimating Average Rates of Change from a Graph
Temperature of a cup of tea, modeled by T(m) = 22 + 68(0.9)m, drawn to scale. Reading approximate points from the graph gives estimated average rates of change of about -4.4°C per minute for the first 10 minutes and about -1.6°C per minute for the next 10. The dashed segments are the secant lines whose slopes are these rates.
Diagram 2: Average Rate of Change as the Slope of a Secant Line
Distance fallen by a dropped stone, d(t) = 4.9t², drawn to scale. The secant line through (1, 4.9) and (3, 44.1) rises 39.2 m over a run of 2 s, so its slope, the average rate of change on 1 ≤ t ≤ 3, is 19.6 m per second.
04
Homework Assignment
~30 min
HSF.IF.B.6 Homework: Average Rate of Change
Directions: Show the fraction (change in output)/(change in input) for every rate. Give each answer with units and write one sentence that interprets it in context. Round to two decimal places when needed. For Part 3, use a graphing calculator or app, and state which points you read from the graph.
Part 1: Functions Given by Equations (Problems 1-2)
For f(x) = 3x² - 2x + 1, find the average rate of change on 1 ≤ x ≤ 4 and on -2 ≤ x ≤ 0. Explain what the sign of each answer tells you about the graph.
The value of a laptop t years after purchase is V(t) = 1400(0.8)t dollars. Find the average rate of change over the first two years and over the next two years. Interpret both and explain why they differ.
Part 2: Functions Given by Tables (Problems 3-4)
Temperatures on a spring day in a mountain city (invented data): 6 a.m. 42°F, 9 a.m. 51°F, 12 p.m. 63°F, 3 p.m. 68°F, 6 p.m. 57°F. Find the average rate of change from 6 a.m. to 12 p.m., from 12 p.m. to 6 p.m., and from 3 p.m. to 6 p.m. When did the temperature fall fastest?
Students enrolled on an online course platform, in thousands (invented data): 2019: 12, 2020: 30, 2021: 41, 2022: 47, 2023: 50. Find the average rate of change from 2019 to 2021 and from 2021 to 2023. Then find it for 2019 to 2023 and explain what that single number hides.
Part 3: Estimating from a Graph (Problems 5-6)
A toy car rolls across a gym floor and slows down. Its distance after x seconds is f(x) = 3√x meters. Graph f for 0 ≤ x ≤ 25. Read points from your graph to estimate the average rate of change on 1 ≤ x ≤ 4 and on 16 ≤ x ≤ 25, then check your estimates with the equation.
A ball is kicked straight up, and its height after t seconds is h(t) = -5t² + 30t meters. Graph h for 0 ≤ t ≤ 6. Estimate the average rate of change on 0 ≤ t ≤ 2, 2 ≤ t ≤ 4 and 4 ≤ t ≤ 6 from the graph and interpret each. Does a rate of 0 mean the ball was not moving?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Calculation
Correct fraction and value for every interval
Correct method with one arithmetic or order error
Method incorrect or missing
Units and Sign
Every rate has correct units and sign
Units or sign missing on some rates
No units given
Interpretation
Each rate explained as an average over the interval in context
Interpretation present but vague or missing "average"
No interpretation
Graph Estimates
Points read are stated and estimates are reasonable
Reasonable estimate, points not stated
Estimate unreasonable or missing
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Choose an answer or work out a short response, then check it against the explanation. The score counter tracks the multiple-choice questions, and Reset quiz starts everything over.
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
What is the average rate of change of f(x) = x² + 3x on 1 ≤ x ≤ 4?
Answer: C
f(1) = 4 and f(4) = 28, so (28 - 4)/(4 - 1) = 24/3 = 8. Choice A is the change in output without dividing by the change in input. Choice B divides by 4, the right endpoint, instead of 4 - 1 = 3. Choice D is f(4)/4.
Question 2 of 20 · Multiple Choice
A function g has g(2) = 11, g(5) = 20 and g(9) = 8. What is the average rate of change of g on 2 ≤ x ≤ 9?
Answer: A
(g(9) - g(2))/(9 - 2) = (8 - 11)/7 = -3/7. Choice B drops the sign: the output decreased overall. Choice C divides the input change by the output change. Choice D is the rate on 5 ≤ x ≤ 9, (8 - 20)/4.
Question 3 of 20 · Multiple Choice
C(x) is the cost in dollars to produce x hundred units. The average rate of change of C on 2 ≤ x ≤ 5 is 150. What does this mean?
Answer: C
The rate has units of dollars per hundred units, because x counts hundreds, and it describes the interval from 200 to 500 units as an average. Choice B ignores that x is in hundreds. Choice A confuses a rate with a value of C.
Question 4 of 20 · Multiple Choice
L(t) is a lake's water level in feet t days after June 1. The average rate of change of L on 0 ≤ t ≤ 30 is -0.2. What does this mean?
Answer: B
A rate of -0.2 feet per day over 30 days means a total change of -0.2 × 30 = -6 feet, spread as an average. Choice C forgets that an average rate allows uneven daily changes. Choice D ignores the "per day" unit. Choice A treats the rate as a level.
Question 5 of 20 · Multiple Choice
On the graph of a car's distance from home, in miles, against time, in hours, you read the points (1, 55) and (3, 175). What is the estimated average rate of change from t = 1 to t = 3?
Answer: C
(175 - 55)/(3 - 1) = 120/2 = 60 miles per hour. Choice A is 175/3, which uses only one point. Choice B is the change in distance without dividing by the 2 hours. Choice D inverts the rate.
Question 6 of 20 · Multiple Choice
A table shows the number of members of a hiking club: year 0: 1,600 and year 2: 1,936. What is the average rate of change over the two years?
Answer: D
(1936 - 1600)/(2 - 0) = 336/2 = 168 members per year. Choice A is the total change, not divided by 2 years. Choice B divides the final value by 2. Choice C divides by 4.
Question 7 of 20 · Multiple Choice
A function has an average rate of change of 0 on 1 ≤ x ≤ 5. Which statement must be true?
Answer: B
A rate of 0 means f(5) - f(1) = 0, so the outputs at the endpoints are equal. The function may go up and down in between, as a thrown ball does, so choices A and D do not have to be true. Choice C confuses a zero rate with zero outputs.
Question 8 of 20 · Multiple Choice
The height of a ball is h(t) = -16t² + 64t + 5 feet after t seconds. What is the average rate of change of h on 1 ≤ t ≤ 3?
Answer: A
h(1) = -16 + 64 + 5 = 53 and h(3) = -144 + 192 + 5 = 53, so (53 - 53)/2 = 0. The ball is at the same height at 1 s and 3 s, going up and then coming down. Choice B is the rate on 1 ≤ t ≤ 2 only, since h(2) = 69, and choice C is the rate on 2 ≤ t ≤ 3. Choice D gives a height, not a rate.
Question 9 of 20 · Multiple Choice
On the graph of a function, the average rate of change from x = a to x = b is equal to which of the following?
Answer: A
The average rate of change (f(b) - f(a))/(b - a) is rise over run between the two points, which is the slope of the secant line through them. Choice B is a single output. Choice D measures length, not steepness.
Question 10 of 20 · Multiple Choice
The graph of a video's total views passes through points you read as (2 days, 1,500 views) and (6 days, 9,500 views). Estimate the average rate of change.
Answer: B
(9500 - 1500)/(6 - 2) = 8000/4 = 2,000 views per day. Choice C is the change in views without dividing by 4 days. Choice A is 9,500/6 and choice D is 9,500/2: both use only one point.
Question 11 of 20 · Multiple Choice
A sprinter's distance in meters at 0, 1, 2, 3 and 4 seconds is 0, 3, 8, 15 and 24. On which 1-second interval is the average rate of change greatest?
Answer: D
The rates are 3, 5, 7 and 9 meters per second, so the sprinter is fastest, on average, from 3 to 4 seconds: she is still speeding up. Choice A is the slowest interval, at 3 meters per second, and choices B and C give 5 and 7 meters per second.
Question 12 of 20 · Multiple Choice
A(t) = 800(1.05)t is an account balance in dollars after t years. What is the average rate of change on 0 ≤ t ≤ 2?
Answer: B
A(0) = 800 and A(2) = 800(1.1025) = 882, so (882 - 800)/2 = $41 per year. Choice A is the total change over 2 years. Choice C is the first year's interest only, 5% of 800. Choice D is 882/2.
Question 13 of 20 · Multiple Choice
V(t) is the volume of water in a tank in liters, and t is in minutes. What are the units of the average rate of change of V?
Answer: C
The rate is change in output over change in input, so its units are output units per input unit: liters per minute. Choice A inverts the fraction. Choice B is the unit of V itself.
Question 14 of 20 · Multiple Choice
A graph of the temperature at a weather station passes through (0, 12) and (8, -4), where time is in hours and temperature in °C. What is the estimated average rate of change?
Answer: D
(-4 - 12)/(8 - 0) = -16/8 = -2°C per hour: the temperature fell 2 degrees per hour on average. Choice A drops the sign. Choice B is the total change. Choice C divides the time change by the temperature change.
Question 15 of 20 · Short Answer
A theater's weekly revenue in dollars when tickets cost x dollars is R(x) = -20x² + 800x. Find the average rate of change of R on 10 ≤ x ≤ 15 and on 25 ≤ x ≤ 30. Interpret both.
R(10) = 6,000 and R(15) = 7,500, so the rate is 1,500/5 = $300 per dollar of ticket price: raising the price from $10 to $15 adds, on average, $300 of revenue for each $1 increase. R(25) = 7,500 and R(30) = 6,000, so the rate is -$300 per dollar: in that price range, raising the price loses revenue on average.
Question 16 of 20 · Short Answer
A gym's membership at the start of month 1 (January), month 3 (March), month 6 (June) and month 9 (September) was 420, 510, 480 and 390. Find the average rate of change from month 1 to 3 and from month 3 to 9, and interpret both.
From month 1 to 3: (510 - 420)/2 = 45 members per month, a gain on average. From month 3 to 9: (390 - 510)/6 = -20 members per month: on average the gym lost 20 members a month from March to September.
Question 17 of 20 · Short Answer
Explain how to estimate the average rate of change of a function from its graph between x = 2 and x = 6. Then use the readings (2, 3.1) and (6, 8.3) to give the estimate.
Find the points on the graph above x = 2 and x = 6, read their y-values as accurately as the grid allows, and divide the change in y by the change in x. Here: (8.3 - 3.1)/(6 - 2) = 5.2/4 = 1.3 (output units per input unit). The result is an estimate because the readings from the graph are approximate.
Question 18 of 20 · Short Answer
Over the interval 0 ≤ t ≤ 4 hours, the average rate of change of distance is 55 miles per hour for car A and 62 miles per hour for car B. Does this mean car B was faster at every moment? Explain, and find how much farther car B traveled.
No. An average rate only compares the total change: car A could have been faster for part of the trip while car B made up the difference later. In 4 hours, car B traveled 62 × 4 = 248 miles and car A 55 × 4 = 220 miles, so car B went 28 miles farther.
Question 19 of 20 · Short Answer
For f(x) = √x, find the average rate of change on 4 ≤ x ≤ 9 and on 9 ≤ x ≤ 16. What do the results show about the graph?
On 4 ≤ x ≤ 9: (3 - 2)/5 = 1/5. On 9 ≤ x ≤ 16: (4 - 3)/7 = 1/7. Both rates are positive, so f increases, but the second is smaller: the graph rises more and more slowly as x grows.
Question 20 of 20 · Short Answer
A hot-air balloon's altitude is 0 m at 0 minutes, 450 m at 10 minutes, 900 m at 25 minutes and 600 m at 40 minutes. Find the average rate of change on 0 ≤ t ≤ 25 and on 25 ≤ t ≤ 40, and interpret both.
On 0 ≤ t ≤ 25: 900/25 = 36 meters per minute, an average climb. On 25 ≤ t ≤ 40: (600 - 900)/15 = -20 meters per minute: the balloon descended 20 m per minute on average.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.IF.B.6 mean?
HSF.IF.B.6 means students can compute the average rate of change of a function over an interval, from an equation or a table, explain what it means in the situation, and estimate a rate of change from a graph. The average rate of change from x = a to x = b is (f(b) - f(a))/(b - a).
Is HSF.IF.B.6 Algebra 1 or Algebra 2?
It is usually introduced in Algebra I, where students compare linear, quadratic and exponential functions, and used again in Algebra II with polynomial, radical and trigonometric models. Precalculus and calculus build on it directly.
What is the difference between rate of change and average rate of change?
For a linear function they are the same number on every interval: the slope. For a nonlinear function the rate changes, so an "average" rate is computed over a specific interval. The rate at a single instant is a calculus idea; in this standard, students estimate rates from graphs and compute averages over intervals.
Is the average rate of change the same as slope?
It is the slope of the secant line through the two endpoints of the interval on the graph. For a line, that is the slope of the line itself. For a curve, different intervals give different secant lines and so different slopes.
What are common mistakes with average rate of change?
A frequent error is dividing the input change by the output change. Others are subtracting in different orders on top and bottom, which flips the sign, dividing by the right endpoint instead of b - a, leaving out units, and reading the answer as an exact change every step rather than an average.
What does a negative average rate of change mean?
It means the output was lower at the end of the interval than at the start, so the quantity decreased on average. An airplane with an average rate of -2,000 feet per minute is losing altitude. It does not say the quantity decreased at every moment.
How do you estimate the rate of change from a graph?
Read two points on the graph as accurately as the grid allows, then compute the change in output over the change in input. Because the readings are approximate, the answer is an estimate, and small differences between students are expected. Drawing the secant line helps students see the rate as a slope.
Why does the interval matter?
For a nonlinear function, different intervals give different average rates, and those differences tell the story: a cooling drink loses heat quickly at first and slowly later, and an investment grows faster in later years. Every rate must be reported with its interval.
How does average rate of change connect to other standards?
It extends slope from 8.F.B.4 to any function. HSF.LE.A.1 uses it to show that linear functions change by equal differences over equal intervals and exponential functions by equal factors. HSS.ID.C.7 interprets the slope of a linear model as a rate, and in calculus the average rate leads to the derivative.
Is average rate of change on the SAT?
Yes, in the sense that rates and slopes in context appear in the Algebra domain of the digital SAT, and nonlinear functions appear in Advanced Math. Students may be asked to find a rate from a table or graph or to interpret one with units.
07
Related Standards
6 standards
These standards connect to HSF.IF.B.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.B.4Prerequisite
Construct a linear function and find its rate of change and initial value
