Quarry School

On a graph, the average rate of change is a slope

Explain it like I am five

On a graph, each input and its output make a point: the input tells you how far across, the output tells you how high. To find the average rate of change between two inputs, locate the points on the graph at those two inputs and connect them with a straight ruler line. The average rate of change is the slope of that line. Picture a hiking trail between two signposts: the trail may wiggle up and down, but the ruler line ignores the wiggles and measures only how much height you gained overall for each step east.

−2−1123−5−4−3−2−1123456(−1, 4)(2, 1)top 5
The curve rises to 5 and then falls. The dashed ruler through the two ends has slope −1; it is not part of the function.
Reminder
  • Signed division. Same signs divide to a positive; different signs divide to a negative. −3−3 = 1 and 3−3 = −1.
  • Subtracting a negative. 2 − (−1) = 2 + 1 = 3. Parentheses keep the negative input together.
  • Slope. Rise over run compares two points: a rise of 9 over a run of 3 gives 93 = 3.
Why it works. The average rate of change formula f(x2)−f(x1)x2−x1 is the slope formula y2−y1x2−x1 in disguise, because on a graph the output f(x) is the height y. So a connecting line rising to the right has a positive slope, a falling one a negative slope, and a flat one zero. What happens between the two points never enters the formula, which is why only the endpoints matter. A curved graph can rise near one endpoint yet have a negative average over the whole interval.
RuleFor [a, b] = [left end, right end], with a < b and both outputs defined, average rate = f(b)−f(a)b−a. This is the slope of the straight ruler through (a, f(a)) and (b, f(b)).
The same idea, five ways
Say it

Say 'the slope of the line joining the two graph points'.

Write it

The average rate compares the ending height with the starting height, per horizontal unit.

In math
  • f(b)−f(a)b−a, a < b
  • riserun
  • Graph words: connect the two endpoint points with a straight ruler.
Like

A ruler laid between two trail signposts measures the overall climb per step east.

See it
−22−4−2246(−1, 4)(2, 1)top 5
The curve rises to 5 and then falls. The dashed ruler through the two ends has slope −1; it is not part of the function.
The same idea, other ways
Follow the graph to a height

To read g(−1), begin at −1 on the horizontal axis, go vertically to the solid curve, and read height 4. The point (−1, 4) and the notation g(−1) = 4 say the same thing.

−22−4−2246(−1, 4)(2, 1)top 5
The curve rises to 5 and then falls. The dashed ruler through the two ends has slope −1; it is not part of the function.
Connect only the two ends

A hiker can climb to a high point in the middle and finish below the starting point. On this curve, the endpoints are at heights 4 and 1. The ruler drops 3 over a horizontal move of 3, so the average rate is −1.

−22−4−2246(−1, 4)(2, 1)top 5
The curve rises to 5 and then falls. The dashed ruler through the two ends has slope −1; it is not part of the function.
Explain it in a sentence

For a straight line the average rate stays the same on every interval because the line has the same steepness throughout. A curve can give different averages: x2 gives 4 on [1, 3] and 8 on [3, 5].

x2 on [1, 3]: 9−12 = 4
x2 on [3, 5]: 25−92 = 8
A longer or differently placed interval can change a curve's average rate.
.1Reading a value off a graph

A graph is another way to store input-output pairs. Go from the input axis to the curve before reading the output axis. Count grid squares using the printed scale; a square can represent more than one unit. A height between grid lines is an estimate.

  • g(t) is the output at input t, rather than g multiplied by t.
  • A point (t, y) on the graph means g(t) = y.
−22−4−2246(−1, 4)(2, 1)top 5
The curve rises to 5 and then falls. The dashed ruler through the two ends has slope −1; it is not part of the function.
Worked exampleReading two heights on a downward parabola, then finding the slope

The solid graph of f(x) = −(x + 1)2 + 4 is drawn on a grid where each square is 1 unit in both directions. Its highest point is (−1, 4). The graph passes exactly through the grid points (−2, 3) and (2, −5). Read the graph to find the average rate of change of f over [−2, 2]. Say what the sign means.

−4−224−6−4−2246(−2, 3)(2, −5)(−1, 4)
Graph of f(x) = −(x + 1)2 + 4 with its highest point at (−1, 4) and the interval endpoints (−2, 3) and (2, −5) marked.
  1. Locate x = −2 (left end) and x = 2 (right end) on the horizontal axis.The interval [a, b] = [−2, 2] has a = −2 < b = 2. These two inputs are where the heights must be read.
  2. From x = −2, move vertically to the solid curve. It meets the curve 3 squares above the x-axis, so f(−2) = 3.The height of the graph at an input is the output. The point (−2, 3) lies exactly on a grid point, so no estimate is needed.
  3. From x = 2, move vertically to the solid curve. It meets the curve 5 squares below the x-axis, so f(2) = −5.The point (2, −5) lies exactly on a grid point. It is below the axis, so the height is negative.
  4. Rise = f(2) − f(−2) = −5 − 3 = −8. Run = 2 − (−2) = 4.Subtract right minus left for both outputs and inputs, in the same order.
  5. Average rate = −84 = −2.The average rate of change is rise divided by run. This is the slope of the ruler through (−2, 3) and (2, −5).
  6. Interpret: the sign is negative, so over [−2, 2] the function falls overall. On average, f drops 2 units for each 1 unit increase in x.A negative slope means the right endpoint is lower than the left endpoint. The graph rises briefly to its top at x = −1, but the overall change across the interval is a decrease.
Answer
The average rate of change of f over [−2, 2] is −2. Overall f decreases, on average 2 units per 1 unit of x.
Check Use the formula. f(−2) = −(−2 + 1)2 + 4 = −1 + 4 = 3, and f(2) = −(2 + 1)2 + 4 = −9 + 4 = −5. These match the graph readings. Then −5−32−(−2) = −84 = −2. Also, the ruler from (−2, 3) drops 8 units over 4 units to the right, which is a slope of −2.

Work to write

  1. f(−2) = 3 (read from graph)
  2. f(2) = −5 (read from graph)
  3. Average rate = f(2)−f(−2)2−(−2) = −5−34
  4. = −84 = −2
  5. Negative: f decreases on average by 2 units per unit of x on [−2, 2]

The average rate of change of f over [−2, 2] is −2. Overall f decreases, on average 2 units per 1 unit of x.

.2Average speed from Anna's table

The table stores distance from home, so the starting 10 miles must be subtracted. In this outward-driving example, the change is also the distance driven during the chosen hours. If a trip turns back, distance from home alone does not measure all miles driven.

  • Average speed over [0, 6] is 47 miles per hour.
  • Over [2, 3], it is 63 miles per hour.
  • The chosen interval can change a travel average.
input t (hours)output D(t) (miles)01015529031534214524062927300↓ evaluate: input given, read the output below it
The highlighted endpoint columns show distance from home at the start and after six hours.
Worked exampleAverage speed of a cyclist between two times in a distance table

A cyclist's total distance from the trailhead is recorded at several times.

Time t (minutes): 10, 25, 40, 55
Distance d(t) (km): 3.2, 8.0, 12.5, 16.1

Find the cyclist's average speed on the interval [10, 40]. Give it in km per minute and in km per hour, and say what the sign means.

input time (minutes)output distance (km)103.22584012.55516.1↓ evaluate: input given, read the output below it
Table of the cyclist's distance from the trailhead at 10, 25, 40 and 55 minutes. The entries at t = 10 and t = 40 are used for the average speed.
  1. Find the interval endpoints in the time row. The left end is a = 10 and the right end is b = 40. Both values appear in the table, and 10 < 40.The rule needs a left input and a right input with a < b, and both outputs must be known.
  2. Read the distance under each time: d(10) = 3.2 km and d(40) = 12.5 km. The points are (10, 3.2) and (40, 12.5).These are the heights of the two points that the straight ruler would join on a distance–time graph.
  3. Find the rise as the right output minus the left output: 12.5 − 3.2 = 9.3 km. Find the run as the right input minus the left input: 40 − 10 = 30 minutes.The numerator of d(b)−d(a)b−a is the change in distance and the denominator is the change in time. Both are taken right minus left.
  4. Divide the rise by the run: 9.330 = 0.31 km per minute.Average rate of change is rise over run, which is the slope of the line through (10, 3.2) and (40, 12.5). The units are km per minute.
  5. Convert to km per hour: 0.31 × 60 = 18.6 km/h.There are 60 minutes in an hour, so the number of km per hour is 60 times the number of km per minute.
  6. Interpret the sign. The rate is positive, so the cyclist's distance from the trailhead increased over the interval.A positive slope means the output rises as the input increases.
Answer
Average speed on [10, 40] = 12.5−3.240−10 = 9.330 = 0.31 km/min, which is 18.6 km/h. The rate is positive, so the cyclist moved farther from the trailhead.
Check Start at 3.2 km and ride at 0.31 km/min for 30 minutes. That adds 0.31 × 30 = 9.3 km, and 3.2 + 9.3 = 12.5 km, which matches d(40). Also, 18.6 km/h is a reasonable cycling speed.

Work to write

  1. a = 10, b = 40
  2. d(10) = 3.2, d(40) = 12.5
  3. average speed = d(40)−d(10)40−10 = 12.5−3.240−10
  4. = 9.330 = 0.31 km/min
  5. 0.31 × 60 = 18.6 km/h
  6. Positive rate: the distance from the trailhead increased.

Average speed on [10, 40] = 12.5−3.240−10 = 9.330 = 0.31 km/min, which is 18.6 km/h. The rate is positive, so the cyclist moved farther from the trailhead.

.3A line compared with a curve

A straight line rises by the same amount for each equal step right. It therefore gives the same average rate on every interval. A curve can have different steepness in different places, so its average may depend on the interval.

  • For f(x) = 2x + 5 the average rate is 2 on every pair of distinct inputs.
  • For f(x) = x2, the averages on [1, 3] and [3, 5] are 4 and 8.
2462468101214161820run 1rise 2(0, 5)(1, 7)(2, 9)(7, 19)
Every unit right adds 2 to the height of this straight line.
Worked exampleAverage rate of change of a straight-line graph

The solid graph of y = f(x) is a straight line. Reading the grid, it passes through the points (−2, 7), (2, 1) and (4, −2). Find the average rate of change of f on the interval [−2, 4], and then on [2, 4]. Interpret the sign, and explain why the two answers agree.

−4−2246−4−22468(−2, 7)(2, 1)(4, −2)
The straight line y = f(x) through (−2, 7), (2, 1) and (4, −2). Its slope, −1.5, is the average rate of change on any interval.
  1. Locate the endpoints of the first interval, x = −2 and x = 4, on the horizontal axis.The interval [a, b] = [−2, 4] has left end a = −2 and right end b = 4, with a < b.
  2. Read the heights above those inputs. At x = −2 the graph is at height 7, so f(−2) = 7. At x = 4 it is at height −2, so f(4) = −2.The average rate uses the outputs at the two ends of the interval. Both points sit on grid intersections, so no estimating is needed.
  3. Find the rise and the run. Rise = f(4) − f(−2) = −2 − 7 = −9. Run = 4 − (−2) = 6.The rule subtracts right minus left in both places, so the order matches.
  4. Divide: average rate = −96 = −32 = −1.5.Average rate = f(b)−f(a)b−a is the slope of the segment joining (−2, 7) and (4, −2).
  5. Repeat for [2, 4]. f(2) = 1 and f(4) = −2. Rise = −2 − 1 = −3. Run = 4 − 2 = 2. Average rate = −32 = −1.5.This is the same method on a different interval of the same line.
  6. Interpret the result. The rate is negative, so f falls 1.5 units for every 1 unit increase in x. Both intervals give −1.5.A straight line has one constant slope. A ruler through any two of its points lies along the line itself, so every interval gives the same average rate.
Answer
The average rate of change is −32 = −1.5 on [−2, 4] and also −1.5 on [2, 4]. The function decreases by 1.5 units per unit of x. The answers agree because the graph is a straight line with slope −1.5.
Check Use the third pair of points, (−2, 7) and (2, 1): 1−72−(−2) = −64 = −1.5. This matches. The equation y = −1.5x + 4 also checks: at x = −2, y = 3 + 4 = 7; at x = 2, y = −3 + 4 = 1; at x = 4, y = −6 + 4 = −2.

Work to write

  1. f(−2) = 7, f(4) = −2
  2. average rate on [−2, 4] = f(4)−f(−2)4−(−2) = −2−76 = −96 = −1.5
  3. f(2) = 1, so average rate on [2, 4] = −2−14−2 = −32 = −1.5
  4. negative: f decreases 1.5 units per unit of x
  5. same answer on both intervals because a straight line has constant slope

The average rate of change is −32 = −1.5 on [−2, 4] and also −1.5 on [2, 4]. The function decreases by 1.5 units per unit of x. The answers agree because the graph is a straight line with slope −1.5.

Strategy: step by step
  1. Locate both interval endpoints on the horizontal axis.
  2. At each input, move vertically to the solid graph and read its height. Count grid steps; estimate if it lies between lines.
  3. Subtract right height minus left height, then right input minus left input.
  4. Divide rise by run and interpret the sign.
Strategy
Read an average rate from a graph
1
Is an endpoint output missing, shown only by a hollow dot?
YesThat output is not attained there; this endpoint quotient is undefined unless another filled point gives the output.
NoRead its solid-graph height.
↓
2
Does a height fall between grid lines?
YesEstimate the height and mark the final rate approximate.
NoUse the exact marked height.
↓
3
Does the curve change direction in the middle?
YesKeep only the two endpoint heights for the average.
NoUse the same endpoint method.
  1. Read two curve heights before doing arithmetic.
  2. Write the paired points or function values so the endpoint order is visible.
  3. Compute vertical change divided by horizontal change.
  4. Describe the average net rise or fall per unit.
Worked exampleAverage rate of change of a parabola from its graph on [−1, 3]

The graph of f(x) = x2 − 1 is drawn on a grid where each square is 1 unit. Read the graph to find the average rate of change of f over [−1, 3]. Say what the sign means.

−224−22468(−1, 0)(3, 8)
Graph of f(x) = x2 − 1 with the interval endpoints (−1, 0) and (3, 8) marked. The ruler through these two points has slope 2.
  1. Locate the endpoints on the horizontal axis: a = −1 on the left and b = 3 on the right.The interval [a, b] runs from the left end to the right end, so a = −1 and b = 3 with a < b.
  2. From x = −1, move vertically to the graph. It meets the curve on the x-axis, so f(−1) = 0. The point is (−1, 0).The output is the height of the graph above the input. Here the height lands exactly on a grid line, so no estimate is needed.
  3. From x = 3, move vertically to the graph. Count up 8 grid steps, so f(3) = 8. The point is (3, 8).Counting whole grid squares gives the exact height. Check with the formula: 32 − 1 = 8.
  4. Rise = f(3) − f(−1) = 8 − 0 = 8. Run = 3 − (−1) = 4.Use right minus left in both the numerator and the denominator. Subtracting −1 is the same as adding 1.
  5. Average rate = 84 = 2. The sign is positive.The rate is rise divided by run. A positive slope means the ruler through the two points goes up from left to right.
Answer
Average rate of change = f(3)−f(−1)3−(−1) = 84 = 2. On average, f increases by 2 units for each 1-unit increase in x over [−1, 3].
Check The line through (−1, 0) with slope 2 is y = 2x + 2. At x = 3 it gives 2(3) + 2 = 8, which is f(3). So the ruler joins both endpoints and its slope is 2. The graph also dips below 0 near x = 0, which shows that the average rate describes only the ruler between the two endpoints, not the shape of the curve between them.

Work to write

  1. a = −1, b = 3
  2. f(−1) = 0, f(3) = 8
  3. rise = 8 − 0 = 8, run = 3 − (−1) = 4
  4. average rate = 84 = 2
  5. Positive: f rises 2 units per unit of x on average over [−1, 3]

Average rate of change = f(3)−f(−1)3−(−1) = 84 = 2. On average, f increases by 2 units for each 1-unit increase in x over [−1, 3].

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Reading two heights on a shifted parabola to find a negative average rate

The solid graph of f(x) = (x − 1)2 − 3 is drawn on a grid where each square is 1 unit in both directions. Its lowest point is (1, −3). The graph passes exactly through the grid points (−1, 1) and (2, −2). Read the graph to find the average rate of change of f over [−1, 2]. Say what the sign means.

−224−4−2246(−1, 1)(2, −2)
The parabola f(x) = (x − 1)2 − 3 on a 1-unit grid, with its lowest point at (1, −3). The interval endpoints are marked at (−1, 1) and (2, −2).
  1. Locate the endpoints x = −1 and x = 2 on the horizontal axis. The left end is a = −1 and the right end is b = 2.The interval [a, b] tells us which two inputs to use, and a < b.
  2. From x = −1, move vertically to the solid curve. It meets the curve 1 square above the x-axis, so f(−1) = 1.The height of the graph above an input is the output at that input. Here the curve crosses exactly at a grid point, so no estimate is needed.
  3. From x = 2, move vertically to the solid curve. It meets the curve 2 squares below the x-axis, so f(2) = −2.A point below the x-axis has a negative height. Counting squares downward gives a negative output.
  4. Rise = f(2) − f(−1) = −2 − 1 = −3. Run = 2 − (−1) = 3.Both differences are taken right end minus left end, in the same order.
  5. Average rate = −33 = −1.The average rate of change is rise divided by run. It is the slope of the straight ruler through (−1, 1) and (2, −2).
  6. Interpret the result: the rate is negative, so f decreased overall on [−1, 2]. On average it dropped 1 unit for each 1 unit of x.A negative slope means the right end is lower than the left end. The curve does fall and then rise inside the interval, but the average rate uses only the two endpoints.
Answer
The average rate of change of f over [−1, 2] is −1. The negative sign means f decreased overall on this interval, by 1 unit of output per unit of input on average.
Check Compute from the formula. f(−1) = (−1 − 1)2 − 3 = 4 − 3 = 1, and f(2) = (2 − 1)2 − 3 = 1 − 3 = −2. Then −2−12−(−1) = −33 = −1. This matches the graph reading.

Work to write

  1. f(−1) = 1 and f(2) = −2 (read from the graph)
  2. Average rate = f(2)−f(−1)2−(−1)
  3. = −2−13 = −33 = −1
  4. Negative, so f decreased overall on [−1, 2], by 1 unit per unit of x on average

The average rate of change of f over [−1, 2] is −1. The negative sign means f decreased overall on this interval, by 1 unit of output per unit of input on average.

Rung 2A straight line has the same average rate on any interval

The graph of the straight line f(x) = −12x + 2 is drawn on a grid where each square is 1 unit in both directions. The line passes exactly through the grid points (−4, 4), (0, 2) and (2, 1). Read the graph to find the average rate of change of f over [−4, 2]. Then find it over [0, 2] and compare the two results. Say what the sign means.

−6−4−2246−2246(−4, 4)(0, 2)(2, 1)
The line f(x) = −12x + 2 on a 1-unit grid, marked at (−4, 4), (0, 2) and (2, 1), the points used for the intervals [−4, 2] and [0, 2].
  1. For [−4, 2], locate x = −4 (left end) and x = 2 (right end) on the horizontal axis.The rule uses a = −4 as the left input and b = 2 as the right input, with a < b.
  2. From x = −4, move vertically to the line. It meets the line 4 squares above the axis, so f(−4) = 4. From x = 2, the line is 1 square above the axis, so f(2) = 1.The height of the graph at each input is the output we need. Both points sit exactly on gridlines, so no estimate is needed.
  3. Rise = f(2) − f(−4) = 1 − 4 = −3. Run = 2 − (−4) = 6.Subtract right minus left in both the outputs and the inputs, so the signs stay consistent.
  4. Average rate = −36 = −12.The average rate is rise divided by run. This is the slope of the ruler through (−4, 4) and (2, 1).
  5. For [0, 2], read f(0) = 2 and f(2) = 1. Then rise = 1 − 2 = −1, run = 2 − 0 = 2, and the average rate = −12 = −12.Use the same steps on a different interval. This tests whether the rate depends on where we measure.
  6. Compare: both intervals give −12. The rate is negative, so f decreases. On average, it falls 12 unit for every 1 unit increase in x.A straight line has a single, constant slope. Every ruler through two of its points lies along the line itself, so every interval gives the same average rate.
Answer
The average rate of change over [−4, 2] is −12, and over [0, 2] it is also −12. The negative sign means f decreases by 12 unit per unit increase in x on each interval.
Check From the formula, f(−4) = −12(−4) + 2 = 4, f(0) = 2 and f(2) = −12(2) + 2 = 1. These match the graph readings. The slope m in f(x) = −12x + 2 is −12, which agrees with both average rates.

Work to write

  1. f(−4) = 4, f(2) = 1
  2. f(2)−f(−4)2−(−4) = 1−46 = −36 = −12
  3. f(0) = 2, so f(2)−f(0)2−0 = −12 = −12
  4. Same rate on both intervals because the graph is a straight line; negative means f is decreasing

The average rate of change over [−4, 2] is −12, and over [0, 2] it is also −12. The negative sign means f decreases by 12 unit per unit increase in x on each interval.

Rung 3Average rate of change read from a graph with a scaled vertical axis

A solid curve y = f(x) is drawn on a grid. Each horizontal square is 1 unit and each vertical square is 2 units. At x = −2 the curve crosses exactly the gridline 5 squares above the x-axis. At x = 4 the curve passes halfway between the gridlines 1 and 2 squares below the x-axis. Find the average rate of change of f on [−2, 4] and say what its sign means.

  1. Mark the interval endpoints on the horizontal axis: a = −2 (left) and b = 4 (right).The rule uses a < b, so a is the left end and b is the right end of [−2, 4].
  2. From x = −2, move vertically to the solid curve. It sits on the gridline 5 squares up, so f(−2) = 5 · 2 = 10.Each vertical square is worth 2 units. The height must be converted from squares to units.
  3. From x = 4, move vertically to the solid curve. It lies halfway between 1 and 2 squares below the axis, which is −1.5 squares. So f(4) = −1.5 · 2 = −3.When the point falls between gridlines, estimate the fraction of a square. Below the axis means a negative height.
  4. Rise = f(4) − f(−2) = −3 − 10 = −13. Run = 4 − (−2) = 6.Subtract right minus left in both the numerator and the denominator so the order matches.
  5. Average rate = −136 ≈ −2.17.Rise divided by run gives the slope of the ruler through (−2, 10) and (4, −3).
  6. Interpret the sign: the value is negative, so on average f decreases by about 2.17 units for each 1 unit increase in x over [−2, 4].A negative slope means the ruler line falls from left to right.
Answer
Average rate of change = −136 ≈ −2.17. The function decreases on average over [−2, 4].
Check Start at (−2, 10) and follow a line of slope −136 for a run of 6. The height changes by 6 · −136 = −13, which gives 10 − 13 = −3. This matches the reading f(4) = −3. The sign is negative, which agrees with the curve ending lower than it started.

Work to write

  1. a = −2, b = 4
  2. f(−2) = 10 (5 squares × 2)
  3. f(4) = −3 (−1.5 squares × 2)
  4. f(4)−f(−2)4−(−2) = −3−106 = −136
  5. ≈ −2.17; negative, so f decreases on average over [−2, 4]

Average rate of change = −136 ≈ −2.17. The function decreases on average over [−2, 4].

Rung 4Average speed of a road trip from a distance table

A car's distance from home, d(t) in miles, was recorded t hours after it left. The table gives t = 0, 1.5, 4 and 5 hours with distances 0, 84, 220 and 280 miles. Find the average speed of the car on the interval [1.5, 5] and say what the sign means.

input time (hours)output distance from home (miles)001.58442205280↓ evaluate: input given, read the output below it
Table of the car's distance from home at four times. The columns for t = 1.5 and t = 5 are the endpoints used for the average speed.
  1. Locate the interval endpoints in the input row: a = 1.5 (left end) and b = 5 (right end).The average rate on [a, b] uses only the two endpoint inputs, with a < b. The entry at t = 4 lies inside the interval and is not needed.
  2. Read the output under each endpoint: d(1.5) = 84 and d(5) = 280.Each table column pairs an input with its output. These two pairs are the points (1.5, 84) and (5, 280) that the ruler would pass through on a graph.
  3. Subtract right minus left in both rows. Rise: d(5) − d(1.5) = 280 − 84 = 196 miles. Run: 5 − 1.5 = 3.5 hours.The formula f(b)−f(a)b−a subtracts in the same order, right end minus left end, on top and on the bottom.
  4. Divide rise by run: 1963.5 = 56 miles per hour.Average rate of change is change in output per unit change in input. Its units are miles per hour, so the result is an average speed.
  5. Interpret the sign. The rate +56 is positive, so the distance from home increased.A positive slope means the output rises as the input increases. The car was moving away from home at an average of 56 miles each hour.
Answer
Average speed on [1.5, 5] = 280−845−1.5 = 1963.5 = 56 miles per hour. The positive sign means the car's distance from home grew.
Check Start at 84 miles at t = 1.5 and add 56 miles per hour for 3.5 hours: 84 + 56 × 3.5 = 84 + 196 = 280. That matches d(5) = 280.

Work to write

  1. a = 1.5, b = 5
  2. d(1.5) = 84, d(5) = 280
  3. average speed = d(5)−d(1.5)5−1.5 = 280−845−1.5
  4. = 1963.5 = 56 miles per hour
  5. Positive: the distance from home increased by 56 miles per hour on average

Average speed on [1.5, 5] = 280−845−1.5 = 1963.5 = 56 miles per hour. The positive sign means the car's distance from home grew.

Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The average rate is the steepness of the curve at the starting point.
The curve rises near −1, but on [−1, 2] it finishes lower than it starts.
✓ Instead: Use the dashed connecting ruler; its average slope is −1.
✗ Not this: The average rate always changes if the interval changes.
The straight line 2x + 5 gives 2 on both [0, 1] and [2, 7].
✓ Instead: A straight line has one slope; a curve's interval averages may differ.
✗ Not this: Read the height of the dashed ruler at an intermediate input as the function output.
The dashed line joins the endpoints but is not the solid curve.
✓ Instead: Read each function value from the solid graph; only the endpoint points are shared.
Tips and tricks
  • Letters are name tags. In [a, b], a is the left input and b the right input; x1 and x2 can name those same jobs.
  • Write what the rate means: 'drops 1 output unit per input unit on [−1, 2]' explains −1.
  • The endpoints must give actual function outputs; a hollow dot by itself leaves that input undefined.
Trap. Putting right-minus-left on top and left-minus-right on the bottom reverses the sign. Keep each input paired with its output and use one order twice.