Average Rate of Change and Secant Lines

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Key Ideas Easy Medium Hard

Average rate of change:

f(b) − f(a)b − a

y − y₁ = m(x − x₁)

What to Remember

Average rate of change

On an interval [a, b], the average rate of change of f is the slope between the two points.

f(b) − f(a)b − a

Read the two y-values from the table. A negative rate means the quantity is decreasing.

Keep the units

The units of the average rate of change are (output units) per (input unit). Liters and minutes become liters per minute.

Secant line

The secant is the line through those two table points. Its slope is the average rate of change from part a.

y − y₁ = m(x − x₁)

You can leave it in point-slope form or rewrite it as y = mx + b.

Overestimate or underestimate

Plug the requested x-value into the secant equation. Then compare to the actual table value.

If the secant value is larger than the table, the approximation is an overestimate. If it is smaller, it is an underestimate.

Level 1

Easy

Read two table values, find the average rate of change, then write and use the secant.

Example

A rain barrel is being filled from a downspout. The table gives the volume V of water in liters after t minutes.

t (minutes)	0	3	6	9	12
V (liters)	8	14	24	32	38

a.) Find the average rate of change on the interval [3, 9].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate V at t = 6. Is this an overestimate or an underestimate?

Average rate of change:

V(9) − V(3)9 − 3

32 − 149 − 3

186

3 liters per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

V − 14 = 3(t − 3)

V − 14 = 3t − 9

V = 3t + 5

Substitute t = 6 into the secant equation.

V = 3(6) + 5

V = 18 + 5

V = 23

The table gives V(6) = 24.

23 < 24, so this is an underestimate.

Answer: a.) 3 liters per minute
b.) V = 3t + 5
c.) 23 liters; underestimate

A sunflower’s height h in inches is recorded each week w.

w (weeks)	0	2	4	6	8
h (inches)	2	6	12	16	18

a.) Find the average rate of change on the interval [2, 8].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate h at w = 4. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

h(8) − h(2)8 − 2

18 − 68 − 2

126

2 inches per week

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

h − 6 = 2(w − 2)

h − 6 = 2w − 4

h = 2w + 2

Substitute w = 4 into the secant equation.

h = 2(4) + 2

h = 8 + 2

h = 10

The table gives h(4) = 12.

10 < 12, so this is an underestimate.

Answer: a.) 2 inches per week
b.) h = 2w + 2
c.) 10 inches; underestimate

A mug of cocoa cools on the counter. The table gives the temperature T in degrees Celsius after t minutes.

t (minutes)	0	5	10	15	20
T (°C)	90	80	68	58	50

a.) Find the average rate of change on the interval [5, 20].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate T at t = 10. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

T(20) − T(5)20 − 5

50 − 8020 − 5

−3015

−2 °C per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

T − 80 = −2(t − 5)

T − 80 = −2t + 10

T = −2t + 90

Substitute t = 10 into the secant equation.

T = −2(10) + 90

T = −20 + 90

T = 70

The table gives T(10) = 68.

70 > 68, so this is an overestimate.

Answer: a.) −2 °C per minute
b.) T = −2t + 90
c.) 70 °C; overestimate

A lemonade stand’s earnings s in dollars after w weeks are shown.

w (weeks)	0	4	8	12	16
s (dollars)	30	50	80	100	110

a.) Find the average rate of change on the interval [4, 16].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate s at w = 8. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

s(16) − s(4)16 − 4

110 − 5016 − 4

6012

5 dollars per week

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

s − 50 = 5(w − 4)

s − 50 = 5w − 20

s = 5w + 30

Substitute w = 8 into the secant equation.

s = 5(8) + 30

s = 40 + 30

s = 70

The table gives s(8) = 80.

70 < 80, so this is an underestimate.

Answer: a.) 5 dollars per week
b.) s = 5w + 30
c.) 70 dollars; underestimate

Maya bikes around a lake. The table gives her distance d in meters after t minutes.

t (minutes)	0	6	12	18	24
d (meters)	0	400	900	1300	1600

a.) Find the average rate of change on the interval [6, 18].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate d at t = 12. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

d(18) − d(6)18 − 6

1300 − 40018 − 6

90012

75 meters per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

d − 400 = 75(t − 6)

d − 400 = 75t − 450

d = 75t − 50

Substitute t = 12 into the secant equation.

d = 75(12) − 50

d = 900 − 50

d = 850

The table gives d(12) = 900.

850 < 900, so this is an underestimate.

Answer: a.) 75 meters per minute
b.) d = 75t − 50
c.) 850 meters; underestimate

A bowl of popcorn is being shared during a movie. The table gives the number of cups p remaining after t minutes.

t (minutes)	0	4	8	12	16
p (cups)	40	34	24	18	12

a.) Find the average rate of change on the interval [4, 12].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate p at t = 8. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

p(12) − p(4)12 − 4

18 − 3412 − 4

−168

−2 cups per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

p − 34 = −2(t − 4)

p − 34 = −2t + 8

p = −2t + 42

Substitute t = 8 into the secant equation.

p = −2(8) + 42

p = −16 + 42

p = 26

The table gives p(8) = 24.

26 > 24, so this is an overestimate.

Answer: a.) −2 cups per minute
b.) p = −2t + 42
c.) 26 cups; overestimate

Level 2

Medium

Longer tables. Same three parts: average rate, secant equation, overestimate or underestimate.

Example

A cistern holds 8,000 gallons of rainwater. A pump is turned on to fill it from a well. The table gives the volume v of water in gallons after t minutes of pumping.

t (minutes)	0	4	8	12	16	20
v (gallons)	500	1400	2500	3900	5000	5800

a.) Find the average rate of change on the interval [4, 16].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate v at t = 12. Is this an overestimate or an underestimate?

Average rate of change:

v(16) − v(4)16 − 4

5000 − 140016 − 4

360012

300 gallons per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

v − 1400 = 300(t − 4)

v − 1400 = 300t − 1200

v = 300t + 200

Substitute t = 12 into the secant equation.

v = 300(12) + 200

v = 3600 + 200

v = 3800

The table gives v(12) = 3900.

3800 < 3900, so this is an underestimate.

Answer: a.) 300 gallons per minute
b.) v = 300t + 200
c.) 3800 gallons; underestimate

A tank holds 6,000 liters of water. At the bottom of the tank, a valve can be opened to drain the water. The tank can be emptied in exactly 30 minutes. Below is a table which gives the volume v of water (in liters) which remain in the tank after t minutes of draining have taken place.

t (minutes)	0	5	10	15	20	25	30
v (liters)	4000	3500	2800	2200	1600	800	0

a.) Find the average rate of change on the interval [5, 25].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate v at t = 15. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

v(25) − v(5)25 − 5

800 − 350025 − 5

−270020

−135 liters per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

v − 3500 = −135(t − 5)

v − 3500 = −135t + 675

v = −135t + 4175

Substitute t = 15 into the secant equation.

v = −135(15) + 4175

v = −2025 + 4175

v = 2150

The table gives v(15) = 2200.

2150 < 2200, so this is an underestimate.

Answer: a.) −135 liters per minute
b.) v = −135t + 4175
c.) 2150 liters; underestimate

A hiking trail climbs from a trailhead. The table gives elevation h in feet after t minutes of walking.

t (minutes)	0	10	20	30	40	50
h (feet)	400	520	700	900	1060	1180

a.) Find the average rate of change on the interval [10, 40].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate h at t = 30. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

h(40) − h(10)40 − 10

1060 − 52040 − 10

54030

18 feet per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

h − 520 = 18(t − 10)

h − 520 = 18t − 180

h = 18t + 340

Substitute t = 30 into the secant equation.

h = 18(30) + 340

h = 540 + 340

h = 880

The table gives h(30) = 900.

880 < 900, so this is an underestimate.

Answer: a.) 18 feet per minute
b.) h = 18t + 340
c.) 880 feet; underestimate

A kiln is heating a batch of mugs. The table gives the temperature T in degrees Fahrenheit after t minutes.

t (minutes)	0	4	8	12	16	20
T (°F)	80	160	260	380	480	560

a.) Find the average rate of change on the interval [4, 20].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate T at t = 12. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

T(20) − T(4)20 − 4

560 − 16020 − 4

40016

25 °F per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

T − 160 = 25(t − 4)

T − 160 = 25t − 100

T = 25t + 60

Substitute t = 12 into the secant equation.

T = 25(12) + 60

T = 300 + 60

T = 360

The table gives T(12) = 380.

360 < 380, so this is an underestimate.

Answer: a.) 25 °F per minute
b.) T = 25t + 60
c.) 360 °F; underestimate

A charity jar is filling during a week-long drive. The table gives the amount s in dollars after d days.

d (days)	0	2	4	6	8	10
s (dollars)	20	80	160	260	340	400

a.) Find the average rate of change on the interval [2, 10].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate s at d = 6. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

s(10) − s(2)10 − 2

400 − 8010 − 2

3208

40 dollars per day

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

s − 80 = 40(d − 2)

s − 80 = 40d − 80

s = 40d

Substitute d = 6 into the secant equation.

s = 40(6)

s = 240

The table gives s(6) = 260.

240 < 260, so this is an underestimate.

Answer: a.) 40 dollars per day
b.) s = 40d
c.) 240 dollars; underestimate

A lawn mower’s remaining fuel g in gallons is recorded after t minutes of mowing.

t (minutes)	0	5	10	15	20	25
g (gallons)	20	18	14	10	3	0

a.) Find the average rate of change on the interval [5, 20].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate g at t = 15. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

g(20) − g(5)20 − 5

3 − 1820 − 5

−1515

−1 gallon per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

g − 18 = −(t − 5)

g − 18 = −t + 5

g = −t + 23

Substitute t = 15 into the secant equation.

g = −(15) + 23

g = −15 + 23

g = 8

The table gives g(15) = 10.

8 < 10, so this is an underestimate.

Answer: a.) −1 gallon per minute
b.) g = −t + 23
c.) 8 gallons; underestimate

Level 3

Hard

More data points, including a fractional slope. Compare the secant to the table carefully.

Example

A storage tank holds 12,000 gallons of oil. At the bottom of the tank, a valve can be opened to drain the oil. The tank can be emptied in exactly 48 minutes. Below is a table which gives the volume v of oil (in gallons) which remain in the tank after t minutes of draining have taken place.

t (minutes)	0	6	12	18	24	30	36	42	48
v (gallons)	5600	5000	4200	3600	3000	2500	2280	1200	0

a.) Find the average rate of change on the interval [12, 36].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate v at t = 24. Is this an overestimate or an underestimate?

Average rate of change:

v(36) − v(12)36 − 12

2280 − 420036 − 12

−192024

−80 gallons per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

v − 4200 = −80(t − 12)

v − 4200 = −80t + 960

v = −80t + 5160

Substitute t = 24 into the secant equation.

v = −80(24) + 5160

v = −1920 + 5160

v = 3240

The table gives v(24) = 3000.

3240 > 3000, so this is an overestimate.

Answer: a.) −80 gallons per minute
b.) v = −80t + 5160
c.) 3240 gallons; overestimate

A phone is in use on a long train ride. The table gives the remaining battery p as a percent after t minutes.

t (minutes)	0	4	8	12	16	20	24	28	32
p (percent)	100	92	80	70	58	44	32	18	4

a.) Find the average rate of change on the interval [8, 24].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate p at t = 16. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

p(24) − p(8)24 − 8

32 − 8024 − 8

−4816

−3 percent per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

p − 80 = −3(t − 8)

p − 80 = −3t + 24

p = −3t + 104

Substitute t = 16 into the secant equation.

p = −3(16) + 104

p = −48 + 104

p = 56

The table gives p(16) = 58.

56 < 58, so this is an underestimate.

Answer: a.) −3 percent per minute
b.) p = −3t + 104
c.) 56 percent; underestimate

A yeast culture is growing in a lab. The table gives the count n in thousands after t hours.

t (hours)	0	2	4	6	8	10	12
n (thousands)	100	140	200	280	400	540	700

a.) Find the average rate of change on the interval [2, 10].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate n at t = 6. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

n(10) − n(2)10 − 2

540 − 14010 − 2

4008

50 thousand cells per hour

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

n − 140 = 50(t − 2)

n − 140 = 50t − 100

n = 50t + 40

Substitute t = 6 into the secant equation.

n = 50(6) + 40

n = 300 + 40

n = 340

The table gives n(6) = 280.

340 > 280, so this is an overestimate.

Answer: a.) 50 thousand cells per hour
b.) n = 50t + 40
c.) 340 thousand cells; overestimate

A bamboo shoot’s height h in centimeters is measured every three hours t.

t (hours)	0	3	6	9	12	15	18
h (centimeters)	4	10	18	28	36	42	46

a.) Find the average rate of change on the interval [3, 15].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate h at t = 9. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

h(15) − h(3)15 − 3

42 − 1015 − 3

3212

83 centimeters per hour

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

h − 10 = 83(t − 3)

h − 10 = 83t − 8

h = 83t + 2

Substitute t = 9 into the secant equation.

h = 83(9) + 2

h = 24 + 2

h = 26

The table gives h(9) = 28.

26 < 28, so this is an underestimate.

Answer: a.) 83 centimeters per hour
b.) h = 83t + 2
c.) 26 centimeters; underestimate

A bus is driving toward a depot. The table gives the remaining distance d in miles after t minutes.

t (minutes)	0	5	10	15	20	25	30	35	40
d (miles)	78	70	60	48	36	26	16	10	0

a.) Find the average rate of change on the interval [10, 35].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate d at t = 20. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

d(35) − d(10)35 − 10

10 − 6035 − 10

−5025

−2 miles per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

d − 60 = −2(t − 10)

d − 60 = −2t + 20

d = −2t + 80

Substitute t = 20 into the secant equation.

d = −2(20) + 80

d = −40 + 80

d = 40

The table gives d(20) = 36.

40 > 36, so this is an overestimate.

Answer: a.) −2 miles per minute
b.) d = −2t + 80
c.) 40 miles; overestimate

A community pool is being filled overnight. The table gives the volume V in gallons after t minutes of filling.

t (minutes)	0	8	16	24	32	40	48
V (gallons)	200	700	1000	1400	1850	2300	2700

a.) Find the average rate of change on the interval [8, 40].

b.) Write the equation of the secant line using the information from part a.

c.) Use your answer from part b to approximate V at t = 24. Is this an overestimate or an underestimate?

Show Answer

Average rate of change:

V(40) − V(8)40 − 8

2300 − 70040 − 8

160032

50 gallons per minute

The slope of the secant is that average rate of change. Point-slope form:

y − y₁ = m(x − x₁)

V − 700 = 50(t − 8)

V − 700 = 50t − 400

V = 50t + 300

Substitute t = 24 into the secant equation.

V = 50(24) + 300

V = 1200 + 300

V = 1500

The table gives V(24) = 1400.

1500 > 1400, so this is an overestimate.

Answer: a.) 50 gallons per minute
b.) V = 50t + 300
c.) 1500 gallons; overestimate

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