Worksheet: Rate of Change

In this worksheet, we will practice finding the rate of change using tables and graphs and solving real-world problems using rate of change.

Q1:

What is the rate of change for the function shown in the given graph?

Q2:

What is the rate of change shown by this graph of a function?

  • A 2 5
  • B βˆ’ 2 5
  • C 1 3
  • D 5 2
  • E βˆ’ 5 2

Q3:

What is the average speed of a sound wave that travels 895 meters in 2.5 seconds?

Q4:

What is the initial value and the rate of change for the function represented by the given table?

π‘₯ -value βˆ’ 1 1 2
𝑦 -value 7 6 βˆ’ 1 6 βˆ’ 5 6
  • Ainitial value = βˆ’ 2 3 , rate of change = 1 2
  • Binitial value = 1 2 , rate of change = βˆ’ 2 3
  • Cinitial value = βˆ’ 1 2 , rate of change = βˆ’ 2 3
  • Dinitial value = 2 , rate of change = βˆ’ 2 3
  • Einitial value = 1 2 , rate of change = 2 3

Q5:

The graph below shows the relation between the cost of a party and the number of people attending. Determine the rate of change.

  • A 7 5 4
  • B 1 7 5 8
  • C25
  • D 4 7 5
  • E 2 7 5 1 2

Q6:

Daniel opened a savings account with $145. Every month, he made the same deposit and made no withdrawals. After 3 months, he had $214. After 6 months, he had $352. After 9 months, he had $559. Determine the rate of change in his savings account.

Q7:

At 5:00, the water level in a pool reaches a height of 10 inches. At 8:00, the water level is at 50 inches. What is the rate of change of the water level per minute?

  • A 3 4 0 of an inch per minute
  • B 5 9 of an inch per minute
  • C 4 0 3 of an inch per minute
  • D 9 2 of an inch per minute
  • E 2 9 of an inch per minute

Q8:

What is the rate of change for the function represented in the following table?

π‘₯ 0 2 3 4 5
𝑦 1 5 7 9 11

Q9:

What is the initial value and the rate of change for the function represented by the given graph?

  • Ainitial value = 1 8 , rate of change = βˆ’ 6
  • Binitial value = βˆ’ 6 , rate of change = 8
  • Cinitial value = 8 , rate of change = βˆ’ 1 6
  • Dinitial value = 8 , rate of change = βˆ’ 6
  • Einitial value = 8 , rate of change = 6

Q10:

This graph illustrates the decay of a radioactive substance over 𝑑 days. Which of the following is the best estimate of the average decay rate from 𝑑 = 5 to 𝑑 = 1 5 ?

  • A βˆ’ 0 . 6 milligrams/day
  • B βˆ’ 1 . 8 milligrams/day
  • C βˆ’ 0 . 4 milligrams/day
  • D βˆ’ 0 . 3 milligrams/day
  • E βˆ’ 0 . 9 milligrams/day

Q11:

If a worker who can paint a house in 300 hours and another who can paint the same house in 100 hours work together, how many minutes do they need to paint the house?

Q12:

A city’s population in the year 1960 was 2 8 7 5 0 0 and 2 7 5 9 0 0 in 1989. Find the average rate of the population growth, in terms of people per year, and explain what it tells us about how the population changes each year.

  • AThe rate is 142. This means that for every year between 1960 and 1989, the population in the city rose by 142 people on average.
  • BThe rate is βˆ’ 2 8 5 . This means that for every year between 1960 and 1989, the population in the city dropped by 285 people on average.
  • CThe rate is 285. This means that for every year between 1960 and 1989, the population in the city rose by 285 people on average.
  • DThe rate is 400. This means that for every year between 1960 and 1989, the population in the city rose by 400 people on average.
  • EThe rate is βˆ’ 4 0 0 . This means that for every year between 1960 and 1989, the population in the city dropped by 400 people on average.

Q13:

The following graph is a plot of the height in meters of a kite over a 4-minute period.

What was the greatest speed, in meters per minute, at which the kite was moving? Was it descending or ascending?

  • A 2 m/minute, descending
  • B 4 m/minute, ascending
  • C 4 m/minute, descending
  • D 3 m/minute, ascending
  • E 3 m/minute, descending

Q14:

A sphere of mass 2.4 kg was moving through a dusty atmosphere. The dust accumulated on its surface at a rate of 125 g/min. How long will it take until the sphere’s mass becomes 4 kg?

Q15:

Tap A takes 240 minutes to fill an aquarium. Tap B takes 80 minutes to fill the same aquarium. Tap C only takes 20 minutes to fill the aquarium. If all three taps were used together, how many seconds would it take to fill the aquarium?

Q16:

A worker painted a wall with an area of 42 m2 in 3 hours. What is his average work rate in m2/h, and how many square meters of wall would he be able to paint at that rate in 2 hours?

  • A 39 m2/h, 546 m2
  • B 14 m2/h, 16 m2
  • C 39 m2/h, 53 m2
  • D 14 m2/h, 28 m2

Q17:

If Scarlett mows her yard in 2 hours and her sister Amelia can mow the yard in 3 hours, then how long will it take them working together to mow the yard?

Q18:

Consider the function β„Ž = { ( 0 , 4 ) , ( 1 , 1 ) , ( 2 , 3 ) , ( 3 , 2 ) , ( 4 , 3 ) } .

Over which interval does the average rate of change of β„Ž have greatest magnitude?

  • A 0 ≀ π‘₯ ≀ 3
  • B 1 ≀ π‘₯ ≀ 2
  • C 3 ≀ π‘₯ ≀ 4
  • D 2 ≀ π‘₯ ≀ 3
  • E 0 ≀ π‘₯ ≀ 1

Over which interval is the average rate of change of β„Ž least?

  • A 0 ≀ π‘₯ ≀ 3
  • B 3 ≀ π‘₯ ≀ 4
  • C 0 ≀ π‘₯ ≀ 1
  • D 2 ≀ π‘₯ ≀ 3
  • E 1 ≀ π‘₯ ≀ 2

Q19:

Consider the function β„Ž = { ( 0 , 4 ) , ( 1 , 1 ) , ( 2 , 3 ) , ( 3 , 2 ) , ( 4 , 3 ) } .

Over which interval does the average rate of change of β„Ž have greatest magnitude?

  • A 1 ≀ π‘₯ ≀ 2
  • B 0 ≀ π‘₯ ≀ 1
  • C 0 ≀ π‘₯ ≀ 3
  • D 3 ≀ π‘₯ ≀ 4
  • E 2 ≀ π‘₯ ≀ 3

Over which interval is the average rate of change of β„Ž greatest?

  • A 0 ≀ π‘₯ ≀ 1
  • B 3 ≀ π‘₯ ≀ 4
  • C 0 ≀ π‘₯ ≀ 3
  • D 2 ≀ π‘₯ ≀ 3
  • E 1 ≀ π‘₯ ≀ 2

Q20:

A phone company charges for service according to the formula: 𝐢 ( 𝑛 ) = 2 4 + 0 . 1 𝑛 , where 𝑛 is the number of minutes used and 𝐢 ( 𝑛 ) is the monthly charge in dollars. Find and interpret the rate of change and the initial value.

  • AThe rate of change is 0.1. For every additional minute used, the monthly charge increases by 0.10 or 24.10 cents. The initial value is 24.1. When there are no minutes used, initially the charge is 24.1.
  • BThe rate of change is 24. For every additional minute used, the monthly charge increases by 24 or 0.01 cents. The initial value is 0.1. When there are no minutes used, initially the charge is 0.1.
  • CThe rate of change is 24. For every additional minute used, the monthly charge increases by 24 or 2.40 cents. The initial value is 2.4. When there are no minutes used, initially the charge is 2.4.
  • DThe rate of change is 2.4. For every additional minute used, the monthly charge increases by 2.40 or 24 cents. The initial value is 24. When there are no minutes used, initially the charge is 24.
  • EThe rate of change is 0.1. For every additional minute used, the monthly charge increases by 0.10 or 24 cents. The initial value is 24. When there are no minutes used, initially the charge is 24.

Q21:

As a raindrop was falling, water vapor condensed on its surface increasing its mass at a rate of 4 milligrams per second. Given that at a certain moment its mass was 0.9 g, find its mass 2 minutes later.

  • A 0.42 g
  • B 1.38 g
  • C 0.66 g
  • D 0.91 g

Q22:

Anthony is depositing money into a bank account. After 3 months, there is $147 in the account, and after 7 months, there is $343 in the account. Find the rate of change of the account.

Q23:

Determine the average rate of change of the function 𝑓 ( π‘₯ ) = 9 π‘₯ + 7  when π‘₯ = π‘₯  .

  • A 9 β„Ž + 1 8 π‘₯ 
  • B 9 β„Ž + 1 8 π‘₯ β„Ž  
  • C 1 8 π‘₯ 
  • D 9 π‘₯ + 1 8 π‘₯ β„Ž + 9 β„Ž + 7    

Q24:

The relation between the cost of wrapping paper and the number of rolls bought is shown in the table. Is the relationship between the two quantities linear? If so, find the constant rate of change.

Number of Rolls 1 2 3 4
Total Cost ($) 3 7 9 12
  • A yes, 4
  • B yes, βˆ’ 4
  • C yes, 3
  • Dno
  • E yes, βˆ’ 3

Q25:

Given the exponential function 𝑦 = 4 ( 1 . 2 1 )   , what is the percentage rate of change? Give your answer to the nearest percent.

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