Worksheet: Slope and Rate of Change

In this worksheet, we will practice interpreting the slope of a straight line as the rate of change of two quantities.

Q1:

Which of the following functions has a rate of change that is more rapid than the function shown in the graph?

  • A 𝑦 = 1 4 π‘₯ βˆ’ 5
  • B 𝑦 = 2 7 π‘₯ βˆ’ 5
  • C 𝑦 = 1 3 π‘₯ βˆ’ 5
  • D 𝑦 = 2 3 π‘₯ βˆ’ 5
  • E 𝑦 = βˆ’ π‘₯ βˆ’ 5

Q2:

Determine which of the given functions has the greatest rate of change.

(a)

π‘₯ βˆ’ 2 βˆ’ 1 0 1 2
𝑦 βˆ’ 2 2 6 10 14

(b) a function where the input is multiplied by 3 and subtracted from 4

(c)

(d) 𝑦=15π‘₯+8

  • Ac
  • BThey have the same rate of change.
  • Ca
  • Db
  • Ed

Q3:

At an amusement park, a booth was selling soft drinks. By 2:00 pm, 37 soft drinks were sold, and, by 4:00 pm, 69 were sold. Assuming that the booth was selling at a linear rate during this period, determine the slope of the line representing this information.

Q4:

Find the rate of change for the following graph.

Q5:

Jennifer and Hannah are playing a video game. The given functions represent the number of levels each of them has finished. Who plays better?

Hannah’s Progress
Time (min) Number of Levels
4 3
12 9
20 15
24 18
  • AJennifer
  • BHannah

Q6:

Which of the following functions has a lower rate of change?

(a)

π‘₯ 9 3 βˆ’ 3 βˆ’ 9
𝑦 2 βˆ’ 2 βˆ’ 6 βˆ’ 1 0

(b) A function with an output that is equal to the input divided by 3 and added to 5.

  • AThey have the same rate of change.
  • B(b)
  • C(a)

Q7:

Find the slopes of ⃖⃗𝐴𝐡, ⃖⃗𝐡𝐢, and ⃖⃗𝐢𝐷.

  • AThe slope of ⃖⃗𝐴𝐡 is 10, the slope of ⃖⃗𝐡𝐢 is 0, and the slope of ⃖⃗𝐢𝐷 is βˆ’10.
  • BThe slope of ⃖⃗𝐴𝐡 is 110, the slope of ⃖⃗𝐡𝐢 is 0, and the slope of ⃖⃗𝐢𝐷 is βˆ’110.
  • CThe slope of ⃖⃗𝐴𝐡 is 10, the slope of ⃖⃗𝐡𝐢 is undefined, and the slope of ⃖⃗𝐢𝐷 is βˆ’10.
  • DThe slope of ⃖⃗𝐴𝐡 is 10, the slope of ⃖⃗𝐡𝐢 is 1, and the slope of ⃖⃗𝐢𝐷 is βˆ’10.
  • EThe slope of ⃖⃗𝐴𝐡 is 110, the slope of ⃖⃗𝐡𝐢 is 1, and the slope of ⃖⃗𝐢𝐷 is βˆ’110.

Q8:

Which of the following best describes the slope of the line on the diagram?

  • Apositive
  • B0
  • Cnegative
  • Dundefined

Q9:

The graph shows the distance covered in kilometers over a 15-hour hike. What were the fastest and slowest speeds recorded?

  • Afastest speed: βˆ’6 km/h, slowest speed: βˆ’13 km/h
  • Bfastest speed: 6 km/h, slowest speed: βˆ’13 km/h
  • Cfastest speed: 3 km/h, slowest speed: 13 km/h
  • Dfastest speed: 6 km/h, slowest speed: 13 km/h
  • Efastest speed: 3 km/h, slowest speed: 16 km/h

Q10:

Use the graph to find out how many action figures each boy has.

Q11:

The scores of a class of students in their end-of-year math and physics examinations can be described by the model 𝑦=31.8+0.43π‘₯, where π‘₯ is the score in math and 𝑦 is the score in physics. What is the interpretation of the value of 0.43 in the model?

  • AFor every additional point scored in the math examination, students’ scores increase by, on average, 31.8 points in their physics examination.
  • BFor every additional point scored in the physics examination, students’ scores increase by, on average, 0.43 additional points in their math examination.
  • CFor every additional point scored in the physics examination, students’ scores decrease by, on average, 0.43 points in their math examination.
  • DFor every additional point scored in the math examination, students’ scores increase by, on average, 0.43 additional points in their physics examination.
  • EFor every additional point scored in the math examination, students’ scores decrease by, on average, 0.43 points in their physics examination.

Q12:

Let 𝑓(π‘₯)=2(3βˆ’π‘₯)(π‘₯+1) and 𝑔 be the quadratic function given by the following table.

π‘₯ βˆ’2 βˆ’1 0 1 2 3
𝑔 ( π‘₯ ) 0 3 0 βˆ’9 βˆ’24 βˆ’45

Which of the following statements is true?

  • AOn the interval 0<π‘₯<3, function 𝑓 has the lower average rate of change.
  • BOn the interval 0≀π‘₯≀3, function 𝑓 has the higher average rate of change.
  • CThe minimum value of 𝑓 is greater than the maximum value of 𝑔.
  • DBoth functions have the same axis of symmetry.
  • EThe sums of zeros of the two functions are the same.

Q13:

The table lists the distance covered in kilometers over a 15-hour hike. During which periods was the average speed below 2 kilometers per hour?

Hour 3 5 9 15
Distance 18 24 28 30
  • Abetween hour 5 and hour 15
  • Bbetween hour 3 and hour 5

Q14:

Using the information in the given table, determine the rate of change in the number of seeds for every orange, and express your answer as a fraction.

Number of Oranges 3 5 7
Number of Seeds 54 90 126
  • A 1 2 1
  • B 3 6 5
  • C 1 1 2
  • D 1 1 8
  • E 1 8 1

Q15:

Which of the following interprets the rate of change of the data in the table?

Remaining Gas (gal) 19 14 9 4
Miles Driven 0 100 200 300
  • AA decrease of 19 gallons per 50 miles
  • BA decrease of 19 gallons per 100 miles
  • CA decrease of 5 gallons per 100 miles
  • DA decrease of 5 gallons per 50 miles

Q16:

A video-on-demand service offers different streaming plans as shown in the given table. Determine, to the nearest cent, the rate of change in cost per minute of video streaming.

Cost $42 $58 $74 $90 $106
Minutes of Video Streaming 5,000 5,500 6,000 6,500 7,000

Q17:

The table shows Sophia’s height at ages 10 and 15. Find the rate of change, in inches per year, of her height between these ages.

Height (in) 53 67
Age (yr) 10 15
  • A 5 1 4 inches per year
  • B 2 4 5 inches per year
  • C 5 3 1 0 inches per year
  • D 1 5 6 7 inches per year
  • E 1 0 5 3 inches per year

Q18:

Using the given table, determine the rate of change in temperature.

Time (in hours) 1:00 PM 2:00 PM 3:00 PM 4:00 PM
Temperature (in degrees) 55 59 63 67

Q19:

The table shows the value of a company’s stock over a 5-day period. Determine the rate of change (dollars per day) between day 1 and day 3.

Value ($) 40.23 45.55 49.87 41.21 45.62
Day 1 2 3 4 5

Q20:

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

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

Q21:

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
  • Ayes, 3
  • Byes, βˆ’4
  • Cyes, βˆ’3
  • Dno
  • Eyes, 4

Q22:

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=15?

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

Q23:

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

Q24:

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?

Q25:

Given the exponential function 𝑦=4(1.21)οŠ±ο—, what is the percentage rate of change? Give your answer to the nearest percent.

  • A 1 7 %
  • B βˆ’ 1 7 %
  • C βˆ’ 2 1 %
  • D 8 4 %
  • E 2 1 %

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