Section 4.3: Average Rate of Change
Honors Analysis
Q Problems
Q1.) Explain why 12 and 20 are congruent mod 8.
6𝑥
Q2.) Simplify: 4𝑥 2 −8𝑥
3 −2
Q3.) Simplify: (2)
Q4.) What is the maximum possible area for a quadrilateral with a perimeter of 56 cm?
Q5.) Write the decimal .636363… as a simplified fraction.
Q6.) The pattern AABBBCCCCAABBBCCCC… continually repeats. What is the 369th letter?
Q7.) An arithmetic sequence is a set of numbers with an adding (or subtracting) pattern. For example, 3, 7, 11, 15… is an
arithmetic sequence with a common difference of 4. What is the 101st term of the sequence?
Q8.) Solve for x: 22𝑥 = 812
Q9.) How many sides does a regular polygon have if its interior angles measure 160 degrees each?
Q10.) Evaluate: ⟦9.8⟧
1. The function ℎ(𝑡) = 10√𝑡 gives the height in feet, h, of a hot air balloon after t seconds.
A) Describe how the height of the balloon changes over time (a graph may be helpful).
B) What is the balloon’s average rate of change on the interval [0, 10] seconds? You may round to the nearest tenth.
Make sure to include appropriate units.
C) What was its average rate of change on the interval [90, 100]? How does this relate to your answer to part A?
2. The table below shows population data for the United States from 1950-1990:
A) To the nearest tenth of a million, what is the average rate of change of the population from 1970-1980? 1980-1990?
What does this tell you? Be sure to include correct units.
B) Use your answer from part A to predict the population of the nation in 1972.
3. The table below shows the value of a car at different ages:
A) Draw a scatter plot of the data.
B) What is the average rate of change of the value on the interval [0, 1]? Include units.
C) What is the average rate of change of value on the interval [2, 5]?
D) Determine the average rate of change for [9, 10].
E) What do you notice about the rate of change of the value?
4 A) Sketch a graph of the function 𝑦 = 𝑥 3 − 9𝑥. Set your graphing calculator to an x interval of [-10, 10] and a y
interval of [-20, 20].
B) To the nearest thousandth, determine the average rate of change of the function on the interval [1.732, 5.732]. What
is the average value on the interval [-1.732, 1.732]?
C) Explain how the results of part B relate to the graph from part A.
D) Would the average rate of change for the interval [-4, -2] be positive or negative? How can you tell from the graph?
E) Calculate the average rate of change for [1.632, 1.832]. Based on the graph, why do you think this result occurs?
5. Organizers of a rock concert have developed the function 𝑃(𝑥) = −50𝑥 2 + 4000𝑥 − 7500, where x represents the
price charged for a ticket in dollars and P represents the total income generated by ticket sales.
A) To the nearest tenth, calculate the average rate of change of income for the interval [10, 20]. Include units. What
does this tell you?
B) Now calculate the average rate of change for income on the interval [35, 40].
C) What is the rate of change for the interval [40, 45]? What does this tell you?
D) Graph the function (use a window with 0 ≤ 𝑥 ≤ 80, 0 ≤ 𝑦 ≤ 80,000). Explain how the graph relates to your earlier
calculations.
6.) The table below shows the velocity of Ben’s car over time after a stop light changes to green.
Time (s)
0
3
6
9
Velocity (mi/hr)
0
15
27
36
12
42
A) Calculate the average rate of change for each 3 second interval. What are the units?
B) In part A you calculated the rate of change of velocity. What is this quantity called?
7. World population can be modeled approximately with the function 𝑃(𝑡) = 6(1.02)𝑡 , where t is the number of years
since 2000 (so t = 0 for 2000) and P is the world population in billions.
A) What is the predicted population for the year 2020?
B) What is the average rate of change of the population for the years in the interval [2000, 2010]? [2010, 2020]?
Include correct units.
C) What is the average rate of change of the population for the years in the interval [2090, 2100]? What is the predicted
population?
D) Describe what you have observed about future population growth. Are there reasons for concern?