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2A. Exponential and Logarithmic Functions
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AP Precalculus

2.2A Change in linear and exponential functions

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Lesson

Introduction

Learning objectives

  • 2.2.A Describe how the input and output values of a function vary together by comparing function values.
  • 2.2.B Describe similarities and differences between linear and exponential functions.

Linear functions and arithmetic sequences

Recall the slope-intercept form of a linear function:

\displaystyle f\left(x\right)=b + mx
\bm{f\left(x\right)}
output
\bm{b}
inital value (y-intercept)
\bm{m}
rate of change (slope)
\bm{x}
input

Similarly, an arithmetic sequence is represented in explicit notation by the formula:

\displaystyle a_n=a_{0}+dn
\bm{n}
term number
\bm{a_n}
nth term
\bm{a_{0}}
initial (first) term
\bm{d}
common difference

The domain of any arithmetic sequence is a subset of the integers. The domain can begin from any non-negative integer but will most often begin at 0 or 1.

We can also utilize point-slope form to help us write arithmetic sequences.

\displaystyle f\left(x\right)=y_i+m\left(x-x_i\right)
\bm{\left(x_i,y_i\right)}
a known point
\bm{m}
slope

For an arithmetic sequence where a term and the common difference are known we can use a similar formula:

\displaystyle a_n=a_k+d\left(n-k\right)
\bm{a_n}
nth term
\bm{a_k}
a known (kth) term
\bm{d}
common difference
\bm{n}
term number (of the nth term)
\bm{k}
term number of the known (kth) term
-4
-3
-2
-1
1
2
3
4
x
-8
-6
-4
-2
2
4
6
8
y
a(n)=3+2(n-1)
-4
-3
-2
-1
1
2
3
4
x
-8
-6
-4
-2
2
4
6
8
y
f(x)=2x+1

Arithmetic sequences are linear functions because both have a constant rate of change (slope or common difference) and an inital value (the first term or y-intercept).

Examples

Example 1

Mandy has already folded 4 shirts before her shift officially begins at the clothing store. Additional information about the number of shirts she has folded at various times during her shift is given in the table.

Time (minutes)015304560
Shirts folded419344964
a

State whether Mandy's number of shirts folded represent an arithmetic sequence.

Worked Solution
Create a strategy

An arithmetic sequence will have a constant rate of change. We can compare the values in the table and see how much the total ticket sales are increasing by each hour.

Apply the idea

Mandy folded 4 shirts before her shift. She then folds 19 - 4 = 15 shirts after 15 minutes, \\34 - 19 = 15 shirts after 30 minutes, 49 - 34 = 15 shirts after 45 minutes, and 64 - 49 = 15 shirts after 60 minutes.

The rate of change is constant and therefore, the ticket sales represent an arithmetic sequence.

b

Determine the linear function that represents the number of shirts Mandy has folded, S(t), as a function of time t (in minutes).

Worked Solution
Create a strategy

We can use the slope-intercept form of a linear equation: S(t) = mt + b, where m is the slope (rate of change) and b is the y-intercept (the number of shirts folded at the start).

Apply the idea

From the table, we can see that when t = 15,\, S(t) = 19. Therefore, we can calculate the slope, m:

\displaystyle m\displaystyle =\displaystyle \dfrac{(19-4)}{(15-0)}Substitute S(t)=19,\,S(0)=4 and t=15
\displaystyle =\displaystyle \dfrac{15}{15}Evaluate the difference
\displaystyle =\displaystyle 1Evaluate

So, the slope (m) is 1. This means that Mandy folds 1 shirt per minute. The y-intercept (b) is the number of shirts folded at the start (t = 0), which is 4. Now we can write the linear function S(t):

S(t) = 1t + 4

c

Calculate the number of shirts Mandy will have folded after 90 minutes.

Worked Solution
Create a strategy

Substitute t=90 into the linear function from part (b).

Apply the idea
\displaystyle S(t)\displaystyle =\displaystyle 1t + 4Write the linear function
\displaystyle =\displaystyle (1 \times 90) + 4Substitute t=90
\displaystyle =\displaystyle 90 + 4Evaluate the product
\displaystyle =\displaystyle 94 \text{ shirts}Evaluate
d

Mandy's shirt folding business is growing. In 2023, she folded approximately 2,000 shirts each month. Between 2023 and 2028, the number of shirts she folded increased at approximately 3.5\% per year. According to this estimate, how many shirts did Mandy fold per month in 2025?

Worked Solution
Create a strategy

We can use the formula: \text{Final Amount} = \text{Initial Amount} \times (1 + \text{Growth Rate})^\text{Number of Years}.

Apply the idea

In this case, the initial amount (number of shirts folded in 2023) is 2000. The growth rate is 3.5\% per year, which we need to convert to a decimal: 3.5\% = 0.035. The number of years between 2023 and 2025 is 2.

\displaystyle \text{Final amount}\displaystyle =\displaystyle 2000 \times (1 + 0.035)^2Substitute the values
\displaystyle =\displaystyle 2000 \times (1.035)^2Evaluate the sum
\displaystyle \approx\displaystyle 2142.45Evaluate

So, according to the estimate, Mandy folded approximately 2142 shirts per month in 2025.

e

What is the ratio of shirt folding from 2025 compared to 2023?

Worked Solution
Create a strategy

Divide the number of shirts in 2025 by the number of shirts in 2023.

Apply the idea
\displaystyle \text{Ratio}\displaystyle =\displaystyle \dfrac{2142}{\text{2000}}Substitute the number of shirts in 2025 and 2023
\displaystyle \approx\displaystyle 1.071Evaluate

The ratio of shirt folding from 2025 compared to 2023 is approximately 1.071.

Idea summary

An arithmetic sequence is a linear function because it has a constant rate of change.

When the first term and common difference are known:

\displaystyle a_n=a_{0}+dn
\bm{n}
term number
\bm{a_n}
nth term
\bm{a_{0}}
initial (first) term
\bm{d}
common difference

When any term and the common difference are known:

\displaystyle a_n=a_k+d\left(n-k\right)
\bm{a_n}
nth term
\bm{a_k}
a known (kth) term
\bm{d}
common difference
\bm{n}
term number (of the nth term)
\bm{k}
term number of the known (kth) term

Outcomes

2.2.A

Construct functions of the real numbers that are comparable to arithmetic and geometric sequences.

2.2.B

Describe similarities and differences between linear and exponential functions.

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