Question

unit 4: polynomial functions
problem 10. the population of deer in south hadley grows according to a linear function. in
2020, there were 120 deer. in 2022, there were 186 deer. this question has three parts:

• (a) find a linear function that models the population of deer in terms of years after 2020.
• (b) how many deer will there be in 2026?
• (c) in about what year will the deer population reach 240?

Explanation:

Part (a)

Step1: Define variables

Let \( x \) be the number of years after 2020, and \( y \) be the population of deer. A linear function has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.
In 2020 (\( x = 0 \)), \( y = 120 \). So when \( x = 0 \), \( y=120 \), which means \( b = 120 \) (since substituting \( x = 0 \) into \( y=mx + b \) gives \( y=b \)).

Step2: Calculate the slope \( m \)

We know two points: when \( x_1 = 0 \) (2020), \( y_1=120 \); when \( x_2 = 2 \) (2022, which is 2 years after 2020), \( y_2 = 186 \).
The formula for the slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Substitute the values: \( m=\frac{186 - 120}{2 - 0}=\frac{66}{2}=33 \).

Step3: Write the linear function

Since \( m = 33 \) and \( b = 120 \), the linear function is \( y=33x + 120 \).

Step1: Determine the value of \( x \) for 2026

2026 is \( 2026 - 2020=6 \) years after 2020, so \( x = 6 \).

Step2: Substitute \( x = 6 \) into the linear function

We use the function \( y = 33x+120 \). Substitute \( x = 6 \):
\( y=33\times6 + 120 \)
\( y = 198+120 \)
\( y=318 \)

Step1: Set \( y = 240 \) in the linear function

We have the function \( y = 33x + 120 \). Set \( y = 240 \), so \( 240=33x + 120 \).

Step2: Solve for \( x \)

Subtract 120 from both sides: \( 240 - 120=33x \)
\( 120 = 33x \)
Then \( x=\frac{120}{33}=\frac{40}{11}\approx3.64 \)

Step3: Find the year

Since \( x \) is the number of years after 2020, the year is \( 2020+\lfloor3.64
floor + 1\) (we round up because the population reaches 240 during the year). \( 2020 + 4=2024 \) (or we can calculate \( 2020+x\approx2020 + 3.64\approx2023.64 \), so approximately 2024).

Answer:

\( y = 33x+120 \)

Part (b)