Let's Break it Down
In a previous lesson, you learned that a linear function always has a rate of change (slope) and an initial value.
Rate of Change
Rate of change \( = m = \frac{\textsf{ Change in Output}(y)}{\textsf{ Change in Input}(x)} = \frac{y_{2} - y_{1}}{x_{2}{- x}_{1}} \)
The initial value is the same as the \( y \)-intercept.
You can use a linear function to model many different types of scenarios that involve a constant rate of change and an initial value. Click each tab to see an example.
In a shoe store, the area of one shoe box is 50 in2, the area of two shoe boxes is 66 in2, and the area of three shoe boxes is 78 in2.
Can you model the area of the shoe boxes using a linear function? Explain.
No, because the rate of change is not constant.
Remember, for a real-world situation to be modeled using a linear function, it must have a constant rate of change and an initial value.
Calculate the rate of change between each set of shoe boxes.
The first rate of change is:
\( m = \frac{66 - 50\textsf{ in}^{2}}{2 - 1\textsf{ boxes}} \)
\( m = \frac{16\textsf{ in}^{2}}{1 \textsf{ box}} \)
The second rate of change is:
\( m = \frac{78 - 66\textsf{ in}^{2}}{3 - 2\textsf{ boxes}} \)
\( m = \frac{12\textsf{ in}^{2}}{1 \textsf{ box}} \)
The rate of change is not constant since \( 16 \neq 12 \).
The table below represents the number of batches of cupcakes a baker can make in relation to time.
| Time, in hours |
Cupcake Batches Baked |
|---|---|
0 |
0 |
2 |
3 |
6 |
9 |
10 |
15 |
14 |
21 |
Can you model this information using a linear function? Explain.
Yes, because the rate of change is constant. The initial value is 0 cupcakes.
Remember, for a real-world situation to be modeled using a linear function, it must have a constant rate of change and an initial value.
Calculate the rate of change in batches of cupcakes per hour.
The first rate of change is:
\( m = \frac{\left( 3 \right) - (0)}{\left( 2 \right) - (0)} = \frac{3\textsf{ batches}}{2 \textsf{ hours}} \)
The second rate of change is:
\( m = \frac{\left( 9 \right) - \left( 3 \right)}{\left( 6 \right) - \left( 2 \right)} = \frac{6}{4} = \frac{3\textsf{ batches}}{2 \textsf{ hours}} \)
The third rate of change is:
\( m = \frac{\left( 15 \right) - (9)}{\left( 10 \right) - (6)} = \frac{6}{4} = \frac{3\textsf{ batches}}{2 \textsf{ hours}} \)
The final rate of change is:
\( m = \frac{\left( 21 \right) - (15)}{\left( 14 \right) - (10)} = \frac{6}{4} = \frac{3\textsf{ batches}}{2 \textsf{ hours}} \)
All the rates of change are identical, so you can use a linear function.
Question
The cost to rent a carpet cleaner is $45 and an additional $5 for every hour. Can you model this information using a linear function? Explain.
Yes. For a real-world situation to be modeled using a linear function, it must have a constant rate of change and an initial value.
The rate of change is 5 (or $5 per hour), and the initial value is 45 (or $45).
