Question

2a) the table shows the distance penelope is from the park as she walks to soccer practice. assume the relationship between the two quantities is linear.

time (min), x	distance (m), y
10	1,380
15	830
20	280

find and interpret the rate of change.

the rate of change is ____________, so penelope is ____________ meters closer to the park every minute.

find and interpret the initial value.

the initial value is ____________, so penelope was initially ____________ meters from the park.

Explanation:

Step1: Calculate rate of change

The rate of change (slope) for a linear relationship is given by $\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points, say $(5, 1930)$ and $(10, 1380)$.
$$\frac{1380 - 1930}{10 - 5} = \frac{-550}{5} = -110$$

Step2: Interpret rate of change

A negative rate of change of -110 means Penelope's distance from the park decreases by 110 meters per minute, so she is 110 meters closer every minute.

Step3: Find the linear equation

The linear equation is $y = mx + b$, where $m = -110$. Using the point $(5, 1930)$:
$$1930 = -110(5) + b$$
$$1930 = -550 + b$$
$$b = 1930 + 550 = 2480$$

Step4: Interpret initial value

The initial value is the y - intercept ($b$), which is 2480. This means initially (at $x = 0$ minutes), Penelope was 2480 meters from the park.

Answer:

The rate of change is $\boldsymbol{-110}$, so Penelope is $\boldsymbol{110}$ meters closer to the park every minute.
The initial value is $\boldsymbol{2480}$, so Penelope was initially $\boldsymbol{2480}$ meters from the park.