Question

Find the equation that models the linear function given in the value table.

x
𝑥
f(x)
−2
−
2
6
−1
−
1
4.5
0 3
1 1.5
2 0


f(x)=
𝑓
(
𝑥
)
=

Answer 1

To find the equation that models the linear function given the table of values, we need to determine the slope (m) and the y-intercept (b) of the linear function in the form \( f(x) = mx + b \).

Let's start by finding the slope (m) using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

We can choose two points from the table to calculate the slope. Let's use the points (-2, 6) and (2, 0).

1. Calculate the change in y (Δy) and the change in x (Δx):
- \( y_1 = 6 \) at \( x_1 = -2 \)
- \( y_2 = 0 \) at \( x_2 = 2 \)

\[
\Delta y = y_2 - y_1 = 0 - 6 = -6
\]
\[
\Delta x = x_2 - x_1 = 2 - (-2) = 2 + 2 = 4
\]

Now calculate the slope (m):

\[
m = \frac{-6}{4} = -\frac{3}{2}
\]

Now, we can use one of the points and the slope to find the y-intercept (b). We can use the point (0, 3), which is one of the points in your table.

Using the slope-intercept form of a linear equation \( f(x) = mx + b \):

When \( x = 0 \), \( f(0) = b \).

So, from the point (0, 3):

\[
f(0) = 3 \implies b = 3
\]

Now we have both the slope and y-intercept:

\[
m = -\frac{3}{2}, \quad b = 3
\]

Thus, the equation that models the linear function is:

\[
f(x) = -\frac{3}{2}x + 3
\]

So, the final answer is:

\[
f(x) = -\frac{3}{2}x + 3
\]