Question

If f(x) is a linear function, f(-2)=-4, and f(4)=-5, find an equation for f(x).
f(x) =

Answer 1

To find the equation of the linear function \( f(x) \), we need to determine the slope \( m \) and the y-intercept \( b \) of the function. We know two points on the line: \( (-2, -4) \) and \( (4, -5) \).

1. **Calculate the slope \( m \)** using the formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \( (x_1, y_1) = (-2, -4) \) and \( (x_2, y_2) = (4, -5) \):
\[
m = \frac{-5 - (-4)}{4 - (-2)} = \frac{-5 + 4}{4 + 2} = \frac{-1}{6}
\]

2. **Use the slope-intercept form** of the linear equation \( f(x) = mx + b \) to find \( b \). We can use one of the points to solve for intercept \( b \). Let's use the point \( (-2, -4) \):
\[
-4 = \left(-\frac{1}{6}\right)(-2) + b
\]
\[
-4 = \frac{2}{6} + b
\]
\[
-4 = \frac{1}{3} + b
\]
Now, isolate \( b \):
\[
b = -4 - \frac{1}{3} = -\frac{12}{3} - \frac{1}{3} = -\frac{13}{3}
\]

3. **Write the linear function** with the found slope and intercept:
\[
f(x) = -\frac{1}{6}x - \frac{13}{3}
\]

Thus, the equation for \( f(x) \) is:

\[
\boxed{f(x) = -\frac{1}{6}x - \frac{13}{3}}
\]