Questions: If f(x) is a linear function, f(-3)=-4, and f(3)=-5, find an equation for f(x) f(x)=

Transcript text: If $f(x)$ is a linear function, $f(-3)=-4$, and $f(3)=-5$, find an equation for $f(x)$ \[ f(x)= \] $\square$ Question Help: Video Submit Question

Solution

Solution Steps

Step 1: Determine the slope of the linear function

A linear function can be expressed in the form: \[ f(x) = mx + b \] where $ m $ is the slope and $ b $ is the y-intercept.

Given two points on the line, $ (-3, -4) $ and $ (3, -5) $, the slope $ m $ can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{-5 - (-4)}{3 - (-3)} = \frac{-1}{6} = -\frac{1}{6} \]

Step 2: Find the y-intercept $ b $

Using the slope $ m = -\frac{1}{6} $ and one of the points, say $ (-3, -4) $, substitute into the linear equation to solve for $ b $: \[ -4 = -\frac{1}{6}(-3) + b \] Simplify: \[ -4 = \frac{1}{2} + b \] Subtract $ \frac{1}{2} $ from both sides: \[ b = -4 - \frac{1}{2} = -\frac{9}{2} \]

Step 3: Write the equation of the linear function

Substitute $ m = -\frac{1}{6} $ and $ b = -\frac{9}{2} $ into the linear equation: \[ f(x) = -\frac{1}{6}x - \frac{9}{2} \]