To solve the problem, we need to find the equation of the linear function $f(x)$ given the points $(-3, 4)$ and $(2, -4)$.

Step 1: General form of a linear function

The equation of a linear function is given by: $f(x) = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

Step 2: Calculate the slope ($m$)

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting $(x_1, y_1) = (-3, 4)$ and $(x_2, y_2) = (2, -4)$: $m = \frac{-4 - 4}{2 - (-3)} = \frac{-8}{5}$ Thus, $m = -\frac{8}{5}$.

Step 3: Substitute into the linear equation

The equation now is: $f(x) = -\frac{8}{5}x + b$

Step 4: Solve for $b$

We substitute one of the given points into the equation to find $b$. Using $(-3, 4)$: $4 = -\frac{8}{5}(-3) + b$ Simplify: $4 = \frac{24}{5} + b$ Subtract $\frac{24}{5}$ from both sides: $b = 4 - \frac{24}{5} = \frac{20}{5} - \frac{24}{5} = -\frac{4}{5}$

Step 5: Final equation

The equation for $f(x)$ is: $f(x) = -\frac{8}{5}x - \frac{4}{5}$

Let me know if you want additional clarifications or have further questions!

Related Questions:

1. What is the slope-intercept form of a linear equation and how does it relate to this problem?
2. How can you verify that this equation satisfies both given points?
3. How would the equation change if $f(x)$ were quadratic instead of linear?
4. Can you rewrite the equation in standard form $Ax + By = C$?
5. How do changes in the slope affect the steepness of a line?

Tip: Always verify your equation by plugging in the given points to ensure the result is correct.