The problem asks you to find a linear function $f(x)$ that satisfies two given conditions: $f(1) = 1$ and $f(3) = -7$. It also requests that you provide the equation in slope-intercept form (i.e., $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept).

Step 1: Find the slope ($m$)

We can calculate the slope of the linear function using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using the points $(1, 1)$ and $(3, -7)$, where $(x_1, y_1) = (1, 1)$ and $(x_2, y_2) = (3, -7)$, we substitute into the slope formula: $m = \frac{-7 - 1}{3 - 1} = \frac{-8}{2} = -4$

Step 2: Use point-slope form to find the equation

Now that we know the slope is $m = -4$, we can use the point-slope form of a linear equation: $y - y_1 = m(x - x_1)$ Using the point $(1, 1)$ and $m = -4$, we plug these values into the formula: $y - 1 = -4(x - 1)$ Simplifying: $y - 1 = -4x + 4$ $y = -4x + 5$

Step 3: Write the equation in slope-intercept form

The equation of the linear function is: $f(x) = -4x + 5$

This is the required linear function in slope-intercept form.

Would you like a detailed explanation of any of these steps or additional examples?

Related Questions:

1. How do you determine the slope from two points on a graph?
2. What is the difference between slope-intercept form and point-slope form?
3. Can we derive the equation using different points on the line?
4. How do we find the y-intercept if the slope and a point are given?
5. How do we graph a linear function from its equation?

Tip: When finding the slope from two points, always subtract in the same direction to avoid sign errors.