Find the linear function with the following properties.
f(-7) = 6
f(0) = -2

To find the linear function \( f(x) = mx + b \) that satisfies the given conditions, we'll determine the values of the slope \( m \) and the y-intercept \( b \) using the given points.

### Step 1: Use the point-slope form of the linear equation
We know two points on the line:
- \( (-7, 6) \)
- \( (0, -2) \)

We can calculate the slope \( m \) using the formula:

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

Substituting the given points:

\[
m = \frac{-2 - 6}{0 - (-7)} = \frac{-8}{7}
\]

So, the slope \( m = -\frac{8}{7} \).

### Step 2: Use the slope-intercept form to find \( b \)
The slope-intercept form of the linear equation is:

\[
f(x) = mx + b
\]

We already have \( m = -\frac{8}{7} \). We can use one of the given points to find \( b \). Let's use \( f(0) = -2 \), which means that when \( x = 0 \), \( f(x) = b \).

Substituting into the equation:

\[
-2 = -\frac{8}{7}(0) + b
\]

\[
b = -2
\]

### Step 3: Write the final linear function
Substituting \( m = -\frac{8}{7} \) and \( b = -2 \) into the equation:

\[
f(x) = -\frac{8}{7}x - 2
\]

So, the linear function is:

\[
f(x) = -\frac{8}{7}x - 2
\]

Would you like any additional details or have any questions? 

Here are some follow-up questions to consider:
1. How would the function change if the point \( f(-7) = 6 \) was different?
2. What is the significance of the slope \( m \) in this context?
3. Can you determine the x-intercept of this function?
4. How would you graph this linear function?
5. What is the value of \( f(7) \) for this function?

**Tip:** When finding the linear function from two points, remember that the y-intercept is simply the value of the function at \( x = 0 \).

Learn how to find the linear function that satisfies f(-7) = 6 and f(0) = -2. This problem involves using the point-slope and slope-intercept forms of a linear equation. Suitable for students in Grades 9-12.

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