We are asked to find the slope $m$ of the linear function $f$, use the table to find the $y$-intercept, and give an equation that defines $f$.

The table gives the following points:

$(-6, -23), (-4, -15), (-2, -7), (0, 1), (2, 9)$

Step 1: Find the slope $m$

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(-6, -23)$ and $(-4, -15)$:

$m = \frac{-15 - (-23)}{-4 - (-6)} = \frac{-15 + 23}{-4 + 6} = \frac{8}{2} = 4$

So, the slope $m = 4$.

Step 2: Find the y-intercept

The y-intercept occurs when $x = 0$. From the table, when $x = 0$, $f(x) = 1$. Therefore, the y-intercept is $1$.

Step 3: Write the equation of the line

The general equation of a line is:

$f(x) = mx + b$

We already know $m = 4$ and the y-intercept $b = 1$. So the equation of the line is:

$f(x) = 4x + 1$

Final Answer:

• Slope $m = 4$
• Y-intercept = 1
• The equation of the line is $f(x) = 4x + 1$.

Would you like more details or have any further questions?

Here are 5 related questions:

1. How do you find the slope using two different points?
2. What is the significance of the y-intercept in a linear equation?
3. Can you verify the equation by plugging in other values from the table?
4. How does changing the slope affect the steepness of a line?
5. What happens to the graph of the line if the y-intercept changes?

Tip: Always check the equation by substituting different points from the table to ensure the accuracy of the equation.