The problem asks to write a linear function $g$ using the given points:

• $g(0) = 9$ (which means the y-intercept is 9),
• $g(8) = 7$ (which gives the point $(8, 7)$).

Step-by-step solution:

1. Identify the form of the linear function:

   A linear function can be written as: $g(x) = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

2. Use the point $g(0) = 9$ to find the y-intercept $b$:

   Since $g(0) = 9$, we know that $b = 9$.

   So the equation becomes: $g(x) = mx + 9$

3. Find the slope $m$ using the points $(0, 9)$ and $(8, 7)$:

   The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting $(x_1, y_1) = (0, 9)$ and $(x_2, y_2) = (8, 7)$: $m = \frac{7 - 9}{8 - 0} = \frac{-2}{8} = -\frac{1}{4}$

4. Write the final equation:

   Now that we know the slope $m = -\frac{1}{4}$ and the y-intercept $b = 9$, the linear function is: $g(x) = -\frac{1}{4}x + 9$

Final Answer:

The linear function is: $g(x) = -\frac{1}{4}x + 9$

Would you like further details on the steps or explanation? Here are 5 related questions to deepen your understanding:

1. How do we know that the slope is negative in this case?
2. Why is the point $g(0) = 9$ called the y-intercept?
3. How can we verify that $g(x) = -\frac{1}{4}x + 9$ passes through $(8, 7)$?
4. What happens if the slope is 0 in a linear equation?
5. How would the equation change if the slope were positive instead?

Tip: When finding the slope, always subtract in the same order for both coordinates to avoid sign mistakes!