The function given is:

$f(x) = -5 + |x - 8|$

The problem asks for a new function $g(x)$ representing the graph of $f(x)$ after translating it 7 units up.

Translation of a Function Vertically:

• When a function is translated up by $k$ units, the new function becomes $g(x) = f(x) + k$.

In this case, since the translation is 7 units up, we add 7 to the function $f(x)$.

$g(x) = f(x) + 7$

Now, substituting $f(x) = -5 + |x - 8|$ into the equation:

$g(x) = (-5 + |x - 8|) + 7$

Simplifying:

$g(x) = |x - 8| + 2$

So the function $g(x)$ is:

$g(x) = |x - 8| + 2$

Do you have any questions, or would you like further clarification?

Here are 5 follow-up questions:

1. How do vertical translations affect the y-coordinates of points on a graph?
2. What would happen if the graph were translated downward instead of upward?
3. How can you perform horizontal translations in function transformations?
4. What is the significance of the absolute value in the given function $f(x)$?
5. How would the function change if both horizontal and vertical translations were applied?

Tip: Always remember that vertical transformations directly affect the output (y-values), while horizontal transformations modify the input (x-values).