Joey already drew a graph of function f(x). He now wants to draw on that same graph the composition g ∘ f(x) where gx=x-8. What will the graph of g(x) look like relative to f(x)?
A.A translation 8 units up
B.A translation 8 units down
C.A translation 8 units to the right
D.A translation 8 units to the left

To determine the graph of the composition \( g \circ f(x) \), where \( g(x) = x - 8 \), relative to \( f(x) \), let's break down how \( g(x) = x - 8 \) affects the function \( f(x) \).

### Steps:
1. **Composition**: The composition \( g \circ f(x) \) means you apply \( f(x) \) first, then apply \( g \) to the result. So:
   \[
   g \circ f(x) = g(f(x)) = f(x) - 8
   \]
2. **Effect on the graph**: The operation \( f(x) - 8 \) shifts the graph of \( f(x) \) vertically downward by 8 units. This is because subtracting a constant from a function translates the graph downward.

Therefore, the graph of \( g \circ f(x) \) will be a **translation 8 units down** relative to the graph of \( f(x) \).

### Correct Answer:
**B. A translation 8 units down**

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Would you like more details on function compositions or translations? 

Here are 5 related questions:
1. What happens when you compose two functions with different transformations?
2. How do horizontal and vertical shifts differ for function graphs?
3. How does function composition affect the domain and range?
4. Can function compositions change the shape of a graph?
5. How would the graph change if \( g(x) = x + 8 \)?

**Tip**: Vertical translations occur when you add or subtract a constant from the function's output.

Joey already drew a graph of function f(x). He now wants to draw on that same graph the composition g ∘ f(x) where g(x) = x - 8. What will the graph of g(x) look like relative to f(x)? A. A translation 8 units up B. A translation 8 units down C. A translation 8 units to the right D. A translation 8 units to the left

Understand how to graph the composition g ∘ f(x) where g(x) = x - 8. Learn how vertical translations affect a function's graph with this detailed explanation, suitable for students in Grades 9-12. This problem explores function composition, graph transformations, and vertical shifts.

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