Find all x- and y-intercepts of the graph of f left parenthesis x right parenthesis equals short dash 2 vertical line x minus 3 vertical line plus 8.

We are given the function \( f(x) = -2 |x - 3| + 8 \), and we need to find its x- and y-intercepts.

### 1. Finding the **y-intercept**:
The y-intercept occurs where \( x = 0 \). So, we substitute \( x = 0 \) into the function and solve for \( f(0) \).

\[
f(0) = -2 |0 - 3| + 8
\]
\[
f(0) = -2 | -3 | + 8 = -2(3) + 8 = -6 + 8 = 2
\]

Thus, the y-intercept is \( (0, 2) \).

### 2. Finding the **x-intercepts**:
The x-intercepts occur where \( f(x) = 0 \). So, we set the function equal to zero and solve for \( x \).

\[
0 = -2 |x - 3| + 8
\]
\[
-2 |x - 3| = -8
\]
\[
|x - 3| = 4
\]

Now, we solve the absolute value equation:
\[
x - 3 = 4 \quad \text{or} \quad x - 3 = -4
\]

Solving both cases:
\[
x = 4 + 3 = 7 \quad \text{or} \quad x = -4 + 3 = -1
\]

Thus, the x-intercepts are \( x = 7 \) and \( x = -1 \). Therefore, the x-intercepts are \( (7, 0) \) and \( (-1, 0) \).

### Summary:
- **y-intercept**: \( (0, 2) \)
- **x-intercepts**: \( (7, 0) \) and \( (-1, 0) \)

Would you like more details on any part of the process or have any other questions?

Here are five related questions you might consider:
1. How do you graph a function involving absolute value, like \( f(x) = -2 |x - 3| + 8 \)?
2. What transformations occur in the graph of \( f(x) = |x - 3| \)?
3. How do you solve absolute value equations generally?
4. What is the vertex of the graph of \( f(x) = -2 |x - 3| + 8 \)?
5. How do the x- and y-intercepts help in sketching the graph of a function?

**Tip**: Absolute value functions create V-shaped graphs, and the turning point (vertex) occurs where the expression inside the absolute value is zero.

Find all x- and y-intercepts of the graph of f(x) = -2 |x - 3| + 8.

Learn how to find the x- and y-intercepts of the absolute value function f(x) = -2 |x - 3| + 8. This step-by-step guide covers solving for intercepts, handling absolute value equations, and understanding key graph transformations. Suitable for students in Grades 9-11.

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