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.