Let's solve the problem step by step. The given equation is:

$y = |x + 4| - 4$

We need to find both the x-intercepts and the y-intercepts. Here's how:

Step 1: Finding the x-intercepts

The x-intercept(s) occur when $y = 0$. Set $y$ to 0 in the equation and solve for $x$:

$0 = |x + 4| - 4$

Add 4 to both sides:

$|x + 4| = 4$

This absolute value equation splits into two cases:

1. $x + 4 = 4$
2. $x + 4 = -4$

Solve both cases:

1. $x + 4 = 4$
   Subtract 4:
   $x = 0$

2. $x + 4 = -4$
   Subtract 4:
   $x = -8$

Thus, the x-intercepts are: $(0, 0) \quad \text{and} \quad (-8, 0)$

Step 2: Finding the y-intercept

The y-intercept occurs when $x = 0$. Substitute $x = 0$ into the equation:

$y = |0 + 4| - 4$

Simplify:

$y = |4| - 4$

$y = 4 - 4 = 0$

Thus, the y-intercept is: $(0, 0)$

Final Answer

• x-intercept(s): $(0, 0)$ and $(-8, 0)$
• y-intercept(s): $(0, 0)$

Would you like me to explain any of the steps further or help with a graph of this equation?

5 Questions for Deeper Understanding

1. What does the absolute value function do to the shape of the graph?
2. How can you confirm that the intercepts are correct by graphing the equation?
3. Can an equation like $|x + 4| - 4$ have more than two x-intercepts? Why or why not?
4. What happens to the graph if the equation changes to $|x + 4| + 4$?
5. How would you solve for intercepts if the equation included $|x + 4|^2 - 4$?

Tip

When solving equations with absolute values, always consider both the positive and negative cases of the absolute value expression!