To find the x- and y-intercepts of the function $f(x) = \frac{1}{2} |x - 1| - 2$, we need to set $f(x)$ equal to zero for the x-intercepts and set $x$ equal to zero for the y-intercept.

Finding the x-intercepts

The x-intercepts occur when $f(x) = 0$: $\frac{1}{2} |x - 1| - 2 = 0$

1. Isolate the absolute value term: $\frac{1}{2} |x - 1| = 2$

2. Multiply both sides by 2: $|x - 1| = 4$

3. This equation has two cases:

   • Case 1: $x - 1 = 4$ $x = 5$

   • Case 2: $x - 1 = -4$ $x = -3$

So, the x-intercepts are at $x = 5$ and $x = -3$. Therefore, the x-intercepts are the points $(5, 0)$ and $(-3, 0)$.

Finding the y-intercept

The y-intercept occurs when $x = 0$: $f(0) = \frac{1}{2} |0 - 1| - 2$

1. Simplify inside the absolute value: $f(0) = \frac{1}{2} | -1 | - 2$
2. Calculate the absolute value and multiply: $f(0) = \frac{1}{2} \cdot 1 - 2 = \frac{1}{2} - 2$
3. Simplify: $f(0) = -\frac{3}{2}$

So, the y-intercept is at the point $\left(0, -\frac{3}{2}\right)$.

Summary

• x-intercepts: $(5, 0)$ and $(-3, 0)$
• y-intercept: $\left(0, -\frac{3}{2}\right)$

Would you like further details on how the absolute value function affects the shape of this graph or any other aspect?

1. How does the absolute value in the function affect the graph's shape?
2. What are other points on the graph to help sketch it?
3. How would you find the vertex of this function?
4. How can you determine the symmetry of the graph?
5. How does the slope affect the function on either side of the absolute value point?

Tip: To find intercepts quickly, remember that setting $f(x) = 0$ finds x-intercepts, and setting $x = 0$ finds the y-intercept.