Common Graphs — Question 5

Question 5 - Solution

We are given: $$f(x) = -|x - 3| + 2$$

Step 1: Parent function: $f(x) = |x|$

Transformations:

• Shift right by 3: $x \to x - 3$

• Reflect over x-axis: negative sign

• Shift up by 2

Step 2: Vertex: $$\text{Vertex at } (3, 2)$$

Step 3: Axis of symmetry: $$x = 3$$

Step 4: Domain and Range $$\text{Domain: } (-\infty, \infty)$$ $$\text{Range: } (-\infty, 2]$$

Step 5: Intercepts

- Y-intercept: Set $x = 0 \Rightarrow f(0) = -|0 - 3| + 2 = -3 + 2 = -1$ - X-intercepts: Solve $f(x) = 0$ $$-|x - 3| + 2 = 0 \Rightarrow |x - 3| = 2 \Rightarrow x - 3 = \pm 2 \Rightarrow x = 1,\ 5$$

Final Answer Summary:

• Vertex: $(3, 2)$

• Axis of symmetry: $x = 3$

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, 2]$

• X-intercepts: $x = 1,\ 5$

• Y-intercept: $y = -1$