Common Graphs — Question 4

Question 4 - Solution

We are given: $$f(x) = \sqrt{x + 2} - 1$$

Step 1: Parent function: $f(x) = \sqrt{x}$

Transformations:

• Horizontal shift left by 2: $x \to x + 2$

• Vertical shift down by 1

Step 2: Domain: $$x + 2 \geq 0 \Rightarrow x \geq -2 \Rightarrow \boxed{\text{Domain: } [-2, \infty)}$$

Step 3: Range: Since square root outputs $\geq 0$, and we subtract 1: $$\boxed{\text{Range: } [-1, \infty)}$$

Step 4: Intercepts

• Y-intercept: $f(0) = \sqrt{2} - 1 \approx 0.414$

• X-intercept: Set $f(x) = 0 \Rightarrow \sqrt{x + 2} = 1 \Rightarrow x + 2 = 1 \Rightarrow x = -1$

Key Point: The graph starts at $(-2, -1)$