Question

sketch the graph of the function
$f(x)=\sqrt{x + 2}+1$

choose the correct graph below
\\(\bigcirc\\) a.
\\(\bigcirc\\) b.
\\(\bigcirc\\) c.
\\(\bigcirc\\) d.

Explanation:

Step 1: Find the domain

For the function \( f(x)=\sqrt{x + 2}+1 \), the expression under the square root must be non - negative. So we solve the inequality \( x+2\geq0 \), which gives \( x\geq - 2 \).

Step 2: Find the y - intercept

To find the y - intercept, we set \( x = 0 \). Then \( f(0)=\sqrt{0 + 2}+1=\sqrt{2}+1\approx1.414 + 1=2.414 \).

Step 3: Find the x - intercept (if it exists)

Set \( y = 0 \), so \( 0=\sqrt{x + 2}+1 \). Then \( \sqrt{x + 2}=- 1 \). But the square root of a number is always non - negative, so there is no x - intercept.

Step 4: Analyze the shape

The parent function is \( y=\sqrt{x} \). The function \( y=\sqrt{x+2} \) is a horizontal shift of the parent function \( y = \sqrt{x} \) to the left by 2 units. Then the function \( y=\sqrt{x + 2}+1 \) is a vertical shift of \( y=\sqrt{x+2} \) up by 1 unit. The graph of \( y=\sqrt{x} \) starts at \( (0,0) \) and increases slowly. After the shifts, our function starts at \( (-2,1) \) (when \( x=-2 \), \( f(-2)=\sqrt{-2 + 2}+1=1 \)) and increases as x increases.

Looking at the options:

• Option A: The starting point seems to be at a positive x - value, which is incorrect as our domain starts at \( x=-2 \).
• Option B: The y - intercept is too low (close to 0), but we know \( f(0)=\sqrt{2}+1\approx2.414 \), so this is incorrect.
• Option C: The starting point is at \( x=-2 \), the y - intercept is around \( \sqrt{2}+1\approx2.414 \), and the graph is increasing, which matches our analysis.
• Option D: The y - intercept is too low (close to 0), so this is incorrect.

Answer:

C