Question

use a table of values to graph the function \\(f(x) = |x + 2| - 3\\). select the correct graph below.

Explanation:

Identify the parent function and transformations

Using the Function Transformations knowledge point

$$
LATEXBLOCK0
$$

Determine the horizontal translation

Using the Horizontal Translation knowledge point

$$
LATEXBLOCK1
$$

Determine the vertical translation

Using the Vertical Translation knowledge point

$$
LATEXBLOCK2
$$

Locate the vertex and key points

We calculate the vertex and key coordinates to identify the correct graph:

• Vertex: The vertex of the parent function \( (0,0) \) shifts left \( 2 \) and down \( 1 \), giving the vertex at \( (-2, -1) \).
• y-intercept: Let \( x = 0 \):

$$ f(0) = |0 + 2| - 1 = 2 - 1 = 1 \implies (0, 1) $$

• x-intercepts: Let \( f(x) = 0 \):

$$ |x + 2| - 1 = 0 \implies |x + 2| = 1 \implies x + 2 = \pm 1 \implies x = -1 \text{ or } x = -3 \implies (-1, 0), (-3, 0) $$

Compare with the given options

We analyze the vertex and intercepts of each graph:

• Graph A: The vertex is at \( (-2, 1) \). This is incorrect.
• Graph B: The vertex is at \( (2, 0) \). This is incorrect.
• Graph C: The vertex is at \( (1, 1) \). This is incorrect.
• Graph D: The vertex is at \( (-2, -1) \), the y-intercept is at \( (0, 1) \), and the x-intercepts are at \( (-3, 0) \) and \( (-1, 0) \). This matches our calculated points perfectly.

Answer:

• (A) V-shaped graph with vertex at (-2, 1)
• (B) V-shaped graph with vertex at (2, 0)
• (C) V-shaped graph with vertex at (1, 1)
• (D) V-shaped graph with vertex at (-2, -1) (Correct answer)