Question

graph $g(x) = 4(x + 4)^2 - 2$.

Explanation:

Step1: Identify the vertex form

The function \( g(x) = 4(x + 4)^2 - 2 \) is in vertex form \( y = a(x - h)^2 + k \), where the vertex is \( (h, k) \). Here, \( h = -4 \) (since \( x + 4 = x - (-4) \)) and \( k = -2 \), so the vertex is \( (-4, -2) \).

Step2: Determine the direction and width

The coefficient \( a = 4 \) is positive, so the parabola opens upward. The absolute value of \( a \) is greater than 1, so the parabola is narrower than the standard \( y = x^2 \).

Step3: Find additional points (optional for graphing)

For example, when \( x = -3 \): \( g(-3) = 4(-3 + 4)^2 - 2 = 4(1)^2 - 2 = 2 \), so the point \( (-3, 2) \) is on the graph. When \( x = -5 \): \( g(-5) = 4(-5 + 4)^2 - 2 = 4(1) - 2 = 2 \), so \( (-5, 2) \) is also on the graph.

To graph: Plot the vertex \( (-4, -2) \), then use the direction (upward) and the width (narrow) along with additional points to sketch the parabola.

Answer:

To graph \( g(x) = 4(x + 4)^2 - 2 \):

1. Plot the vertex at \( (-4, -2) \).
2. Since \( a = 4 > 0 \), the parabola opens upward.
3. Use the narrow width (due to \( |a| = 4 > 1 \)) and additional points (e.g., \( (-3, 2) \), \( (-5, 2) \)) to draw the parabola symmetric about the vertical line \( x = -4 \).