Question

given that (f(x)) is the original function, what is true about the transformation, (g(x))?

• (g(x)) is reflected about the (y)-axis.
• (g(x)) is reflected about the (y)-axis and shifted up by 4 units.
• (g(x)) is reflected about the (x)-axis.
• (g(x)) is reflected about the (x)-axis and shifted up by 4 units.

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Identify the vertex of the original function \( f(x) \)

Looking at the graph of \( f(x) \):

• The vertex (lowest point) of the parabola is located at \( (-4, -2) \).
• The parabola opens upwards.

Step 2: Identify the vertex of the transformed function \( g(x) \)

Looking at the graph of \( g(x) \):

• The vertex (highest point) of the parabola is located at \( (-4, 2) \).
• The parabola opens downwards.

Step 3: Analyze the transformations

1. Reflection:

• The parabola changes from opening upwards to opening downwards. This indicates a reflection across the \( x \)-axis:

$$ y = -f(x) $$

• Reflecting the original vertex \( (-4, -2) \) across the \( x \)-axis changes the sign of the \( y \)-coordinate, resulting in the point \( (-4, 2) \).

1. Vertical Shift:

• Since the reflected vertex \( (-4, 2) \) matches the vertex of \( g(x) \) exactly, no vertical shift is required after the reflection.
• Therefore, the transformation is solely a reflection about the \( x \)-axis.

Answer:

\( g(x) \) is reflected about the \( x \)-axis.