Question

what transformation(s) have been applied to function (f(x)) to get (g(x))? check all that apply.

translation
reflection
dilation
rotation

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Analyze orientation (Reflection)

The original parabola \( f(x) \) opens upward.
The transformed parabola \( g(x) \) opens downward.

This change in orientation indicates a vertical reflection across a horizontal line.

Step 2: Analyze size and shape (Dilation)

Identify the vertex and key points of both functions:

• For \( f(x) \), the vertex is at \( (-6, -4) \). Moving \( 1 \) unit horizontally to the left or right from the vertex leads to a point \( 1 \) unit up at \( (-5, -3) \) and \( (-7, -3) \).
• For \( g(x) \), the vertex is at \( (0, 3) \). Moving \( 1 \) unit horizontally to the left or right from the vertex leads to a point \( 1 \) unit down at \( (1, 2) \) and \( (-1, 2) \).

Since the width and rate of opening of the parabola remain identical, there is no stretching or compressing. Thus, there is no dilation.

Step 3: Analyze position (Translation)

The vertex of \( f(x) \) is at \( (-6, -4) \).
The vertex of \( g(x) \) is at \( (0, 3) \).

The shift from \( -6 \) to \( 0 \) horizontally and from \( -4 \) to \( 3 \) vertically (after accounting for the reflection) shows that the graph has been shifted (translated) both horizontally and vertically.

Answer:

• translation
• reflection