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 the orientation (reflection)

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

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

Step 2: Analyze the position (translation)

The vertex of \( f(x) \) is located at:

$$ (-6, -3) $$

The vertex of \( g(x) \) is located at:

$$ (0, 3) $$

Since the position of the vertex has shifted both horizontally and vertically, a translation has occurred.

Step 3: Analyze the shape (dilation)

For \( f(x) \), moving \( 1 \) unit horizontally from the vertex \( (-6, -3) \) to \( x = -5 \) or \( x = -7 \) results in a vertical change of \( 1 \) unit upward to \( y = -2 \).
For \( g(x) \), moving \( 1 \) unit horizontally from the vertex \( (0, 3) \) to \( x = 1 \) or \( x = -1 \) results in a vertical change of \( 1 \) unit downward to \( y = 2 \).

Since the rate of opening remains the same (no stretching or compressing), there is no dilation.

Answer:

• translation
• reflection