Which of the following correctly identifies the transformations undergone from f(x)=x2

Question

to 4f(x+3)=4(x+3)2
?(1 point)
Responses

translated to the right and stretched vertically
translated to the right and stretched vertically

translated to the left and stretched vertically
translated to the left and stretched vertically

translated to the left and compressed vertically
translated to the left and compressed vertically

translated to the right and compressed vertically

Answer 1

To analyze the transformation from the function \( f(x) = x^2 \) to \( 4f(x+3) = 4(x+3)^2 \), let’s break it down step by step.

1. Starting with \( f(x) = x^2 \):

   • This is the basic function.

2. Transformation to \( f(x+3) \):

   • The expression \( x + 3 \) indicates a horizontal translation to the left by 3 units. So, \( f(x+3) = (x+3)^2 \).

3. Transformation to \( 4f(x+3) \):

   • The factor of 4 outside the function indicates a vertical stretch by a factor of 4. Thus, \( 4f(x+3) = 4(x+3)^2 \).

Putting it all together, the overall transformation from \( f(x) = x^2 \) to \( 4(x+3)^2 \) involves:

• A translation to the left by 3 units (due to the \( +3 \) inside the function).
• A vertical stretch by a factor of 4 (due to the \( 4 \) outside the function).

Therefore, the correct response is:

translated to the left and stretched vertically.