Questions: What kind of transformation converts the graph of f(x)=(x+1)^2-6 into the graph of g(x)=4(x+1)^2-6? Horizontal shrink Vertical shrink Horizontal stretch Vertical stretch

What kind of transformation converts the graph of f(x)=(x+1)^2-6 into the graph of g(x)=4(x+1)^2-6?
Horizontal shrink
Vertical shrink
Horizontal stretch
Vertical stretch

Transcript text: What kind of transformation converts the graph of $f(x)=(x+1)^{2}-6$ into the graph of $g(x)=4(x+1)^{2}-6$ ? Horizontal shrink Vertical shrink Horizontal stretch Vertical stretch

Solution

Solution Steps

To determine the transformation that converts the graph of $ f(x) = (x+1)^2 - 6 $ into the graph of $ g(x) = 4(x+1)^2 - 6 $, we need to compare the two functions. The transformation involves multiplying the function by a factor of 4, which affects the vertical dimension. This indicates a vertical stretch.

Step 1: Identify the Functions

We have two functions:

$ f(x) = (x+1)^2 - 6 $
$ g(x) = 4(x+1)^2 - 6 $

Step 2: Analyze the Transformation

To convert $ f(x) $ into $ g(x) $, we observe that the term $ (x+1)^2 $ is multiplied by 4 in $ g(x) $. This indicates a transformation that affects the vertical scaling of the graph.

Step 3: Determine the Type of Transformation

The transformation from $ f(x) $ to $ g(x) $ involves multiplying the output of $ f(x) $ by 4. Since the coefficient of the quadratic term in $ g(x) $ is greater than that in $ f(x) $ (specifically, $ 4 > 1 $), this indicates a vertical stretch.

Final Answer

The transformation that converts the graph of $ f(x) $ into the graph of $ g(x) $ is a vertical stretch. Thus, the answer is \$\boxed{\text{Vertical stretch}}\$.

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