Bot GPT 4-o mini To answer the first question, we start with the function \( f(x) = |x^2| \). To stretch the function vertically by a factor of 3, we multiply the entire function by 3. The equation of the transformed function will be:
\[ f(x) = 3|x^2| \]
So the correct response is:
f(x) = 3|x^2|
For the second question, if we have the dotted line representing a vertically stretched function, \( g(x) = 3|x^2| \), and we want to find the factor \( k \) that would cause a vertical compression back to the original function \( f(x) = |x^2| \), we can express that as:
\[ f(x) = k \cdot g(x) \]
Given that \( g(x) = 3|x^2| \) and the original function \( f(x) = |x^2| \):
Setting them equal:
\[ |x^2| = k \cdot (3|x^2|) \]
To find \( k \):
\[ k \cdot 3|x^2| = |x^2| \Rightarrow k = \frac{|x^2|}{3|x^2|} \]
For \( |x^2| \neq 0 \):
\[ k = \frac{1}{3} \]
So the answer for \( k \) is:
k = \(\frac{1}{3}\)
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