Use transformations to explain how the graph of $$ f $$ can be found by using the graph of $$ y=x^{2} $$. $$ f(x)=\frac{1}{3}(x+4)^{2} $$ Which of the following explains the transformations to find the graph of $$ f $$ ? Shifting of $$ y=x^{2} $$ to the right 4 units and vertical stretching of the resulting graph by a factor of $$ \frac{1}{3} $$. Shifting of $$ y=x^{2} $$ to the left 4 units and horizontal stretching of the resulting graph by a factor of $$ \frac{1}{3} $$. Shifting of $$ y=x^{2} $$ to the left 4 units and vertical shrinking of the ressulting graph by a factor of $$ \frac{1}{3} $$. Shits and horizontal shrinking of the resulting graph by a factor of $$ \frac{1}{2} $$.

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UpStudy AI · Accepted Answer

1. Start with the parent function $$ y = x^2 $$.

2. The expression $$ (x+4)^2 $$ indicates a horizontal shift. Since $$ x $$ is replaced by $$ x+4 $$, the graph shifts to the left by 4 units (not to the right).

3. The multiplication by $$ \frac{1}{3} $$ outside the squared term affects the vertical direction. Multiplying by a factor less than 1 causes a vertical shrink (compression), not a stretch.

Thus, the correct description is:

- Shifting of $$ y=x^{2} $$ to the left 4 units and vertical shrinking of the resulting graph by a factor of $$ \frac{1}{3} $$.
The graph of $$ f(x) = \frac{1}{3}(x+4)^2 $$ is obtained by shifting the graph of $$ y = x^2 $$ to the left by 4 units and then shrinking it vertically by a factor of $$ \frac{1}{3} $$.

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