Use transformations of the standard cubic function, , to graph the function
.
U. The graph of should be horizontally shifted to the left by 5 units,
stretched vertically by a factor of , and shifted vertically up by 2 units.
C. The graph of should be horizontally shifted to the left by 5 units,
shrunk vertically by a factor of , and shifted vertically down by 2 units.
D. The graph of should be horizontally shifted to the right by 5 units,
Use the graphing tool to graph the function.

Ask by Wyatt Jimenez. in the United States

Mar 25,2025

Question

Use transformations of the standard cubic function, , to graph the function
.
U. The graph of should be horizontally shifted to the left by 5 units,
stretched vertically by a factor of , and shifted vertically up by 2 units.
C. The graph of should be horizontally shifted to the left by 5 units,
shrunk vertically by a factor of , and shifted vertically down by 2 units.
D. The graph of should be horizontally shifted to the right by 5 units,
Use the graphing tool to graph the function.

Ask by Wyatt Jimenez. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

Step 1: Start with the standard cubic function  
$$
f(x)=x^3.
$$

Step 2: Identify the transformation inside the function  
The given function is  
$$
h(x)=\frac{1}{3}(x+5)^3-2.
$$
Here, the expression $$x+5$$ implies a horizontal shift. In general, $$f(x+c)$$ shifts the graph to the left by $$c$$ units. Thus, $$x+5$$ shifts the graph left by 5 units.

Step 3: Identify the vertical transformation (scaling)  
The factor $$\frac{1}{3}$$ outside of the cube implies a vertical scaling. Multiplying the output by $$\frac{1}{3}$$ shrinks the graph vertically by a factor of $$\frac{1}{3}$$.

Step 4: Identify the vertical translation  
The final term $$-2$$ indicates a vertical shift downward by 2 units.

Step 5: Combine the transformations  
- Horizontally shifted left by 5 units.
- Vertically shrunk by a factor of $$\frac{1}{3}$$.
- Vertically shifted downward by 2 units.

The correct description of the transformation is provided in option C:

The graph of $$ f(x)=x^{3} $$ should be horizontally shifted to the left by 5 units, shrunk vertically by a factor of $$\frac{1}{3}$$, and shifted vertically down by 2 units.
The graph of $$ f(x)=x^{3} $$ should be shifted left by 5 units, shrunk vertically by a factor of $$ \frac{1}{3} $$, and shifted down by 2 units.

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