Complete a chart then sketch the graph of $$ f(x)=x^{2} $$ Complete a chart then sketch the graph of $$ f(x)=\frac{1}{4} x^{2} $$ Compare the 2 sketches. What Transformation occurs as a result of the coefficient of $$ \frac{1}{4} $$ ? Vertical Shift Horizontal Shift Vertical Stretch (More Narrow) Vertical Shrink (Wider)

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**Step 1. Completing the Table**

For $$ f(x)=x^2 $$:

| $$ x $$ | $$ x^2 $$ |
|:-------:|:---------:|
|   -3    |    9      |
|   -2    |    4      |
|   -1    |    1      |
|    0    |    0      |
|    1    |    1      |
|    2    |    4      |
|    3    |    9      |

For $$ f(x)=\frac{1}{4}x^2 $$:

| $$ x $$ | $$\frac{1}{4}x^2$$ |
|:-------:|:------------------:|
|   -3    |   $$\frac{9}{4} = 2.25$$  |
|   -2    |   $$\frac{4}{4} = 1$$       |
|   -1    |   $$\frac{1}{4} = 0.25$$    |
|    0    |          0         |
|    1    |   $$\frac{1}{4} = 0.25$$    |
|    2    |   $$\frac{4}{4} = 1$$       |
|    3    |   $$\frac{9}{4} = 2.25$$  |

**Step 2. Sketching the Graphs**

- The graph of $$ f(x)=x^2 $$ is the standard parabola opening upward.
- The graph of $$ f(x)=\frac{1}{4}x^2 $$ is also a parabola opening upward, but its $$ y $$-values are reduced by a factor of $$ \frac{1}{4} $$.

**Step 3. Comparing the Graphs**

Compared to the graph of $$ f(x)=x^2 $$, the graph of $$ f(x)=\frac{1}{4}x^2 $$:

- Has the same vertex at the origin.
- Is wider or more spread out.

**Step 4. Determining the Transformation**

The coefficient $$ \frac{1}{4} $$ causes each $$ y $$-value to be $$ \frac{1}{4} $$ of what it would be in $$ f(x)=x^2 $$. This transformation is a **vertical shrink** (wider) of the graph, not a vertical stretch, horizontal shift, or vertical shift.

**Answer:** Vertical Shrink (Wider)
The coefficient $$ \frac{1}{4} $$ causes the graph of $$ f(x)=\frac{1}{4}x^2 $$ to be a vertical shrink (wider) compared to $$ f(x)=x^2 $$.

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