1. **State the problem:** We are given the quadratic functions $y=5x^2$ and $y=\frac{1}{4}x^2$ and need to fill in tables of values by squaring $x$, multiplying by the coefficient $a$, and then plotting points. 2. **Formula and rules:** The general form is $y = a x^2$. - Step 1: Square the $x$ value: $x^2$. - Step 2: Multiply by $a$: $y = a \times x^2$. 3. **Complete the table for $y=5x^2$:** | $x$ | $x^2$ | $5 \times x^2$ | Point $(x,y)$ | |---|---|---|---| | -2 | 4 | 20 | $(-2, 20)$ | | -1 | 1 | 5 | $(-1, 5)$ | | 0 | 0 | 0 | $(0, 0)$ | | 1 | 1 | 5 | $(1, 5)$ | | 2 | 4 | 20 | $(2, 20)$ | 4. **Vocabulary:** Since $a=5 > 1$, the graph shows a **vertical stretch** (it is narrower than $y=x^2$). 5. **Vertex:** The vertex of $y=5x^2$ is at the origin $(0,0)$. 6. **Complete the table for $y=\frac{1}{4}x^2$:** | $x$ | $x^2$ | $\frac{1}{4} \times x^2$ | Point $(x,y)$ | |---|---|---|---| | -2 | 4 | 1 | $(-2, 1)$ | | -1 | 1 | $\frac{1}{4}$ | $(-1, \frac{1}{4})$ | | 0 | 0 | 0 | $(0, 0)$ | | 1 | 1 | $\frac{1}{4}$ | $(1, \frac{1}{4})$ | | 2 | 4 | 1 | $(2, 1)$ | 7. **Vocabulary:** Since $a=\frac{1}{4} < 1$, the graph shows a **vertical compression** (it is wider than $y=x^2$). 8. **Vertex:** The vertex of $y=\frac{1}{4}x^2$ is also at the origin $(0,0)$. This completes the tables and vocabulary for both functions.