Question

the graph of $y=x^2$ is shown on each grid in parts (a) and (b) below
(a) use transformations to get the graph of $y=(-2x)^2$.
(b) use transformations to get the graph of $y=\frac{1}{2}x^2$.

Explanation:

Step1: Simplify the function for (a)

Note that $(-2x)^2 = (2x)^2$, so $y=(-2x)^2 = 4x^2$. This is a horizontal compression of $y=x^2$ by a factor of $\frac{1}{2}$ (or vertical stretch by 4).

Step2: Identify key points for (a)

For $y=x^2$, key points are $(-2,4), (-1,1), (0,0), (1,1), (2,4)$. For horizontal compression by $\frac{1}{2}$, replace $x$ with $2x$, so new points are $(-1,4), (-\frac{1}{2},1), (0,0), (\frac{1}{2},1), (1,4)$.

Step3: Analyze transformation for (b)

$y=\frac{1}{2}x^2$ is a vertical compression of $y=x^2$ by a factor of $\frac{1}{2}$.

Step4: Identify key points for (b)

Take key points of $y=x^2$: $(-2,4), (-1,1), (0,0), (1,1), (2,4)$. Multiply each y-value by $\frac{1}{2}$: $(-2,2), (-1,\frac{1}{2}), (0,0), (1,\frac{1}{2}), (2,2)$.

Answer:

(a) Plot the points $(-1,4), (-\frac{1}{2},1), (0,0), (\frac{1}{2},1), (1,4)$ and draw a parabola through them (a narrow parabola opening upwards, compressed horizontally compared to $y=x^2$).
(b) Plot the points $(-2,2), (-1,\frac{1}{2}), (0,0), (1,\frac{1}{2}), (2,2)$ and draw a parabola through them (a wider parabola opening upwards, compressed vertically compared to $y=x^2$).