Question

describe the dilation in each function. then graph the function.

1. $y = 5x$
2. $y = \frac{1}{2}|x|$
3. $y = 2x^2$

Explanation:

Step1: Analyze $y=5x$ dilation

Parent function is $y=x$. The coefficient $5>1$, so it's a vertical stretch by factor $5$.

Step2: Analyze $y=\frac{1}{2}|x|$ dilation

Parent function is $y=|x|$. The coefficient $0<\frac{1}{2}<1$, so it's a vertical compression by factor $\frac{1}{2}$.

Step3: Analyze $y=2x^2$ dilation

Parent function is $y=x^2$. The coefficient $2>1$, so it's a vertical stretch by factor $2$.

Step4: Graph $y=5x$

Plot points: when $x=0$, $y=0$; $x=1$, $y=5$; $x=-1$, $y=-5$. Draw a straight line through these points.

Step5: Graph $y=\frac{1}{2}|x|$

Plot points: when $x=0$, $y=0$; $x=2$, $y=1$; $x=-2$, $y=1$. Draw a V-shaped graph through these points.

Step6: Graph $y=2x^2$

Plot points: when $x=0$, $y=0$; $x=1$, $y=2$; $x=-1$, $y=2$; $x=2$, $y=8$; $x=-2$, $y=8$. Draw a parabola through these points.

Answer:

1. For $y=5x$: Vertical stretch by a factor of $5$ relative to the parent function $y=x$. The graph is a steep straight line passing through the origin.
2. For $y=\frac{1}{2}|x|$: Vertical compression by a factor of $\frac{1}{2}$ relative to the parent function $y=|x|$. The graph is a wider V-shape passing through the origin.
3. For $y=2x^2$: Vertical stretch by a factor of $2$ relative to the parent function $y=x^2$. The graph is a narrower upward-opening parabola with vertex at the origin.