How do you sketch the graph of y=-(2x)^2 and describe the transformation?

Dec 24, 2017

graph{-(2x)^2 [-6.146, 6.34, -5.25, 0.994]}

Transformations compared to $y = {x}^{2}$:

1. Reflection in the x-axis
2. Horizontal compression by a factor of 1/2

Explanation:

For a function $y = f \left(x\right)$, a transformed graph of $f$ has the equation $y = a f \left(b \left(x - h\right)\right) + k$.

$a$ represents the vertical stretches (by a factor of $| a |$) and any x-axis reflections (if $a < 0$)

$b$ represents the horizontal stretches (by a factor of $| \frac{1}{b} |$) and any y-axis reflections (if $h < 0$)

$h$ represents horizontal translations ($h > 0$ means translate right; $h < 0$ means translate left)

$k$ represents vertical translations ($k > 0$ means translate up; $k < 0$ means translate down)

So if $y = {x}^{2}$, then $y = - f \left(2 x\right) = - {\left(2 x\right)}^{2}$

Reflection in x-axis, then horizontal compression by a factor of 1/2.

To graph this, graph $y = {x}^{2}$ first, then reflect it in the y-axis (multiply all y-coordinates of points by $- 1$). Then horizontally compress it by a factor of 1/2 by multiplying all x-coordinates of the points by $\frac{1}{2}$.