# Which steps transform the graph of y=x^2 to y=-2(x- 2)^2+ 2?

Jun 25, 2018
1. Reflection over the $y$-axis.
2. Vertical stretch by a factor of two.
3. Horizontal translation right two units.
4. Vertical translation up two units.

#### Explanation:

The transformation $g \left(x\right)$ of a polynomial function $f \left(x\right)$ takes the form:

$g \left(x\right) = a f \left[k \left(x - d\right)\right] + c$

$a$ is the factor of vertical stretch or compression. If $a$ is negative, then the transformed graph is reflected over the $y$-axis.

$\frac{1}{k}$ is the factor of horizontal stretch or compression. If $k$ is negative, then the transformed graph is reflected over the $x$-axis.

$d$ is the horizontal translation.
$c$ is the vertical translation.

When looking at a transformation, the steps are applied moving from the left side of the equation to the right.

In $y = - 2 {\left(x - 2\right)}^{2} + 2$:

$a = - 2$. The graph is reflected over the $y$-axis and stretched vertically by a factor of 2.

$d$ = 2. The graph is translated two units to the right.
$c$ = 2. The graph is translated two units upwards.