Question

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

Answer 1

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(x)` of a polynomial function `f(x)` takes the form:

`g(x) = af[k(x-d)]+c`

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

`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(x-2)^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.