Question

How do you graph `y=-2(3+x)^2 +4`?

Answer 1

Calculate the vertex and the `x`- and `y`-intercepts and then sketch the graph.

Explanation:

`y=-2(3+x)^2+4`

Step 1. Convert your equation to vertex form.

The vertex form is `y = a(x-h)^2 +k`.

Rearrange your equation to `y = -2(x+3)^2 +4`.

Then `a=-2`, `h=-3`, and `k=4`.

Step 2. Find the vertex.

The vertex is at (`h,k`) or (`-3,4`).

Step 3. Find the `y`-intercept.

Set `x=0` and solve for `y`.

`y=-2(3+x)^2+4 = -2(3)^2+4 = -2(9) +4 = -18+4 = -14`

The `y`-intercept is at (`0,-14`).

Step 4. Find the `x`-intercept(s).

Set `y=0` and solve for `x`.

`0 = -2(3+x)^2+4`

`(3+x)^2 -2 =0`

`((3+x)+sqrt2)( (3+x)-sqrt2) = 0`

`3+x+sqrt2=0` and `3+x-sqrt2=0`

`x=-3-sqrt2≈ -4.4` and `x=-3+sqrt2≈-1.6`

Step 5. Draw your axes and plot the four points.

Step 6. Draw a smooth parabola that passes through the four points.

And you have your graph.