Question

How do you sketch the graph of `y=2(x+4)^2-3` and describe the transformation?

Answer 1

See Explanation. Transformations are vertical stretch by a factor of 2, horizontal shift 4 units left, vertical shift 3 units down.

Explanation:

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

`a` is the vertical stretch. If it is big, there is a lot of stretch. If it it less than 1, there is compression.

`h` is the horizontal shift. Notice the "`-`" in the equation.
This means a `h-value` of `+4` as seen in the question is really a `-4`, shifting the graph left.

`k` is the vertical shift. It moves the graph up/down. Positive `k` moves the graph up while negative `k` move the graph down. Simple.

Now the graphs. I do one transformation each time to show the steps.

`y=x^2`
graph{x^2 [-4.75, 5.25, -0.98, 4.02]}

`y=2x^2`
graph{2x^2 [-4.75, 5.25, -0.98, 4.02]}

`y=2(x+4)^2`
graph{2(x+4)^2 [-8.25, 1.75, -0.9, 4.1]}

`y=2(x+4)^2-3`
graph{2(x+4)^2-3 [-8.04, 1.96, -3.22, 1.78]}