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

Jan 9, 2018

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.
$y = a {\left(x - h\right)}^{2} + k$

$a$ 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 $h - v a l u e$ 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]}

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

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

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