# How do you sketch the graph of y=(x-2)^2-7 and describe the transformation?

Sep 9, 2017

See below.

#### Explanation:

The function is in the form: $a {\left(x - h\right)}^{2} + k$
Where $h$ is the axis of symmetry, and $k$ is the maximum or minimum value of the function. This is known as the vertex of the parabola.

From example: vertex is at $\left(2 , - 7\right)$

We now need to find roots and $y$ axis intercept. This will then give us a sufficient number of plotting points.

Expand $y = {\left(x - 2\right)}^{2} - 7$ , and equate it to $0$

${x}^{2} - 4 x - 3 = 0$

Solution by quadratic formula gives roots:

$\left(2 + \sqrt{7} , 0\right)$ and $\left(2 - \sqrt{7} , 0\right)$

$y$ axis intercept is where $x = 0$

$y = {\left(0\right)}^{2} - 4 \left(0\right) - 3$

$\left(0 , - 3\right)$

So all plotting points are:

$\left(2 , - 7\right)$ ,$\left(2 + \sqrt{7} , 0\right)$ , $\left(2 - \sqrt{7} , 0\right)$ , $\left(0 , - 3\right)$

Graph:
graph{x^2 -4x -3 [-5, 10, -12.8, 20]}

This can be viewed as the graph of $y = {x}^{2}$ translated 2 units to the right and 7 units in the $- y$ direction.