# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function y = x^2 +6x + 2?

May 13, 2017

Differentiate the expression to find an equation for its slope and set the slope equal to $0$ to find the minimum. Use the coordinates of the minimum point to find the line of symmetry, and some other points as guides to graph the function.

Minimum: $\left(- 3 , - 7\right)$
Line of symmetry: $x = - 3$
Y-intercept: $y = 2$
X-intercepts: $x = - 5.65$ and $x = - 0.35$

#### Explanation:

It is simplest to take the elements of this question in a different order, and this is often a useful tip in answering questions in math and science. The order things are listed in in the question might not be the best order to do them in.

It's start by finding the maximum or minimum, but even before that I'd think about it a bit. Ignore the complexity for the moment, and this equation has $y = {x}^{2}$ in it and no higher powers of $x$. That means it's going to be a parabola and that is opening will be upward. So we already know that it will have a minimum, not a maximum, and have some idea what it will look like.

A minimum (or maximum) is a point where the slope is $0$: the curve was going down and turns and starts going up, or vice versa, and for just a moment the slope (gradient) is zero.

How to we find the slope of a curve? Using differentiation. The first derivative of a curve is its gradient:

$y = {x}^{2} + 6 x + 2$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x + 6$

(f you've been assigned this question, you should already know how to differentiate an expression like this!)

Then we can simply set that equal to $0$, since the slope is zero at the minimum:

$2 x + 6 = 0$
$2 x = - 6$

$x = - 3$

The x-coordinate of the minimum is $- 3$, and that means the equation of the line of symmetry will be $x = - 3$, so we've already solved one other part of the question.

To find the y-coordinate of the minimum, simply plug this value, $x = - 3$, back into the original equation:

$y = {x}^{2} + 6 x + 2 = {\left(- 3\right)}^{2} + 6 \left(- 3\right) + 2 = 9 - 18 + 2 = - 7$

So the minimum is at the point $\left(- 3 , - 7\right)$.

We know the location of the minimum and the line of symmetry. It might be helpful to know the locations of a few more points to help us graph the function. The x-intercepts and y-intercepts are useful ones. The y-axis is the line $x = 0$, so we can substitute $0$ for $x$ in the original expression:

$y = {\left(0\right)}^{2} + 6 \left(0\right) + 2 = 2$, so the y-intercept is $y = 2$.

We do the same trick to find the x-intercepts, by setting $y$ to $0$. Remember when we imagined the shape of the graph? It's an upward-opening parabola with the minimum below the x-axis, so there will be 2 x-intercepts.

$0 = {x}^{2} + 6 x + 2$

This is a quadratic equation, and we can solve it with the quadratic formula or other means.

That yields $x = - 5.65$ and $x = - 0.35$.

Those points and understanding the shape of a parabola should be enough to graph the function.