# How do you find the vertex for y=x^2+6x+7?

Aug 24, 2017

Use the 'complete the square form' i.e. (x±1/2b)^2-n

#### Explanation:

Okay, so:
in order to get to the complete the square form, you need to halve the 'b' coefficient of your ax^2±bx±c equation. So, in this case, halve 6 to get:

${\left(x + 3\right)}^{2}$

right?

Let's expand the bracket, shall we?

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

Now, you'll notice that the 'c' isn't 7, but 9. That's where the 'n' comes in. The 'n' is supposed to be a simple calculation that gets your 'c' right; if your 'n' becomes a little bit too complicated, double check if you've done the halving/ expanding correctly!

$= {x}^{2} + 6 x + 9$

We can get the 'c' to 7 by subtracting 2. The -2 becomes 'n'. So we get:

$= {x}^{2} + 6 x + 9 - 2$
$= {x}^{2} + 6 x + 7$

To get:

${\left(x + 3\right)}^{2} - 2$

So now that you have your complete the square form done, think of what number gets ONLY the bracket to equal to 0. In this case, -3. Thus, -3 becomes your x value for the vertex.

$\left(- 3 + 3\right) - 2$

$= 0 - 2$

However, you have the remaining 'n' part to think of as well, but since the bracket is already equal to 0,

$0 - 2 = - 2$

your 'n' value becomes the y value for the vertex! So the vertex of the graph is:

(-3, -2)

Hope this helps!

Aug 24, 2017

An alternative to finding the vertex. See explanation for steps.

Vertex has a coordinate of $\left(- 3 , - 2\right)$

#### Explanation:

$y = {x}^{2} + 6 x + 7$ is a quadratic equation.

Graphing $x$ and $y$ on a cartesian plane would give a curve.
In this case, the graph is curving up due to the fact that ${x}^{2}$ has a positive coefficient.

Since it is a graph curving up, we know that the gradient at points on the graph would be increasing from a negative number when the value of x increases.

At the vertex, or in other words, the minimum point (in this case since it curves up) has a gradient of 0.

Why?
Because the minimum point is a point where the gradient changes from negative to positive.

Firstly, to find its vertex, we need to form a gradient function, $\frac{\mathrm{dy}}{\mathrm{dx}}$

Differentiate $y = {x}^{2} + 6 x + 7$
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x + 6$

We know that at vertex, gradient or $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$
$\therefore 2 x + 6 = 0$

$x = - \frac{6}{2} = - 3$

We have the x-coordinate of the vertex. To find the y-coordinate, simply plug in $x = - 3$ into $y = {x}^{2} + 6 x + 7$

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

The vertex has a coordinate of $\left(- 3 , - 2\right)$