How do you find the vertex and the intercepts for f(x) = -x^2 + 8x + 2?

Apr 9, 2017

Vertex: $\left(4 , 18\right)$
Y-intercept: $\left(0 , 2\right)$
X-interecpt: $\left(4 + 3 \sqrt{2} , 0\right)$ and $\left(4 - 3 \sqrt{2} , 0\right)$

Explanation:

The vertex is pretty easy. We just need to complete the square .

First, the leading coefficient of the polynomial must be $1$, so we need to factor the $- 1$. That leaves us with $y = - 1 \left({x}^{2} - 8 x - 2\right)$. Now, the purpose of completing the square is to find a constant that makes ${x}^{2} - 8 x$ a perfect square. To that, we use this formula: $c = {\left(\frac{1}{2} \cdot b\right)}^{2}$ or ${\left(\frac{1}{2} \cdot 8\right)}^{2}$, which is $16$.

Now we know that we have to add $16$ to make it a pefect square, but because we cannot just add something on one side of the equation, we need to "get rid of it" too. We could add $16$ on both sides, or we can just add $16$ and then subtract it immediately, which is the same thing. Either way works :)
$y = - 1 \left({x}^{2} - 8 x \textcolor{g r e e n}{+ 16 - 16} - 2\right)$
$y = - \left(\left({x}^{2} - 8 x + 16\right) - 16 - - 2\right)$

${x}^{2} - 8 x + 16$ is a perfect square, so let's symplify it

$y = - \left({\left(x - 4\right)}^{2} - 16 - 2\right)$
$y = - \left({\left(x - 4\right)}^{2} - 18\right)$

Now we just distribute the negative:
$y = - {\left(x - 4\right)}^{2} + 18$

The equation is now in vertex form.

It's easy to find the vertex from this point:
$y = - {\left(x - \textcolor{red}{4}\right)}^{2} + \textcolor{p u r p \le}{18}$
$\left(\textcolor{red}{4} , \textcolor{p u r p \le}{18}\right)$.

Finding the $x$-interecpt means setting $y = 0$ and solving for $x$:

$0 = - {\left(x - 4\right)}^{2} + 18$
$- 18 = - {\left(x - 4\right)}^{2}$
$18 = {\left(x - 4\right)}^{2}$
$\pm \sqrt{18} = x - 4$
$4 \pm \sqrt{18} = x$
or $x = 4 \pm 3 \sqrt{2}$

Those are the exact values. If you want the estimated values, they're $x \approx 8.243$ and $x \approx - .0 .243$

To find the $y$-intercept we just set $x = 0$ and solve for $y$:
$y = - {\left(0 - 4\right)}^{2} + 18$
$y = - {\left(- 4\right)}^{2} + 18$
$y = - 16 + 18$
$y = 2$