# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function y = x^2 - 2x - 15?

Aug 14, 2017

Refer to the explanation.

#### Explanation:

$y = {x}^{2} - 2 x - 15$ is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c$,

where:

$a = 1$, $b = - 2$, and $c = - 15$.

Vertex

The maximum or minimum point of a parabola. Since $a > 0$, the vertex is the minimum and the parabola opens upward.

The vertex form for a quadratic equation is:

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

where:

$h$ is the axis of symmetry and $\left(h , k\right)$ is the vertex.
To determine $h$ from the standard form, use the formula: $h = \frac{- b}{2 a}$.

$h = \frac{- \left(- 2\right)}{2 \cdot 1} = \frac{2}{2} = \textcolor{red}{1}$

To determine $k$ from the standard form, substitute $y$ for $k$, and substitute the value of $h$ for $x$ and solve for $k$.

$k = {\textcolor{red}{1}}^{2} - 2 \left(\textcolor{red}{1}\right) - 15$

$k = 1 - 2 - 15$

k=color(blue)(-16

The vertex is $\left(\textcolor{red}{1} , \textcolor{b l u e}{- 16}\right)$.

We will also need to find the x-intercepts by substituting $0$ for $y$ and solving for $x$.

X-Intercepts

The values of $x$ when $y = 0$.

Factor:

$0 = {x}^{2} - 2 x - 15$

Find two numbers that when add equal $- 2$, and when multiplied equal $- 15$. The numbers $3$ and $- 5$ meet the requirements.

$0 = \left(x + \textcolor{p u r p \le}{3}\right) \left(x \textcolor{m a \ge n t a}{- 5}\right)$

$0 = \left(x + \textcolor{p u r p \le}{3}\right)$

$- \textcolor{p u r p \le}{3} = x$

Switch sides.

$x = - \textcolor{p u r p \le}{3}$

$0 = \left(x \textcolor{m a \ge n t a}{- 5}\right)$

$\textcolor{m a \ge n t a}{5} = x$

Switch sides.

$x = \textcolor{m a \ge n t a}{5}$

The x-intercepts are $\left(- 3 , 0\right)$ and $\left(5 , 0\right)$.

Summary

The axis of symmetry is $\textcolor{red}{1}$.

The vertex is $\left(\textcolor{red}{1} , \textcolor{b l u e}{- 16}\right)$

The x-intercepts are $\left(\textcolor{p u r p \le}{- 3} , 0\right)$ and $\left(\textcolor{m a \ge n t a}{5} , 0\right)$.

Plot these points and sketch a parabola through the points. Do not connect the dots.

graph{y=x^2-2x-15 [-15.57, 16.47, -16.77, -0.75]}