# How do you find the vertex and intercepts for y = x^2 – 4x + 9?

Jun 15, 2018

vertex: (2, 5)

x-intercept: none

y-intercept: (0, 9)

#### Explanation:

$y = {x}^{2} - 4 x + 9$

The equation is a quadratic equation in standard form, or $y = \textcolor{red}{a} {x}^{2} + \textcolor{g r e e n}{b} x + \textcolor{b l u e}{c}$.

The vertex is the minimum or maximum point of a parabola . To find the $x$ value of the vertex, we use the formula ${x}_{v} = - \frac{\textcolor{g r e e n}{b}}{2 \textcolor{red}{a}}$, where ${x}_{v}$ is the x-value of the vertex.

We know that $\textcolor{red}{a = 1}$ and $\textcolor{g r e e n}{b = - 4}$, so we can plug them into the formula:
${x}_{v} = \frac{- \left(- 4\right)}{2 \left(1\right)} = \frac{4}{2} = 2$

To find the $y$-value, we just plug in the $x$ value back into the equation:
$y = {\left(2\right)}^{2} - 4 \left(2\right) + 9$

Simplify:
$y = 4 - 8 + 9$

$y = - 4 + 9$

$y = 5$

Therefore, the vertex is at $\left(2 , 5\right)$

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Now to find the intercepts.

The $x$-intercept is the value of $x$ when $y$ equals to zero.

To find it, just plug in $0$ for $y$ in the equation:
$0 = {x}^{2} - 4 x + 9$

Since we cannot factor ${x}^{2} - 4 x + 9$, we will use the quadratic formula $x = \frac{- \textcolor{g r e e n}{b} \pm \sqrt{{\textcolor{g r e e n}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{b l u e}{c}}}{2 \textcolor{red}{a}}$

We already know that $\textcolor{red}{a = 1}$, $\textcolor{g r e e n}{b = - 4}$, and $\textcolor{b l u e}{c = 9}$, so let's plug them into the formula:
x = (-(color(green)(-4)) +- sqrt(color(green)((-4))^2 - 4(color(red)(1))(color(blue)(9))))/(2(color(red)(1))

Now simplify:
$x = \frac{4 \pm \sqrt{16 - 36}}{2}$

$x = \frac{4 \pm \sqrt{- 20}}{2}$

Since we cannot do a square root of a negative number (it becomes imaginary), that means there are no $x$-intercepts .

The $y$-intercept is the value of $y$ when $x$ equals to zero.

To find it, just plug in $0$ for all the $x$'s in the equation:
$y = {\left(0\right)}^{2} - 4 \left(0\right) + 9$

Simplify:
$y = 0 - 0 + 9$

$y = 9$

Now we write it as a coordinate, so it becomes $\left(0 , 9\right)$.

In summary,

vertex: (2, 5)

x-intercept: none

y-intercept: (0, 9)

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Here is a graph of this quadratic equation:

As you can see, there is no $x$-intercept, and the vertex and $y$-intercept are shown there.

For another explanation/example of finding the vertex and intercepts of a standard equation, feel free to watch this video:

Hope this helps!