How do you find the vertex and intercepts for y =1/2x^2 +2x - 8?

Jun 26, 2018

Vertex: $\left(- 2 , - 10\right)$
y-intercept: $- 8$
x-intercepts $- 2 \left(1 \pm \sqrt{5}\right)$

Explanation:

Finding the vertex
Notice that our target will be the vertex form $y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$ with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

Given
$y = \frac{1}{2} {x}^{2} + 2 x - 8$

Isolate the terms involving $x$
$y + 8 = \frac{1}{2} {x}^{2} + 2 x$

Modify so the coefficient of ${x}^{2}$ is $1$
$2 \left(y + 8\right) = {x}^{2} + 4 x$

Noting that if ${x}^{2} + 4 x$ are the first two terms of an expanded binomial ${\left(x + a\right)}^{2} = \left({x}^{2} + 2 a x + {a}^{2}\right)$ then $4 x = 2 a x \Rightarrow {a}^{2} = {2}^{2}$
So we will need to add ${2}^{2}$ (to both sides) to "complete the square"
$2 \left(y + 8\right) + {2}^{2} = {x}^{2} + 4 x + {2}^{2}$
or
$2 \left(y + 8\right) + 4 = {\left(x + 2\right)}^{2}$

Now we reverse the process to isolate $y$ and leave the result in vertex form:
$2 \left(y - 8\right) = {\left(x + 2\right)}^{2} - 4$

$\left(y - 8\right) = \textcolor{g r e e n}{\frac{1}{2}} {\left(x + 2\right)}^{2} - 2$

$y = \textcolor{g r e e n}{\frac{1}{2}} {\left(x + 2\right)}^{2} - 10$

y=color(green)(1/2)(x-color(red)(""(-2)))^2+color(blue)(""(-10))

which is the vertex form with vertex at (color(red)(-2),color(blue)(""(-10)))

y-intercept
The y-intercept is the value of $y$ when $x = 0$

$y = \frac{1}{2} {x}^{2} + 2 x - 8$ with $x = 0$
$\textcolor{w h i t e}{\text{XXX}} \Rightarrow {y}_{\left(x = 0\right)} = - 8$

x-intercepts
Similarly the x-intercept values are the values of $x$ for which $y = 0$

So we need to solve $\frac{1}{2} {x}^{2} + 2 x - 8 = 0$
or (simplified by multiplying both sides by $2$
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} + 4 x - 16 = 0$

Using the quadratic formula (ask if you need this), we can find
$\textcolor{w h i t e}{\text{XXX}} x = - 2 \left(1 \pm \sqrt{5}\right)$

graph{(1/2)x^2+2x-8 [-13.55, 11.76, -10.93, 1.73]}