# How do you graph the function, label the vertex, axis of symmetry, and x-intercepts. y = -2x^2 + 10x - 1?

Feb 18, 2018

Assumption: You mean 'what are the critical points'
Have a look at https://socratic.org/s/aNwB6wzt
as a method guide for completing the square

#### Explanation:

$\textcolor{b l u e}{\text{General shape}}$

As the $2 {x}^{2}$ term is negative then the graph is of form $\textcolor{b l u e}{\bigcap \text{ thus the vertex is a maximum}}$

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$\textcolor{b l u e}{\text{Determine the vertex}}$
Write as: $y = - 2 \left({x}^{2} - \frac{10}{2}\right) - 1$

Then
$\textcolor{red}{{x}_{\text{vertex}} = \left(- \frac{1}{2}\right) \times \left(- \frac{10}{2}\right) = + 2.5}$

Substitute for $x$

$\textcolor{red}{{y}_{\text{vertex}} = - 2 {\left(2.5\right)}^{2} + 10 \left(2.5\right) - 1 = 11.5}$

color(red)("Vertex" ->(x,y)=(2.5,11.5)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the y-intercept}}$

Set $x = 0$ giving

${y}_{\text{intercept}} = - 2 {\left(0\right)}^{2} + 10 \left(0\right) - 1$

$\textcolor{red}{{y}_{\text{intercept}} = - 1}$

So the y-intercept is the constant $c$ in $y = a {x}^{2} + b x + c$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the x-intercept}}$

There are 3 methods

Method 1: Factorise.
This will only work if the x-intercepts are whole numbers

Method 2: Use the formular $\to x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
Where $y = a {x}^{2} + b x + c$
Note that if ${b}^{2} - 4 a c$ is negative then the graph does not cross the x-axis. ${b}^{2} - 4 a c$ has the name off 'determinate'

Method 3: Complete the square
The advantage of this approach is that you can also (almost) directly read of the coordinates of the vertex.

Using Method 3:

$y = - 2 {x}^{2} + 10 x - 1 \textcolor{w h i t e}{\text{ddd")->color(white)("ddd}} y = \textcolor{m a \ge n t a}{- 2} {\left(x \textcolor{g r e e n}{- 2.5}\right)}^{2} \textcolor{p u r p \le}{+ k} - 1$

Set $\textcolor{m a \ge n t a}{- 2} {\left(\textcolor{g r e e n}{- 2.5}\right)}^{2} + \textcolor{p u r p \le}{k} = 0$

$- 12.5 + k = 0 \textcolor{w h i t e}{\text{d")=>color(white)("d}} k = + 12.5$

$y = - 2 {x}^{2} + 10 x - 1 \textcolor{w h i t e}{\text{ddd")->color(white)("ddd}} y = \textcolor{m a \ge n t a}{- 2} {\left(x \textcolor{g r e e n}{- 2.5}\right)}^{2} \textcolor{p u r p \le}{+ 12.5} - 1$

$\textcolor{w h i t e}{\text{dddddddddddddddddddd")->color(white)("ddd}} y = - 2 {\left(x - 2.5\right)}^{2} + 11.5$

Reading off from $y = - 2 {\left(x - 2.5\right)}^{2} + 11.5$ we have

${x}_{\text{vertex}} = \left(- 1\right) \times \left(- 2.5\right) = + 2.5$
${y}_{\text{Vertex}} = + 11.5 \to \frac{23}{2}$

To find x-intercepts set $y = 0 = - 2 {\left(x - 2.5\right)}^{2} + 11.5$

$+ 2 {\left(x - 2.5\right)}^{2} = + 11.5$

${\left(x - 2.5\right)}^{2} = 5.75$

$x - 2.5 = \pm \sqrt{5.75}$

$x = 2.5 \pm \sqrt{5.75}$

$x \approx 0.1 \mathmr{and} x \approx 4.9$ to 1 decimal place