# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function y = 7 – 6x – x^2?

Jan 19, 2017

Answer given in detail so you can see where everything comes from.

y_("intercept")=c=+7" "x_("intercept")->x=+1" and "x=-7

The coefficient of ${x}^{2}$ is negative so the graph is of form $\cap$ thus the vertex is a maximum.

$\text{vertex } \to \left(x , y\right) = \left(- 3 , 16\right)$

#### Explanation:

Conventional format:$\to y = a {x}^{2} + b x + c$

So we have: $\text{ } y = - {x}^{2} - 6 x + 7. \ldots \ldots \ldots \ldots \ldots E q u a t i o n \left(1\right)$

$\textcolor{b l u e}{\text{Determining the x-intercepts}}$

This factorises making the calculations more straight forward.

To make $- {x}^{2}$ we need: $\left(- x\right) \times \left(+ x\right)$. Also, where the graph crosses the x-axis, we have the value $y = 0$. So we write:

(-x+?)(+x+?)=0

I spot that $1 \times 7 = 7 \text{ and that } 7 - 1 = 6$ but we need the bigger value to be negative as $- 7 + 1 = - 6 \text{ to give us } - 6 x$

If we place $+ 7$ at (-x+?)(x+7) we end up with $- 7 x$

Try out: $\textcolor{b l u e}{\left(- x + 1\right)} \textcolor{b r o w n}{\left(x + 7\right)} = 0 \ldots \ldots \ldots E q u a t i o n \left(2\right)$

CHECK:
Multiply the right hand brackets by everything in the left hand brackets giving:

" "color(brown)( color(blue)(-x)(x+7)color(blue)(" "+" "1)(x+7))
$\text{ "-x^2-7x" "+" } x + 7$

$= - {x}^{2} - 6 x + 7$ as required so this is the correct factorisation

Using Equation(2):

$\text{ "(-x+1)(x+7)=0 => x=+1" and } x = - 7$

Are solutions for this condition
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$\textcolor{b l u e}{\text{Determining the axis of symmetry}}$

This will be in the middle of the x-intercepts.

${x}_{\text{symmetry}} = \frac{1 - 7}{2} = \frac{- 6}{2} = - 3$
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$\textcolor{b l u e}{\text{Determining the vertex}}$

${x}_{\text{vertex")=x_("symmetry}} = - 3$

Substitute $x = 3$ into Equation(1)

y_("vertex")=-(-3)^2-6(-3)+7" "=" "+16

$\text{vertex } \to \left(x , y\right) = \left(- 3 , 16\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determining the y-intercept}}$

Consider: $y = a {x}^{2} + b x + c \text{ "->" } y = - {x}^{2} - 6 x + 7$

${y}_{\text{intercept}} = c = + 7$ 