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

1 Answer
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 nn thus the vertex is a maximum.

"vertex "->(x,y)=(-3,16)

Explanation:

Conventional format:-> y=ax^2+bx+c

So we have: " " y=-x^2-6x+7................Equation(1)

color(blue)("Determining the x-intercepts")

This factorises making the calculations more straight forward.

To make -x^2 we need: (-x)xx(+x). Also, where the graph crosses the x-axis, we have the value y=0. So we write:

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

I spot that 1xx7=7" and that "7-1=6 but we need the bigger value to be negative as -7+1=-6" to give us "-6x

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

Try out: color(blue)((-x+1))color(brown)((x+7))=0 .........Equation(2)

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))
" "-x^2-7x" "+" "x+7

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

Using Equation(2):

" "(-x+1)(x+7)=0 => x=+1" and "x=-7

Are solutions for this condition
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

color(blue)("Determining the axis of symmetry")

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

x_("symmetry")=(1-7)/2=(-6)/2=-3
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

color(blue)("Determining the vertex")

x_("vertex")=x_("symmetry") = -3

Substitute x=3 into Equation(1)

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

"vertex "->(x,y)=(-3,16)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Determining the y-intercept")

Consider: y=ax^2+bx+c" "->" "y=-x^2-6x+7

y_("intercept")=c=+7

Tony BTony B