Question

How do you graph `y=x^2-2x-8`?

Answer 1

Refer explanation.

Explanation:

• METHOD 1: ALGEBRA

`y =x^2-2x-8` is quadratic in `x`. `a=1,b=-2,c=-8`

As coefficient of `x^2` is positive so, its graph will be mouth opening upward parabola.

Check discriminant of the quadratic to examine the nature of the roots.
`D=b^2-4ac`
`D=4+32=36`
As `D>0`, the roots of quadratic will be real and unequal. Also, we can find roots of `y=0` that are `x=4,-2`

Here, `y=-8` at`x=0`

The minima of the above quadatic is at `x=-b/(2a)=2/2=1`
The value of quadratic at minima `= -D/(4a) =-36/4= -9`

By analysing above all points graph will be
graph{x^2-2x-8 [-25.65, 25.65, -12.83, 12.82]}

• METHOD 2:CALCULUS

Find `dy/dx` and `(d^2y)/dx^2` . Also check the nature of the graph by derivative tests.