Question

How do you graph `y=x^3 + 3x^2 - 6x`?

Answer 1

graph{y=x^3+3x^2-6x [-10, 10, -5.08, 4.92]}

Explanation:

Make a table for `x` and `y`, and find some solutions that satisfy the equation `y=x^3+3x^2-6x`.

Plot `4` or `5` `(x,y)` dots on a coordinate plane.

Connect the dots.

Finished graph should look like this:

graph{y=x^3+3x^2-6x [-10, 10, -5.08, 4.92]}

Answer 2

Refer Explanation section

Explanation:

Given -

`y=x^3+3x^2-6x`

There is no constant term in the function. Hence, the curve passes through the origin.

Moreover, the turning points in the curve are very very important. This will help us to decide the range of `x`values we have to select.

`dy/dx=3x^2+6x-6`
`dy/dx=>3x^2+6x-6=0`
We shall try to solve using completing the square method.
`3x^2+6x=6`
`x^2+2x=2` -- [devide all the terms by 3]
`x^2+2x+1=2+1` -- [take half of the coefficient of x, square it and add it to both sides]

`(x+1)^2=3`
`x+1=+-sqrt3`
`x=+-sqrt3-1`
`x=1.732-1=0.732`
`x=-1.732-1=-2.732`

At `x=0.732` and `x=-2.732` the curve turns.
Select a range of `x` values that includes these two values.