Question

How do you find the important parts of the equation to graph the function `y=(x-2)(x+9)`?

Answer 1

x-intercepts: `2` and `(-9)`
y-intercept: `(-18)`
vertex: `(-3 1/2, -30 1/4)`

Explanation:

x-intercepts
the values of `x` which cause `y` to be `=0`
Based on the given form: `y=(x-2)(x+9)`
these values are `x=2` and `x=-9`

y-intercept
the value of `y` when `x=0`
`y=(0-2)(x+9) = -18`

vertex
Converting into vertex form: `y=m(x-a)^2+b`
`y=(x-2)(x+9) = x^2+7x-18`
complete the square
`y=x^2+7x+(7/2)^2-18-(7/2)^2`

`color(white)("X") = 1(x-(-7/2))^2+(-121/4)`
which is the vertex form with vertex at `(-7/2,-121/4) = (-3 1/2, -30 1/4)`

We can check the graph to see that these values are reasonable:
graph{(x-2)(x+9) [-60, 60, -40, 25]}