Question

How do you graph `f(x) = -(x-2)(x + 5)`?

Answer 1

By finding the extremum and the two `x`-intercepts. And plotting them.

Explanation:

This is a Parabola. And one way to graph Parabolas is to find three strategic points:
`color(red)((1))` The extremum :

And the extremum occurs when the slope is zero. So, we solve to equation `f'(x)=0`

`=>-(x-2)*1-(x+5)*1=0`

`=>-2x-3=0`

`=>x=-3/2`

Next plug in `x=-3/2` into `f(x)` to get the value of `y`

`y=f(3/2)=-(-3/2-2)(-3/2+5)=(7/2)(7/2)=49/4`

So the extremum is `(-3/2,49/4)`

`color(red)((2))` The roots(the `x`-intercept) :

We solve the equation `f(x)=0`

`=>-(x-2)(x+5)=0`

`=>x=2" "` and `" "x=-5`

Hence the intercepts are : `(2,0)` and `" "(-5,0)`

Plot these three points and link them up to obtain a sketch of the graph of `f(x)`.