Question

What are the important points needed to graph `f(x) = -(x + 2)(x-5)`?

Answer 1

Graph of `f(x)` is a parabola with `x-` intercepts `(-2, 0) and (5, 0)` and an absolute maximum at `(1.5, 12.25)`

Explanation:

`f(x) =-(x+2)(x-5)`

The first two 'important points' are the zeros of `f(x)`. These occur where `f(x)=0` - I.e. the `x-`intercepts of the function.

To find the zeros: `-(x+2)(x-5) =0`

`:.x=-2 or 5`

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

Expanding `f(x)`

`f(x) = -x^2+3x+10`

`f(x)` is a quadratic function of the form `ax^2+bx+c`. Such a function is represented graphically as a parabola.

The vertex of the parabola occurs at `x=(-b)/(2a)`

i.e where `x=(-3)/-2 = 3/2 = 1.5`

Since `a<0` the vertex will be at the absolute maximum `f(x)`

`:.f_max = f(3/2) = -(3/2)^2+3(3/2)+10`

`= -9/4 + 9/2 +10 = 9/4+10 = 12.25`

Hence another 'important point' is: `f_max = (1.5, 12.25)`

We can see these points of the graph of `f(x)` below.

graph{-(x+2)(x-5) [-36.52, 36.52, -18.27, 18.27]}