Question

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

Answer 1

There are quite a number of tools we can use to graph functions.

First, we know this is a polynomial. We also can see that it is degree two, since there are two first degree terms (and if we multiplied it through we would get `x^2` but nothing higher).

Now that we know it is second degree, we know that means it could be a concave or a convex parabola. That is determined by the sign of the coefficient of `x^2`, i.e. if the number before `x^2` is positive or negative. We can see pretty easily that the coefficient will be `-1`, so it will be a convex parabola, i.e. going down.

From here, this equation is factored. This means that we can easily see what the zeros are. They happen when `x-2 = 0` or `x+5=0`, i.e. `x in {-5, 2}`.

Now that we know all of this, we can plot the function. The function goes up to `x=-5` and then in between it maxes out and then it falls down, through `x = 2` and down to infinity. This should resemble this
graph{-(x-2)(x+5) [-8, 8, -10, 20]}