Question

How do you graph the inequality `y<=x^2-4`?

Answer 1

See below

Explanation:

First consider the limiting case: `y = x^2-4`

This is a parabola with a absolute minimum value
(since the coefficient of `x^2 >0`)

Since there is no term in `x`, `y_min = y(o) =-4`

Considering `y = (x+2)(x-2)`

`y` will vave zeros at `x=-2 and x=2`

From the results above, it is possible to graph `y` in the limiting case.

Now, turning to the inequality: `y <= x^2-4`

This may be represented graphically as the entire area of the `xy-`plane that is below and on the limiting case graph. The area that satisifies the inequality is shown shaded below extended to `(-oo, +oo)` on both axes.

graph{y<=x^2-4 [-10, 10, -5, 5]}