What are the main steps of determining the inequality by its given graph?

Question

What are the main steps of determining the inequality by its given graph? How can I determine the following inequality? I am not asking for the ready answer, I just want to know which major steps in this and similar tasks I should do to come up with the right answer.

Answer 1

First of all you should look at the boundary, which as you probably know is a parabola, and its axis of symmetry is a vertical line, so the equation will look something like $$y=ax^2+bx+c \\a,b,c\in\mathbb{R}$$

So now that you estabilished the boundary by finding the equation of the parabola, which I am supposing you should do by knowing the axis and vertex, we should estabilish the inequality. First of all, notice that the curve bounds two areas of the plane, the area above the curve and the area below. How do we find which is the one you want? Well, the two areas will be $y>ax^2+bx+c$ and $y<ax^2+bx+c$ . You can take a point inside the area you want, for example, in this case your book highlights it for you,$(0,-3)$, and you verify which of the two inequalities it satisfies, and this will tell you the inequality.