Note that a parabola that opens downwards and its translated down lies entirely below the x-axis.
Example Function: y=- 12x^2-2
Practice makes perfect
Let's start by reviewing the form of the graph of the simplest quadratic function possible, y=x^2. Its graph is a parabola that has the vertex at the origin.
Now, recall that for a more general quadratic function, y=ax^2+c, its graph is a transformation of the one shown above, which depends on the values of the parameters a and c.
The parameter a can shrink or stretch the graph as well as reflect it across the x-axis.
&∙ If a>0, the parabola opens upwards.
&∙ If a<0, the parabola opens downwards.
&∙ If 1<|a| the parabola is verticaly streched.
&∙ If 0<|a|<1 the parabola is verticaly shrink.
The parameter c translates the graph of the function vertically.
&∙ If c>0, the parabola is translated upwards by c units.
&∙ If c<0, the parabola is translated downwards by |c| units.
Since we are required for the graph to lie below the x-axis, we can choose a<0. This way the parabola will open downwards. Furthermore, if we choose c<0, the parabola will be shifted downwards. This is enough for the graph to lie entirely below the x-axis. Let's see an example: y= - 12x^2-2.
Note that this is just an example, as there are infinitely many quadratic functions satisfying the exercise's condition.
