a Let's start by reviewing the effects of the parameters

a

and

c

in a quadratic function of the form

y = a x^{2} + c .

If $a > 0,$ the parabola opens upwards.
If $a < 0,$ the parabola opens downwards.
If $c > 0,$ the graph is translated upwards by $c$ units.
If $c < 0,$ the graph is translated downwards by $∣ c ∣$ units.

Note that the parameter $a$ can also shrink or stretch the graph of the function vertically. However, the graph is still a parabola. Remember that the vertex of the parabola $y = a x^{2}$ is at the origin, intersecting the $x -$ axis at just one point. Let's consider the case $a > 0 .$

If we translate the same graph downwards then it intersects the $x -$ axis twice. For this to happen, the parameters should be $a > 0$ and $c < 0 .$

However, if the parabola opens downwards we have to translate it upwards instead. This happens when $a < 0$ and $c > 0 .$

With what we discussed so far, we now have enough information to complete the exercise's sentence.

The graph of $y = a x^{2} + c$ intersects the $x -$ axis in two places when $a$ and $c$ have different signs.

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