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=ax^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=ax^2+c intersects the x-axis in two places when a and c have different signs.
b This time we need the parabola to not intersect the x-axis. Note that if the parabola opens upwards it is enough to translate it upwards. This happens when a>0 and c>0.
On the other hand, if the parabola opens downwards we need to translate it down. This happens when a<0 and c<0.
With this information we can complete the exercise's sentence.
The graph of y=ax^2+c does not intersect the x-axis in two places when a and c have the same sign.
