If the point nine, zero is the vertex of the graph of the quadratic function 𝑓, what is the solution set of the equation 𝑓 of 𝑥 is equal to zero?
In this question, we’re given some information about a quadratic function 𝑓. We’re told that the point nine, zero is the vertex of the graph of this function. We need to use this information to determine the solution set of the equation 𝑓 of 𝑥 is equal to zero. And to do this, let’s start by recalling what we mean by the vertex of a quadratic function 𝑓.
We can start by recalling the graph of any quadratic function will have a parabolic shape, and there are two possible orientations for this parabolic arc to take. First, when the leading coefficient is negative, we know the parabolic arc will open downwards. Second, if the leading coefficient is positive, then we know the parabolic arc will open upwards. So, these are the two possible general shapes that the graph of the function 𝑓 can take. And there’s something interesting we can note about both of these shapes. There’s a single turning point; we call this the vertex of the quadratic.
And there’s a few interesting properties worth noting about the vertex of a parabola. First, the line of symmetry of this parabola passes through the vertex. Second, if the quadratic opens downwards, then the vertex occurs at the maximum output of the function. And similarly, if the quadratic opens upwards, then the vertex occurs at the minimum output of the function. And we can use this to determine information about the graph of the function 𝑓. However, we also need to link this to the solution set of the equation 𝑓 of 𝑥 is equal to zero. And we can do this by noting if 𝑥 is a solution to this equation, so 𝑓 evaluated at 𝑥 is equal to zero, the output of the function is zero. So, the 𝑦-coordinate of this point on the curve will be zero. It will be an 𝑥-intercept of the graph.
Therefore, we can answer this question by determining the possible 𝑥-intercepts of the graph of our function 𝑓 of 𝑥 by knowing its vertex is the point nine, zero. So, let’s now sketch some possible graphs of the function 𝑦 is equal to 𝑓 of 𝑥. We’ll start by marking the vertex on our graph; it’s the point nine, zero.
One possible graph of the function 𝑦 is equal to 𝑓 of 𝑥 might look like the following; it’s an upward-opening parabola with vertex at the point nine, zero. However, we do need to know there are infinitely many quadratics with vertex at the point nine, zero. For example, the graph of the function could also look like the following. It could be a wider parabola; it could also be a narrower parabola. Similarly, since we’re given no information about the function 𝑓 of 𝑥 other than the coordinates of its vertex, the parabola could also open downwards.
However, if we consider the 𝑥-intercepts of all of these possible graphs, we can notice something interesting. All of these graphs only have a single 𝑥-intercept, the vertex of the parabola. And this fact ties in with the properties of parabolas we discussed previously. In the parabola which opens downwards, the vertex occurs at the maximum output of the function, and in a parabola which opens upwards, the vertex occurs at the minimum output of the function. So, in both cases, we only get a single 𝑥-intercept to the point nine, zero.
Therefore, there’s only one solution to this equation; our value of 𝑥 needs to be equal to nine. And to write this as a solution set, this is the set of all possible solutions to the equation, and this gives us the set whose only element is nine. And it’s worth noting this property holds in general for any parabola. If the quadratic has a vertex which lies on the 𝑥-axis, then this is the only solution to the equation 𝑓 of 𝑥 is equal to zero. It will have a single 𝑥-intercept.
Therefore, we were able to show if nine, zero is the vertex of the graph of a quadratic function 𝑓, then the solution set of the equation 𝑓 of 𝑥 is equal to zero is the set whose only element is nine.