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 equals zero?
We know that quadratic function graphs are symmetrical about the vertex. We can take the point we’re given and sketch it on a coordinate grid. Here’s the point nine, zero. And because nine, zero is the vertex, it will either be the minimum value or the maximum value for this quadratic function. We don’t have enough information to determine if this is the minimum or the maximum value. If the parabola opens upward, this will be a minimum value. If the parabola opens downward, this will be a maximum value. However, in either case, this function will only cross the 𝑥-axis at the point nine, zero.
If the vertex falls on the 𝑥-axis, the quadratic has one solution. The places where 𝑓 of 𝑥 equals zero are the solutions. There is only one place here where 𝑓 of 𝑥 equals zero. 𝑓 of nine will equal zero, as that is the only solution. Then the solution set for this equation will be nine. Written in set notation looks like this, the curly brackets and the value nine inside.