Find the coordinates of the vertex of the graph of 𝑓 of 𝑥 equals 𝑥 squared minus six 𝑥 minus four. State the value of the function at the vertex and determine whether it is a maximum or minimum.
For any parabola, 𝑓 of 𝑥 equals 𝑎𝑥 squared plus b𝑥 plus 𝑐 and the vertex ℎ, 𝑘 can be found using ℎ equals negative 𝑏 over two 𝑎 and 𝑘 equals 𝑓 of ℎ. So if this is our function, 𝑎 equals one, 𝑏 equals negative six, and 𝑐 equals negative four. So to find the vertex ℎ, 𝑘, we will find ℎ by negative 𝑏 over two 𝑎. So here, we will multiply negative one times negative six over two times one, which is six over two, which is equal to three. So that is ℎ.
So now we need to find 𝑘. And 𝑘 is equal to 𝑓 of ℎ. We have to plug in the value of ℎ into our function. So it’s equal to 𝑓 of three. So we will take our function and plug in three. And we get negative 13. So the vertex is three, negative 13.
So now we have to decide if it’s a minimum or maximum value. Since 𝑎 is a positive leading coefficient, our graph should open upward. So our vertex will be at the bottom which means it would be a minimum.
Therefore, the vertex again is three, negative 13. The value of the function at the vertex is negative 13. And it’s a minimum value.