Determine the quadratic function 𝑓 with the following properties. Its graph has a vertex at three, negative 17. And 𝑓 of four equals five.
We’ve been given a vertex and a solution point, which means we should think about the vertex form of quadratic functions. Which is 𝑓 of 𝑥 equals 𝑎 times 𝑥 minus ℎ squared plus 𝑘, where ℎ, 𝑘 is the vertex, and 𝑎 is some constant value. Using this form 𝑓 of 𝑥 equals 𝑎 times 𝑥 minus ℎ squared plus 𝑘, we can plug in three for ℎ and negative 17 for 𝑘. Which gives us 𝑓 of 𝑥 equals 𝑎 times 𝑥 minus three squared plus negative 17. Plus negative 17 can be simplified to, say, minus 17.
But in order to find 𝑎, we’ll need to use the point we were given. If 𝑓 of four equals five, then we plug in four for 𝑥 and five for 𝑓 of 𝑥. Four minus three is one. One squared is one. 𝑎 times one equals 𝑎. So, we have five equals 𝑎 minus 17. And if we add 17 to both sides, we see that 𝑎 equals 22. And so, we can go back to our original equation and plug in 22 for 𝑎. And we get the equation 𝑓 of 𝑥 equals 22 times 𝑥 minus three squared minus 17.