Video: Using the Vertex Form of a Quadratic Function

Determine the quadratic function 𝑓 with the following properties: Its graph has a vertex at (3, βˆ’17). 𝑓(4) = 5.

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Video Transcript

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 π‘₯ equals π‘Ž times π‘₯ minus β„Ž squared plus π‘˜, where β„Ž, π‘˜ is the vertex, and π‘Ž is some constant value. Using this form 𝑓 of π‘₯ equals π‘Ž times π‘₯ minus β„Ž squared plus π‘˜, we can plug in three for β„Ž and negative 17 for π‘˜. Which gives us 𝑓 of π‘₯ equals π‘Ž times π‘₯ minus three squared plus negative 17. Plus negative 17 can be simplified to, say, minus 17.

But in order to find π‘Ž, we’ll need to use the point we were given. If 𝑓 of four equals five, then we plug in four for π‘₯ and five for 𝑓 of π‘₯. Four minus three is one. One squared is one. π‘Ž times one equals π‘Ž. So, we have five equals π‘Ž minus 17. And if we add 17 to both sides, we see that π‘Ž equals 22. And so, we can go back to our original equation and plug in 22 for π‘Ž. And we get the equation 𝑓 of π‘₯ equals 22 times π‘₯ minus three squared minus 17.

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