Question Video: Determining a Quadratic Equation in Vertex Form from its Graph Mathematics

Write the quadratic equation represented by the graph shown.

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Video Transcript

Write the quadratic equation represented by the graph shown.

We have the general form for a quadratic equation π of π₯ equals ππ₯ squared plus ππ₯ plus π and the vertex form π times π₯ minus β squared plus π. To find the equation, weβll consider some features of the graph. The shape of this graph opens downward. And that means we know that our π-value will be less than zero. It will be negative. We have a π¦-intercept at the point zero, two. When it comes to the roots, we canβt know with a great deal of accuracy what the roots are. So weβll leave them there for now. The vertex here is a maximum. And we do know where itβs located, at the point one, three.

Because we know the vertex, we can start with our vertex form. Our vertex is β, π. And so we plug in one for β and three for π. The only thing weβre missing now is this π-variable. We know that itβs less than zero, but we donβt know exactly what it is. To find it, we can plug in another point from the graph that we already know. If we plug in zero for π₯ and two for π of π₯, weβll be able to solve for our π-value. Zero minus one squared is one; one times π is π, which means π plus three equals two. And two minus three is negative one, so we can say that π equals negative one.

And weβll go back and plug that in. Instead of having negative one, we can just write the negative sign and say that π of π₯ equals negative π₯ minus one squared plus three. This is the vertex form of the graph we have. If we wanted to write this in the general form, we could expand this π₯ minus one squared, which would give us the negative of π₯ squared minus two π₯ plus one plus three. We distribute the negative. And when we combine like terms, we have the general form of negative π₯ squared plus two π₯ plus two. Both of these forms are the quadratic represented by this graph.