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

Write the quadratic equation represented by the graph shown.


Video Transcript

Write the quadratic equation represented by the graph shown.

We have the general form for a quadratic equation 𝑓 of π‘₯ equals π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 and the vertex form π‘Ž times π‘₯ minus β„Ž squared plus π‘˜. To find the equation, we’ll consider some features of the graph. The shape of this graph opens downward. And that means we know that our π‘Ž-value will be less than zero. It will be negative. We have a 𝑦-intercept at the point zero, two. When it comes to the roots, we can’t know with a great deal of accuracy what the roots are. So we’ll leave them there for now. The vertex here is a maximum. And we do know where it’s located, at the point one, three.

Because we know the vertex, we can start with our vertex form. Our vertex is β„Ž, π‘˜. And so we plug in one for β„Ž and three for π‘˜. The only thing we’re missing now is this π‘Ž-variable. We know that it’s less than zero, but we don’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 𝑓 of π‘₯, we’ll be able to solve for our π‘Ž-value. Zero minus one squared is one; one times π‘Ž is π‘Ž, which means π‘Ž plus three equals two. And two minus three is negative one, so we can say that π‘Ž equals negative one.

And we’ll go back and plug that in. Instead of having negative one, we can just write the negative sign and say that 𝑓 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.

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