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

Write the quadratic equation represented by the graph shown. Give your answer in factored form.

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Video Transcript

Write the quadratic equation represented by the graph shown. Give your answer in factored form.

Let’s begin by examining the graph we’ve been given. We might first notice that the vertex or turning point of this graph has coordinates one, negative nine. This gives us some idea of what the completed square form equation of this graph might look like. An equation of the form π‘Ž π‘₯ plus π‘˜ squared plus β„Ž has a vertex negative π‘˜, β„Ž. So we let negative π‘˜ be equal to one and β„Ž be equal to negative nine. And we see that the equation of our graph is 𝑦 equals some constant π‘Ž times π‘₯ minus one all squared minus nine.

So how do we find the value of π‘Ž? Well, in fact, we can choose the coordinate of any point that lies on this curve and substitute that in. A really straightforward one is the coordinate four, zero. The π‘₯-coordinate is four, and the 𝑦-coordinate is zero. And so our equation becomes zero equals π‘Ž times four minus one squared minus nine. Well, four minus one squared is three squared, which is nine. So our equation becomes zero equals nine π‘Ž minus nine. We add nine to both sides of this equation. And finally, we’ll divide through by nine. And when we do, we find that π‘Ž is equal to one. Substituting this back into the equation π‘Ž times π‘₯ minus one all squared minus nine, and we find that the equation of this quadratic is 𝑦 equals π‘₯ minus one squared minus nine.

Now, in fact, we’re told to give this in factored form. So what next? Well, we’re simply going to distribute the parentheses, simplify, and then factor. π‘₯ minus one all squared is π‘₯ minus one times π‘₯ minus one. Distributing the parentheses, and we get π‘₯ squared minus π‘₯ minus π‘₯ plus one. And so, our equation becomes 𝑦 equals π‘₯ squared minus two π‘₯ minus eight. To factor this, we simply find a pair of numbers that have a product of negative eight and sum to make negative two. That’s negative four and two. And so, in factored form, the quadratic equation represented by the graph shown is 𝑦 equals π‘₯ minus four times π‘₯ plus two.

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