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

Video Transcript

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

To do this, let’s consider some features of the graph of quadratic functions like shape, 𝑦-intercept, roots, and vertex. The shape of this parabola opens upward. And that tells us that our 𝑎-value will be positive, greater than zero. We have a 𝑦-intercept located at the point zero, zero. To find the roots, we look for the place where the graph crosses the 𝑥-axis. Here we have a root at zero, zero and at five, zero. The vertex of this graph has been labeled. It’s a minimum coordinate located at two and a half, negative 6.25.

What we wanna do now is think a little bit more about the roots. The roots are sometimes called solutions. They’re the place where our function equals zero. We have solutions when 𝑥 equals zero and when 𝑥 equals five. We can take these solutions and turn them into factors. Because 𝑥 is the same thing as 𝑥 minus zero, 𝑥 equals zero is a factor. And for factor two, we would want to say 𝑥 minus five equals zero. Here are two factors. However, we still don’t know what our 𝑎-value is because there are many equations that have the factors 𝑥 and 𝑥 minus five. And that means we should have 𝑓 of 𝑥 equals 𝑎 times 𝑥 times 𝑥 minus five.

To solve for 𝑎, we can plug in coordinates of points we know fall on this graph. We know that when 𝑥 equals 2.5, 𝑓 of 𝑥 equals negative 6.25. Two and a half minus five is negative two and a half. Two and a half times negative two and a half equals negative 6.25. And if we divide both sides of the equation by negative 6.25, we see that 𝑎 equals one. And if 𝑎 equals one, then our factored form is 𝑓 of 𝑥 equals 𝑥 times 𝑥 minus five. We always want to check for this 𝑎-value because as I’ve just drawn on top of this graph — here’s an equation that has the same factors. However, in this case, the 𝑎 would have to be less than zero because the graph opens downward, which means it’s not enough just to know the factors. You need to also check for the 𝑎-value.