Question Video: Determining a Quadratic Equation from its Graph Mathematics • Second Year of Secondary School

Video Transcript

Write the quadratic equation represented by the graph shown.

In order to write this equation, let’s consider some features of the graph of quadratic functions, shape, 𝑦-intercept, the roots, and the vertex. We also have the general form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 or the vertex form 𝑓 of 𝑥 equals 𝑎 times 𝑥 minus ℎ squared plus 𝑘. Considering the shape, we see that this parabola opens downward. And that means the 𝑎-value will be less than zero. It will be negative. The 𝑦-intercept is located here at zero, zero, which mean that the roots here will just be one root at zero, zero. And it happens to be the vertex, which is the maximum point of this function.

If we use vertex form, we know that the vertex is found at the point ℎ, 𝑘. And so we can take this vertex of zero, zero and plug it into that general vertex form. When we simplify that, we find out that the function is 𝑎 times 𝑥 squared. And that means we need to know what 𝑎 is. We know that the 𝑎-value is negative. But in order to find what it exactly is, we need to consider another point from the graph. We could use one of the other points we know from the graph. For example, we know that the graph crosses the point two, negative four. So we plug in two for 𝑥 and negative four for 𝑓 of 𝑥. And then we have negative four equals four 𝑎.

From there, we divide both sides of the equation by four. And we see that negative one equals 𝑎 or 𝑎 equals negative one. And we plug that value back in for 𝑎, which gives us negative one times 𝑥 squared. And we can simplify that to just be negative 𝑥 squared. So we found the quadratic equation represented by the graph shown to be 𝑓 of 𝑥 equals negative 𝑥 squared.