# Question Video: Determining a Quadratic Equation from its Graph Mathematics

Write the quadratic equation represented by the graph shown.

02:30

### Video Transcript

Write the quadratic equation represented by the graph shown.

We’re told that this is going to be a quadratic equation. And in fact, observing the general shape of the graph, it’s a parabola. So, we know it’s definitely the graph of a quadratic function. This is of the form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 is not equal to zero.

Now, in fact, our job is to identify the values of 𝑎, 𝑏, and 𝑐. And we can quite quickly determine the value of 𝑏. The function of the form 𝑎𝑥 squared plus 𝑐, or indeed 𝑘𝑥 squared plus 𝑐, has a line of symmetry 𝑥 equals zero; it’s symmetrical about the 𝑦-axis. We see our graph is indeed symmetrical about the 𝑦-axis, so we can deduce that the value of 𝑏 must be equal to zero. And so, our function is of the form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑐.

And next, we can use information about its 𝑦-intercept to work out the value of 𝑐. Given a function of the form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑐, its 𝑦-intercept is zero, 𝑐. Now, our 𝑦-intercept is zero, two. It passes through the 𝑦-axis at two. And so, 𝑐 must be equal to two, meaning we can write our function as 𝑓 of 𝑥 equals 𝑎𝑥 squared plus two.

But how do we find the value of 𝑎? Well, we can choose a point that lies on the given curve and substitute the 𝑥- and 𝑦-values of this coordinate into the function. Let’s choose the point with coordinates two, negative two. As an ordered pair, this is 𝑥, 𝑦 or 𝑥, 𝑓 of 𝑥. And so, 𝑓 of 𝑥 equals 𝑎𝑥 squared plus two becomes negative two equals 𝑎 times two squared plus two. Two squared is four. So, the equation becomes negative two equals four 𝑎 plus two. Then subtracting two from both sides of this equation, we get negative four equals four 𝑎. And then we divide through by four, giving us 𝑎 is equal to negative one.

And so, we could write our function as 𝑓 of 𝑥 equals negative one 𝑥 squared plus two, or we could write it as 𝑦 equals negative 𝑥 squared plus two. And in fact, since the parabola in our diagram is n shaped, we were expecting the coefficient of 𝑥 squared to be negative. The equation of the curve is 𝑦 equals negative 𝑥 squared plus two.