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 π‘₯ equals π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐, where π‘Ž, 𝑏, and 𝑐 are real numbers and π‘Ž is not equal to zero.

Now, in fact, our job is to identify the values of π‘Ž, 𝑏, and 𝑐. And we can quite quickly determine the value of 𝑏. The function of the form π‘Žπ‘₯ squared plus 𝑐, or indeed π‘˜π‘₯ squared plus 𝑐, has a line of symmetry π‘₯ 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 π‘₯ equals π‘Žπ‘₯ 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 π‘₯ equals π‘Žπ‘₯ 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 π‘₯ equals π‘Žπ‘₯ squared plus two.

But how do we find the value of π‘Ž? Well, we can choose a point that lies on the given curve and substitute the π‘₯- 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 π‘₯, 𝑦 or π‘₯, 𝑓 of π‘₯. And so, 𝑓 of π‘₯ equals π‘Žπ‘₯ squared plus two becomes negative two equals π‘Ž times two squared plus two. Two squared is four. So, the equation becomes negative two equals four π‘Ž plus two. Then subtracting two from both sides of this equation, we get negative four equals four π‘Ž. And then we divide through by four, giving us π‘Ž is equal to negative one.

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

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