Question Video: Identifying the Rule of a Quadratic Function from Its Graph Mathematics • Third Year of Preparatory School

Video Transcript

Which of the following is the equation of the function drawn on the graph? (a) 𝑓 of 𝑥 equals negative 𝑥 squared plus eight. (b) 𝑓 of 𝑥 equals 𝑥 squared minus eight. (c) 𝑓 of 𝑥 equals negative 𝑥 squared minus eight. (d) 𝑓 of 𝑥 equals 𝑥 squared plus eight. Or (e) 𝑓 of 𝑥 equals 𝑥 minus eight.

Let’s begin by identifying some key features of the function drawn on the graph. First, we observe that it is a parabola, and in fact it is a positive or U-shaped parabola which opens upwards. This tells us that the function drawn on the graph is a quadratic function. All parabolas are symmetric with a vertical line of symmetry through their vertex. And we observe that the line of symmetry for this function is the line 𝑥 equals zero or the 𝑦-axis. Together These two pieces of information tell us that the function drawn on the graph has an equation of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared plus 𝑐. And what we need to do is determine the values of the constants 𝑘 and 𝑐.

Now the value of 𝑐 is the 𝑦-intercept of the graph. We know this because the 𝑦-intercept occurs when 𝑥 is equal to zero. And if we are to evaluate the function 𝑓 of zero, we obtain 𝑘 multiplied by zero squared plus 𝑐, which is simply 𝑐. From the graph, we can see that the 𝑦-intercept of this function is negative eight, so the value of 𝑐 is negative eight. We therefore have that 𝑓 of 𝑥 is equal to 𝑘𝑥 squared minus eight. But we need to determine the value of 𝑘. To do this, we can use the coordinates of any other point on the curve.

So let’s consider the point with coordinates three, one. In other words, when 𝑥 is equal to three, 𝑓 of 𝑥 must be equal to one. Substituting 𝑥 is equal to three and 𝑓 of 𝑥 is equal to one into our equation, we have one is equal to 𝑘 multiplied by three squared minus eight. That simplifies to one is equal to nine 𝑘 minus eight. And then we add eight to each side of the equation to give nine is equal to nine 𝑘. By dividing both sides of this equation by nine, it follows that 𝑘 is equal to one.

So we know that 𝑓 of 𝑥 is of the form 𝑘𝑥 squared plus 𝑐. And we found that 𝑘 is equal to one and 𝑐 is equal to negative eight. So the equation of the function drawn on the graph is 𝑓 of 𝑥 equals 𝑥 squared minus eight, which is option (b).

Let’s just briefly consider the four other options we were given. Looking at option (e) 𝑓 of 𝑥 equals 𝑥 minus eight, this represents the linear function. So the graph of this function would be a straight line. And so we know straight away that this one is not the correct function. Looking at options (a) and (d), the value of 𝑐, the constant term in each function, is positive eight. And so these two functions would each have a 𝑦-intercept of positive, not negative, eight. Looking at option (c), this does have the correct 𝑦-intercept, but the coefficient of 𝑥 squared is negative one when it should be positive one. In fact, this function would be a parabola with a 𝑦-intercept of negative eight, but as the coefficient of 𝑥 squared is negative, it would be a parabola that opens downwards.

We can of course check our answer by picking any other point on the graph, for example, the point with coordinates negative four, eight. Substituting 𝑥 equals negative four into function (b), we have 𝑓 of negative four equals negative four squared minus eight. That’s 16 minus eight, which is equal to eight. And so this confirms that our answer is correct. The equation of the function drawn on the graph is 𝑓 of 𝑥 equals 𝑥 squared minus eight.