Question Video: Deciding Whether a Particular Graphical Property Implies That a Function Is a Quadratic Mathematics • Third Year of Preparatory School

Video Transcript

If the equation 𝑓 of 𝑥 is equal to zero has two solutions, does this mean that 𝑓 is a quadratic function that intercepts the 𝑥-axis at two points?

To answer this question, we first note that if the equation 𝑓 of 𝑥 is equal to zero has two solutions 𝑠 sub one and 𝑠 sub two, then its solution set 𝑆 has two elements. And this is where 𝑠 sub one and 𝑠 sub two are distinct real numbers. We also know that for a quadratic function of the form 𝑓 of 𝑥 is equal to 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, if the equation 𝑓 of 𝑥 is equal to zero has two solutions 𝑥 is equal to 𝑠 sub one and 𝑥 is 𝑠 sub two, then 𝑠 sub one and 𝑠 sub two are two distinct roots of the function 𝑓 of 𝑥. And let’s consider what this means graphically.

We know that the graph of a quadratic function is a parabola, that is, a curve that’s either U shaped or n shaped. And it’s symmetric about a vertical line. The sign of the leading coefficient 𝑎 — that’s the coefficient of 𝑥 squared — tells us whether the curve is going to be U shaped or n shaped. If 𝑎 is less than zero, the curve is n shaped as shown. If, on the other hand, 𝑎 is greater than zero as shown in the second graph, the curve is U shaped.

Recall also that the number of solutions to the equation 𝑓 of 𝑥 is equal to zero tells us whether and if so at how many points the curve intercepts the 𝑥-axis. If there are two distinct solutions 𝑠 sub one and 𝑠 sub two, then the function has two distinct roots and the curve intercepts and in this case crosses the 𝑥-axis twice. If there’s one repeated solution so that 𝑓 of 𝑥 is equal to 𝑘 multiplied by 𝑥 minus 𝑠 sub one squared, where 𝑘 is nonzero, then the graph intercepts the 𝑥-axis only once, touching it at 𝑥 is equal to 𝑠 sub one. In this case, we say that 𝑠 sub one is a repeated root.

The solution set 𝑠 then has only one element and that’s 𝑠 sub one. And if there are no real solutions to 𝑓 of 𝑥 is equal to zero, then the graph does not intercept the 𝑥-axis and we say that the solution set is the empty set.

Okay, so now making some space, we’ve established in fact that if 𝑓 of 𝑥 is a quadratic function whose graph intercepts the 𝑥-axis at two points, then the equation 𝑓 of 𝑥 is equal to zero has two real solutions. But this does not quite match the given statement in the question. In fact, the question statement is the reverse of this. Let’s read this again. This says that if 𝑓 of 𝑥 is equal to zero has two solutions, does this mean that 𝑓 is a quadratic function that intersects the 𝑥-axis at two points? In fact, it turns out that there are other functions 𝑓 of 𝑥 that are not quadratics, but for which there are exactly two solutions to 𝑓 of 𝑥 is equal to zero.

We can see some examples of these as shown. We know that a quadratic has exactly one turning point. However, we see that while each of the graphs of the functions shown intercept the 𝑥-axis at two points, they each have more than one turning point and so they cannot be quadratic functions. And hence, the fact that the equation 𝑓 of 𝑥 is equal to zero has two solutions for a function 𝑓 of 𝑥 does not necessarily mean that 𝑓 of 𝑥 is a quadratic function. Our answer is therefore no, 𝑓 of 𝑥 may not necessarily be a quadratic function.