Video: Finding the Equation of a Curve From a Sketch

Video Transcript

Which of the following functions is graphed in the given figure? Is it A) 𝑦 equals negative 𝑥 minus seven cubed, B) 𝑦 equals 𝑥 plus seven cubed, C) 𝑦 equals negative 𝑥 plus seven cubed, D) 𝑦 equals 𝑥 minus seven cubed, or E) 𝑦 equals 𝑥 cubed plus seven?

The main feature marked on our graph is the 𝑥-intercept at 𝑥 equals seven. If we call this the graph of 𝑦 equals 𝑓 of 𝑥, then when 𝑥 is seven, 𝑦, which is 𝑓 of seven, is equal to zero. We can go through our options, eliminating those for which 𝑓 of seven is not zero.

So starting with option A, 𝑦 is negative 𝑥 minus seven cubed. So we try 𝑓 of 𝑥 equals negative 𝑥 minus seven cubed. Replacing 𝑥 by seven, we get that 𝑓 of seven is equal to negative seven minus seven cubed, which is just zero. So for option A, 𝑓 of seven is zero as required. Of course, this doesn’t mean that option A is right. It just means that we can’t rule it out yet.

We move on to option B, where 𝑓 of 𝑥 is 𝑥 plus seven cubed. And so 𝑓 of seven is seven plus seven cubed. This turns out to be 2744. That exact value is not important. What is important is that 𝑓 of seven does not equal zero as it should. For this reason, we can eliminate option B as a possibility.

It’s the same for option C. The only difference from option B is this minus sign. And putting this minus sign into the calculations on the right-hand side, we see that, for option C, 𝑓 of seven is equal to negative 2744 and again not zero. We can therefore also eliminate option C.

How about option D, where 𝑓 of 𝑥 is 𝑥 minus seven cubed? Well then 𝑓 of seven is seven minus seven cubed, which is zero. And this is what we need to be true. So we can’t eliminate option D.

And finally, for option E, 𝑓 of 𝑥 is 𝑥 cubed plus seven. And so 𝑓 of seven is seven cubed plus seven, which is not zero. Option E is also eliminated. Because we saw from the graph that 𝑓 of seven must be zero, we could eliminate options B, C, and E for which 𝑓 of seven wasn’t zero.

We’re left with two options: A) 𝑦 equals negative 𝑥 minus seven cubed and D) just 𝑦 equals 𝑥 minus seven cubed. Given that 𝑓 of seven is zero and all the options were polynomial functions, it’s not surprising that both remaining functions are functions which have factors of 𝑥 minus seven. And it’s not surprising to see that the exponent of this factor is three.

If the exponent were one, we’d have something which looks very much like a straight line through the 𝑥-intercept seven. And if the exponent were two, we’d get a curve which touched but didn’t cross the 𝑥-axis at the 𝑥-intercept seven. This would be true for any even exponent with the curve getting less and less like a parabola as the exponent increased.

What we actually see in this graph is something like an S-shape, which we associate with a cubic. But, actually, you will get a very similar shaped curve for any odd exponent greater than one.

Anyway, the exponents in options A and D are the same. In fact, the only difference between options A and D is this minus sign here. And so we’re going to have to use this minus sign to decide between these two options.

Looking at our graph, we can see that if 𝑥 one is less than or equal to 𝑥 two, then 𝑓 of 𝑥 one is less than or equal to 𝑓 of 𝑥 two. This is the definition of an increasing function. As 𝑥 increases, 𝑓 of 𝑥 increases. Is this true for the function in the first option?

Let’s try some values, choosing 𝑥 one to be seven and 𝑥 two to be eight. Then certainly 𝑥 one is less than or equal to 𝑥 two. We already know for this option that 𝑓 of seven is zero. Replacing 𝑥 by eight and then simplifying, we see that 𝑓 of eight is negative one. 𝑓 of seven is therefore not less than or equal to 𝑓 of eight. And so the function in option A is not increasing. We therefore eliminate option A and are left with only option D.

Another way to see why we must eliminate option A is to look at the graph and ask yourself, Does it make sense that 𝑓 of eight is equal to negative one? The graph is just a sketch. So we can’t read off 𝑓 of eight. But we can see that, for values of 𝑥 greater than seven, it looks like 𝑓 of 𝑥 is positive. And negative one certainly isn’t positive.

Having eliminated all the other options, our answer must be option D) 𝑦 equals 𝑥 minus seven cubed. You can check that, for this option, 𝑓 of eight is one, which is much more reasonable. Certainly, it’s greater than zero. And you can also check that this function is an increasing function. As 𝑥 increases, 𝑥 minus seven increases. And as 𝑥 minus seven increases, 𝑥 minus seven cubed increases.

If you know about graph transformations, you’ll know that the graph of 𝑦 equals 𝑥 minus seven cubed should be the graph of 𝑦 equals 𝑥 cubed translated seven units to the right. And this is indeed what we see in the figure.