Question Video: Finding the Equation of a Curve From a Sketch Mathematics • 10th Grade

Which of the following functions is graphed in the given figure? [A] π¦ = β(π₯ β7)Β³ [B] π¦ = (π₯ + 7)Β³ [C] π¦ = β(π₯ + 7)Β³ [D] π¦ = (π₯ β 7)Β³ [E] π¦ = π₯Β³ + 7?

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Video Transcript

Which of the following functions is graphed in the given figure? Is it A) π¦ equals negative π₯ minus seven cubed, B) π¦ equals π₯ plus seven cubed, C) π¦ equals negative π₯ plus seven cubed, D) π¦ equals π₯ minus seven cubed, or E) π¦ equals π₯ cubed plus seven?

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

So starting with option A, π¦ is negative π₯ minus seven cubed. So we try π of π₯ equals negative π₯ minus seven cubed. Replacing π₯ by seven, we get that π of seven is equal to negative seven minus seven cubed, which is just zero. So for option A, π of seven is zero as required. Of course, this doesnβt mean that option A is right. It just means that we canβt rule it out yet.

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

Itβ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, π of seven is equal to negative 2744 and again not zero. We can therefore also eliminate option C.

How about option D, where π of π₯ is π₯ minus seven cubed? Well then π of seven is seven minus seven cubed, which is zero. And this is what we need to be true. So we canβt eliminate option D.

And finally, for option E, π of π₯ is π₯ cubed plus seven. And so π of seven is seven cubed plus seven, which is not zero. Option E is also eliminated. Because we saw from the graph that π of seven must be zero, we could eliminate options B, C, and E for which π of seven wasnβt zero.

Weβre left with two options: A) π¦ equals negative π₯ minus seven cubed and D) just π¦ equals π₯ minus seven cubed. Given that π of seven is zero and all the options were polynomial functions, itβs not surprising that both remaining functions are functions which have factors of π₯ minus seven. And itβs not surprising to see that the exponent of this factor is three.

If the exponent were one, weβd have something which looks very much like a straight line through the π₯-intercept seven. And if the exponent were two, weβd get a curve which touched but didnβt cross the π₯-axis at the π₯-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 weβre going to have to use this minus sign to decide between these two options.

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

Letβs try some values, choosing π₯ one to be seven and π₯ two to be eight. Then certainly π₯ one is less than or equal to π₯ two. We already know for this option that π of seven is zero. Replacing π₯ by eight and then simplifying, we see that π of eight is negative one. π of seven is therefore not less than or equal to π 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 π of eight is equal to negative one? The graph is just a sketch. So we canβt read off π of eight. But we can see that, for values of π₯ greater than seven, it looks like π of π₯ is positive. And negative one certainly isnβt positive.

Having eliminated all the other options, our answer must be option D) π¦ equals π₯ minus seven cubed. You can check that, for this option, π of eight is one, which is much more reasonable. Certainly, itβs greater than zero. And you can also check that this function is an increasing function. As π₯ increases, π₯ minus seven increases. And as π₯ minus seven increases, π₯ minus seven cubed increases.

If you know about graph transformations, youβll know that the graph of π¦ equals π₯ minus seven cubed should be the graph of π¦ equals π₯ cubed translated seven units to the right. And this is indeed what we see in the figure.