Question Video: Finding the Equation of a Curve From a Sketch | Nagwa Question Video: Finding the Equation of a Curve From a Sketch | Nagwa

Question Video: Finding the Equation of a Curve From a Sketch Mathematics

Which of the following functions is graphed in the given figure? [A] 𝑦 = βˆ’(π‘₯ βˆ’7)Β³ [B] 𝑦 = (π‘₯ + 7)Β³ [C] 𝑦 = βˆ’(π‘₯ + 7)Β³ [D] 𝑦 = (π‘₯ βˆ’ 7)Β³ [E] 𝑦 = π‘₯Β³ + 7?

05:47

Video Transcript

Which of the following functions is graphed in the given figure? Is it A) 𝑦 equals negative π‘₯ minus seven cubed, B) 𝑦 equals π‘₯ plus seven cubed, C) 𝑦 equals negative π‘₯ plus seven cubed, D) 𝑦 equals π‘₯ minus seven cubed, or E) 𝑦 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 𝑦 equals 𝑓 of π‘₯, then when π‘₯ 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 π‘₯ minus seven cubed. So we try 𝑓 of π‘₯ equals negative π‘₯ minus seven cubed. Replacing π‘₯ 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 π‘₯ is π‘₯ 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 π‘₯ is π‘₯ 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 π‘₯ is π‘₯ 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 π‘₯ minus seven cubed and D) just 𝑦 equals π‘₯ 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 π‘₯ 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 π‘₯-intercept seven. And if the exponent were two, we’d get a curve which touched but didn’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 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 π‘₯ one is less than or equal to π‘₯ two, then 𝑓 of π‘₯ one is less than or equal to 𝑓 of π‘₯ two. This is the definition of an increasing function. As π‘₯ increases, 𝑓 of π‘₯ increases. Is this true for the function in the first option?

Let’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 𝑓 of seven is zero. Replacing π‘₯ 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 π‘₯ greater than seven, it looks like 𝑓 of π‘₯ is positive. And negative one certainly isn’t positive.

Having eliminated all the other options, our answer must be option D) 𝑦 equals π‘₯ 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 π‘₯ increases, π‘₯ minus seven increases. And as π‘₯ minus seven increases, π‘₯ minus seven cubed increases.

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

Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

  • Interactive Sessions
  • Chat & Messaging
  • Realistic Exam Questions

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy