Trigonometry: How to determine d in f(θ)=a sinθ +d

[GuavaEater]

Hi there, I have the following question: the range of a trigonometric function f(θ)=a sinθ +d is 5.0 ≤ f(θ) ≤ 26.0 . Correct to the nearest tenth, the value of d is ______.

From what I understand, the y-intercept of sin functions is at the midpoint of a period. So I counted the number between the original minimum range value which should be 1, and the new minimum, 5.

I came to the conclusion that d = 6 based on this. Is this correct? If not, where should I start?

Thanks.

 

[ksdhart2]

Unfortunately, your answer is incorrect. How I'd approach the problem is to carefully reread the problem text and see what information I can extract from it. You're given that f(θ) is a variant of the sine function, and you know the basic shape and patterns involved in a sine function. Because the sine function is periodic, we can limit ourselves to only looking at the portion on [0, 2pi]. You're then also given the minimum and maximum values of f(θ). Because you're not adding or multiplying the argument of the sine function, the period and what we might call the "critical values" remain the same.

Putting it all together, the maximum of f(θ) will still occur at θ = pi/2, and the minimum at θ = 3pi/2. That gives two equations to solve: \(\displaystyle a \cdot sin \left( \dfrac{\pi}{2} \right) + d = 26\) and \(\displaystyle a \cdot sin \left( \dfrac{3\pi}{2} \right) + d = 5\). Can you finish up from here?

 

[MancPaul]

The way I look at it

a is the amplitude - so the function f(θ)=a sinθ ranges between -a ≤ f(θ) ≤ a

the d in the function f(θ)=a sinθ +d just translates the above from -a to 5 and from a to 26

so d must translate the value zero to the midpoint of 5 and 20 which is 12.5

Eh voila!! d must equal 12.5

 

[Harry_the_cat]

I'd consider what the a and the d do to the basic graph of y= sin(x). The a affects the amplitude and the d involves a vertical shift. Can you visualise the new graph sitting up between the y-values of 5 and 26? What is the new amplitude (that is a) and how far up from the x-axis has it moved (that is d)?

 

[ksdhart2]

a is the amplitude - so the function f(θ)=a sinθ ranges between -a ≤ f(θ) ≤ a

the d in the function f(θ)=a sinθ +d just translates the above from -a to 5 and from a to 26

so d must translate the value zero to the midpoint of 5 and 20 which is 12.5

Eh voila!! d must equal 12.5

That's a good strategy, and it does work out. The only problem is that you've made a typo so your final answer is incorrect.

Additionally, thank you for volunteering as a tutor on this website. However, although I appreciate your willingness to volunteer your time, in the future please do not give out fully worked solutions. On Free Math Help, we prefer to give the students hints and encourage them to learn and understand the material. I have found that giving solutions (regardless of whether or not steps and work are shown) often encourages students to take the easy way out and copy down the answer without understanding the rationale - then they find themselves stumped again on the very next problem which is the same but with different numbers, and must ask for help again.

 

[GuavaEater]

Retrying the question

I retried this with the advice given on here. What I did this time was I deduced that a + d = the maximum range, and that because this is a sine function, a will be the midpoint between 5 and 26. So I went 26-5 = 21, then divided that by 2, which gave me 10.5, which I believe to be my a value. With the previous knowledge that a + d = maximum range, I rearranged a(10.5) + d = max(26.0), into d = 26.0 - 10.5, which equaled 15.5, which should be my d value. I also could have went minimum value (5) + a(10.5) = d, as I'm adding my offset from the original midpoint (5) in to determine the vertical offset. If there's anything wrong with this way ,or if there's a better way let me know! Thanks!

 

[ksdhart2]

Yes, that's correct. Well done! If you're ever uncertain about an answer, you can always check it yourself by plugging your answer into the givens. In this case, if you graph the equation a sin(theta) + d, where a = 10.5 and d = 15.5, you'll see that the maximum is at 26 and the minimum is at 5. This meets the criteria the problem specified, so your answer is correct.