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Problem 32 Hard Difficulty

(a) Using pencil and paper, not a graphing utility, determine the amplitude, period, and (where appropriate) phase shift for each function.
(b) Use a graphing utility to graph each function for two complete cycles. [In choosing an appropriate viewing rectangle, you will need to use the information obtained in part (a).]
(c) Use the graphing utility to estimate the coordinates of the highest and the lowest points on the graph.
(d) Use the information obtained in part (a) to specify the exact values for the coordinates that you estimated in part (c).
$$y=-2.5 \cos (3 x+4)$$

Answer

(a.) Amplitude $=2.5$
period $==\frac{2 \pi}{3}$
phase shift $=\frac{-4}{3}$
(b.) click for graph
(c.) the highest points are : $\mathrm{A}(1.8,2.5), \mathrm{F}(-0.3,2.5)$
the lowest points are : (0.76,-2.5),(-1.33,-2.5)
(d.) the highest points are : $\mathrm{A}(1.81,2.5), \mathrm{F}(-0.29,2.5)$
the lowest points are : (0.76,-2.5),(-1.33,-2.5),(2.86,-2.5)

Discussion

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Top Algebra Educators
Catherine R.

Missouri State University

Alayna H.

McMaster University

Kayleah T.

Harvey Mudd College

Michael J.

Idaho State University

Video Transcript

So we have the equation. Why is equal to negative 2.5 co sign off three X plus four. This equation can be rewritten as why is equal to negative 2.5 course sign three times X plus four or three. So for part, 1/4 thing really confused. Tell the amplitude that the aptitude is 2.5. We eat to find opinion. So the pian is two pi over three and then the feet shift is negative for were three. And for part B, you can see the graph right here, and you can see that from the graph that DPD is to pirate three can also see that the amplitude is to fight fight and you can see that the coordinates off the maximum point are one point into your eight comic and put five and the corners off the minimum point are, as you 0.761 common negative 2.5. So that's Ah, the quarters off the highest and lowest points. So for point for part C, you could just refer to the graph and you can get the coordinates now. What we would do is to say on the graph as well. Now, for part B, What we want to do is a party with you will find the exact corners on those points for the maximum and minimum. So we are going to start with the graph off. Why is equal to negative? 2.5 course. Sign off three X. So this will have a peon off two pi over three. So let's ah, craft this So the graph would look like this. Where the coordinates off this point here, This would be to pi over three and this would be pi over city. So there's a face shift and the next maximum would be here and this would be at high. So there's a face shift off 4/3. So what that means is that we will decrease each Xcor print by ah, 4/3. So when we take this first pirate three, it becomes negative. Use a red pen to draw this car. So when it shifted, then this gets shifted on to something like this. So now this gets shifted by four or three and this shifts get shifted by for over three. So the coordinates off the maximum point would be Hi minus 4/3 and the export wife order, It would be to put five and the corners off the minimum point. Would that be to pile over three minus four or earthy and the white ordinary quickie negative 2.5?

Other Schools
Top Algebra Educators
Catherine R.

Missouri State University

Alayna H.

McMaster University

Kayleah T.

Harvey Mudd College

Michael J.

Idaho State University