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# (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).$$y=-2.5 \cos \left(\frac{1}{3} x+4\right)$$

## the lowest points are : $\mathrm{A}(-12,-2.5), \mathrm{B}(6.85,-2.5), \mathrm{C}(25.7,-2.5)$

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have the equation. Why is equal to negative 2.5 co sign off 1/3 X plus four. This can be rewritten as why is equal to negative 2.5 co sign off one orthe e X plus 12. And from here for part A. What we can do is we confined the amplitude, which is a complaint. Five. Then we can find a period, which is two pi invited by one of earthy, which comes equal to six pipe, and then the phase shift is negative. 12. So, for part B, you can see that the graph is drawn here and the graph shows at the Peter de 65 job also shows that the amplitude is 2.5, and on the graph, you can also see the corners of the maximum and minimum points. The maximum point has approximate coordinates 16.274 and 2.5, and the minimum has approximate coordinates equal to 6.85 and negative 2.5. Now let's find the exact values off these maximum and minimum points. So for partly, we start with the graph or why is equal to negative 2.5 co signed off 1/3 X. So what this craft will look like would be something like this. So the first car minimum since the period is six pie. So the first minimum after zero occurs at six by and the first maximum occurs at three pipe, and that means that the next maximum would occur at nine pie. Now, if we want to do, why is equal to negative 2.5 course sign off 1/3 X plus 12 than what we need to do is we need to take each of these peaks and lowest points, and we need to move them to the left by an amount equal to 12 units. So each X coordinates gets moved to the left by 12 units. So this 1st 1 will actually become negative. So we're not interested in this. This would then become a 12. This would become six by minus 12 and this one would become nine by minus 12. So let's right the coordinates. So for the maximum point, the corners would be nine high minus 12 comma 2.5, and for the minimum point, the coordinates would be in a six by minus 12 comma. Negative. Complete fuck. So those would be the exact coordinates off the maximum and minimum points.

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