Graphing Sine and Cosine Answers
Identify the transformations needed to graph the cosine function y = –0.5cos(x) – 3 from the parent cosine function. Check all that apply.
d = ⇒ 1
e =
f =
The maximum height reached by the barnacleis m.
PENDING
The boat is traveling at a rate of 1 meter per second.How long does it take the barnacle to get back to its starting point?
a = ,
The data from the previous table is plotted to the right. The distance the boat traveled is on the x-axis, and the vertical position of the barnacle is on the y-axis.The graph follows the pattern.
This is the pattern for a function.
Question text not available
When the graph is below the x-axis, the barnacle is underwater. The graph dips below the x-axis three times between x = 0 and x = 20, so the barnacle goes underwater three times. Every time the barnacle goes underwater, the boat travels meters (one half the circumference of the wheel) before the barnacle comes back up, so the barnacle covers 3 meters underwater between x = 0 and x = 20.
Find the following (take ground level to be 0):Gum’s minimum height: ft Gum’s maximum height: ft
a = [___], b = [___], c = [___], d = [___], e = [___]
1, 3, -2, -12, -8
Graph the following points on the graphing calculator:Explain how to use the graph to write an equation to model the gum’s height. Be sure to identify the pattern of the points in your explanation, and identify the values of a and k.
The pattern is min-zero-max-zero-min, which is the pattern for a cosine function of the form y = acos(x) + k, but is reflected over the x-axis (so a < 0). The amplitude is |a|, so a = –1 and |a| = 1. The midline is exactly between the max and the min, y = (6 + 8)/2 = 7, so k = 7. The equation is y = –cos(x) + 7.
Graph the function y = –cos(x) + 7 and explain how to find the number of times the gum returns to the wall as it travels a distance of 60 feet.
The graph of y = –cos(x) + 7 is a cosine wave reflected across the x-axis with a vertical shift of 7 units. The minimum value of the function is 6 (found at y = -1 + 7), which represents the height of the wall. To find the number of times the gum returns to the wall, count the number of minimum points on the graph for the interval 0 < x ≤ 60. Since the period is 2π ≈ 6.28, you can divide the total distance by the period (60 / 6.28 ≈ 9.55). There are 9 full cycles, meaning the gum returns to the wall 9 times.
b = ,
When the boat has traveled 10 meters, the height of the barnacle is approximately:
Check each of the following that you included in your explanation.When the graph is below the x-axis, the barnacle is underwater.The graph dips below the x-axis three times between x = 0 and x = 20, so the barnacle goes underwater three times.Every time the barnacle goes underwater, the boat travels meters (one half the circumference of the wheel) before the barnacle comes back up, so the barnacle covers 3 meters underwater between x = 0 and x = 20.
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