Question 1078505
the general formula is:


y = acos(b(x-c))+d


a is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift.


your equation is y = cos(pi * x) - 2


shift it up 3 units and it becomes y = cos(pi * x) + 1


shift it to the left 1 unit and it becomes y = cos(pi * (x+1) + 1


stretch it vertically by a factor of 2 and it becomes y = 2 * cos(pi * (x+1)) + 1


my first graph is the original function.


<img src = "http://theo.x10hosting.com/2017/042701.jpg" alt="$$$" </>


my second graph shifts the function from the first graph up by 3 units.


<img src = "http://theo.x10hosting.com/2017/042702.jpg" alt="$$$" </>


my third graph shifts the function from the second graph to the left 1 unit.


<img src = "http://theo.x10hosting.com/2017/042703.jpg" alt="$$$" </>


my fourth graph vertically expands the third graph by a factor of 2.


<img src = "http://theo.x10hosting.com/2017/042704.jpg" alt="$$$" </>


the fourth graph is your solution.


that looks like your selection d.


note that:


frequency is equal to 2pi / period.


since the frequency of your first graph is pi radians, then the period is 2pi/pi = 2 radians.


you get one complete cycle of the cosine wave in 2 radians as shown on the graph.


in the first graph:


the horizontal range goes from x = 0 to x = 2
the center line is y = -2
the vertical range goes from y = -1 to y = -3


in the second graph:


the horizontal range remains at x = 0 to x = 2
the center line moves up 3 units to y = 1
the vertical range becomes y = 2 to y = 0


this is because the graph has been shifted up 3 units.


in the third graph:


the horizontal range becomes x = -1 to x = 1
the center line remains at y = 1
the vertical range remains at y = 2 to y = 0


this is because the graph has been shifted to the left 1 unit.


in the fourth graph:


the horizontal range remains at x = -1 to x = 1
the center line remains at y = 1
the vertical range becomes y = 3 to y = -1


this is because the vertical range has been multiplied by a factor of 2.


here's a reference that looks pretty complete.


<a href= "http://www.mathguide.com/lessons2/GraphingTrig.html" target = "_blank">http://www.mathguide.com/lessons2/GraphingTrig.html</a>