The Graphs of Sine and Cosine - University High School
The Graphs of Sine, Cosine, and Tangent
Sketch the graphs of y = sin(x) and y = cos (x) on the following axes. Set your calculator to radians and set the zoom to ZOOMTRIG. We are currently only interested from 0 to 2[pic].
Label the points of interest on your graphs—x-intercepts, y-intercepts, maximums and minimums. This can be accomplished by using the unit circle (You will need a decimal approximation for the radians to see the value on your calculator).
Example: The calculator shows an x-intercept of (3.141592, 0). You notice that the [pic]= 0. So on the graph, a point should be (3.141592, 0) and it is. Label your points in terms of Pi.
[pic][pic]
In general, what will happen to the graph of a function if it is multiplied by a positive constant? Ex. y = f(x) will compare how to [pic]?
____________________________________________________________________________
____________________________________________________________________________
Considering what you know already, the graph of [pic]will compare how to the graph of [pic] when a > 0 ?____________________________________________________
Will all trig functions behave similarly?_________________________________________
Graph the following and label all points of interest: (in terms of Pi) (On next page)
1. [pic]
2. [pic]
[pic][pic]
Describe how the graph behavior for [pic]in comparison to [pic]?
____________________________________________________________________________
____________________________________________________________________________
Graph to verify your conjecture: Label all points of interest
The number out in front of the trig functions seems to determine the _________________
of the function.
If the coefficient out in front is negative the graph is a ___________________________
over the ____________________________
The amplitude of the sine and cosine functions {[pic];[pic]} is the largest value of y (output) that is produced for all x (inputs).
Investigate: Give the amplitude of the functions sine or cosine in terms of a. What is the amplitude of the following functions?
1. [pic]
2. [pic]
Explain what you notice about the relationship between amplitude and the sign (not sine)of a: ______________________________________________________________________
________________________________________________________________________
WHY? Give a clear reason: __________________________________________________
_______________________________________________________________________
Other Types of Stretching
Compare the graph of [pic]and the graph of [pic].
Use the same axes label the points of interest in terms of Pi.
*** Note the scale on what has been provided has changed.***
Based on the results of the last question how will the graph of [pic] compare to
y =cos(x) Graph both below. Label points of interest.
The period of a trig function is the angle measure after which the graph begins to repeat.
We have already investigated graphs that shift horizontally. Recall the graph of y = f(x-2) will be shifted 2 units right when compared to y =f(x).
Therefore, since f was any function the same will follow for trig functions.
Graph [pic] and [pic]
Would this always be the case? We have looked at trig functions that we were able to stretch out or shrink. What if we were to combine a horizontal shift with a horizontal stretch?
Consider [pic] You know that the graph will be
shifted___________________& stretched_________________________
Graph the y = cos (x). How far do you need to move the maximum point say at (0,1) until it matches up with the new graph?________________________________________________
This is called the phase shift. Given the following [pic] Determine the
1. amplitude __________________
2. period_____________________
3. phase shift__________________
TANGENT
What is the tan[pic]?_______________________
What should be observed in the graph of tangent at the x-value of [pic]?_______________________
Where else will these occur from [pic]?____________________________
Sketch the graph of tangent of x. on the interval [pic]
[pic]
What is the amplitude of tan(x)? _______________________________________
What is the period of tan(x)? __________________________________________
What is the midpoint between any two consecutive asymptotes?______________
Complete the following Table
x-value |[pic] |-1.5707 |-1.57 |-1.4 |-1.1 |0 |1.1 |1.4 |1.57 |1.5707 |[pic] | |Tan(x) | | | | | | | | | | | | |
Describe the behavior of the tangent curve as x approaches [pic] and [pic].
________________________________________________________________________________________________________________________________________________
Based on what we know of period, what will be the period of the function [pic] ? ______________________________
Verify Graphically.
[pic]
If we graph [pic], what will be the difference if any? ________________________________________________________________________
What if we graph [pic], conjecture about the graph. ________________________________________________________________________
Plot both along with the tan (x) .
[pic]
Were your thoughts correct?_________________________________________________
The interval for tangent is usually written [pic] < x < [pic]. Without graphing where will the vertical asymptotes of y = tan (.5x) hint: solve an inequality ____________________________________________________________
What is the period?___________________________________________________
-----------------------
[pic]
[pic]
Write a concise summary describing the period of the trig functions of the form [pic] or [pic] where a>0
____________________________________________________________________________________________________________________________________
[pic]
[pic]
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