Graphs of Secant, Cosecant, Tangent, and Cotangent

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These are the problems that come from Homework Five and are focused around understanding the graphs of secant, cosecant, tangent, and cotangent.

The first problems are about gathering information about the function if you are given an explicit formula. You should definitely make sure you can also produce a graph of one period of each of these functions and check it against something like Desmos.

1 :

If $f(x)=\sage {p1disp}$ find each of the following:

$\text {Amplitude:}=\answer {\sage {p1ans1}}$ $\text {Period:}=\answer {\sage {p1ans2}}$ $\text {Frequency:}=\answer {\sage {p1ans3}}$ $\text {Phase Shift:}=\answer {\sage {p1ans4}}$

2 :

If $f(x)=\sage {p2disp}$ find each of the following:

$\text {Amplitude:}=\answer {\sage {p2ans1}}$ $\text {Period:}=\answer {\sage {p2ans2}}$ $\text {Frequency:}=\answer {\sage {p2ans3}}$ $\text {Phase Shift:}=\answer {\sage {p2ans4}}$

3 :

If $f(x)=\sage {p3disp}$ find each of the following:

$\text {Amplitude:}=\answer {\sage {p3ans1}}$ $\text {Period:}=\answer {\sage {p3ans2}}$ $\text {Frequency:}=\answer {\sage {p3ans3}}$ $\text {Phase Shift:}=\answer {\sage {p3ans4}}$

4 :

If $f(x)=\sage {p4disp}$ find each of the following:

$\text {Amplitude:}=\answer {\sage {p4ans1}}$ $\text {Period:}=\answer {\sage {p4ans2}}$ $\text {Frequency:}=\answer {\sage {p4ans3}}$ $\text {Phase Shift:}=\answer {\sage {p4ans4}}$

These problems are about extracting most of those same features from the graph itself. (Note that I say most. There is a change between the two sets of problems and there is also a reason that this swap was made.)

5 :

Find the amplitude, period, frequency, and vertical shift of the function whose graph is given below. If $a=\sage {p5disp1}$ , $b=\sage {p5disp2}$ , $c=\sage {p5disp3}$ , and $d=\sage {p5disp4}$ .

$\text {Amplitude:}=\answer {\sage {p5a}}$ $\text {Period:}=\answer {\sage {p5pe}}$ $\text {Frequency:}=\answer {\sage {p5fr}}$ $\text {Vertical Shift:}=\answer {\sage {p5v}}$

6 :

Find the amplitude, period, frequency, and vertical shift of the function whose graph is given below. If $a=-1$ , $b=\sage {p6disp1}$ , $c=\sage {p6disp2}$ , $d=\sage {p6disp3}$ , $e=\sage {p6disp4}$ and $f=\sage {p6disp5}$ . (Note: With a few assumptions made about what the points $(c,d)$ and $(e,f)$ represent as far as the period of tangent or cotangent is concerned, we don’t actually need to know the locations of the asymptotes. They are included here to make things more explicit.)

$\text {Amplitude:}=\answer {\sage {p6a}}$ $\text {Period:}=\answer {\sage {p6pe}}$ $\text {Frequency:}=\answer {\sage {p6fr}}$ $\text {Vertical Shift:}=\answer {\sage {p6v}}$