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Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

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Section TF-2 Graphs of Trigonometric Functions

Worksheet Graphing Trigonometric Functions

For this outcome Specifically, for this outcome, students should be able to:
  • Given a trigonometric function, identify its period, domain, amplitude, phase shift, vertical shift, and reflection properties.
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  • Given a trigonometric function, graph it accurately using its period, domain, symmetry properties, and other attributes.
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  • Given a trigonometric function find the key points or asymptotes on the principal period.
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Periods, Domains, and Symmetry
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Trigonometric functions have specific periods, domains, and symmetry properties that are essential for graphing and analyzing them. We say that a graph is periodic if it repeats itself at regular intervals. The period of a trigonometric function is the length of one complete cycle of the function. Sine and cosine functions have a period of \(2\pi\text{.}\) Secant and cosecant functions have a period of \(2\pi\text{.}\) Tangent and cotangent functions have a period of \(\pi\text{.}\)
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The domain of a trigonometric function is the set of all possible input values (x-values) for which the function is defined. Sine and cosine functions have a domain of all real numbers. Secant and cosecant functions have a domain of all real numbers except where the function is undefined (i.e., where the denominator is zero). Tangent and cotangent functions have a domain of all real numbers except where the function is undefined (i.e., where the denominator is zero).
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Graphs of Sine and Cosine
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The general form of a sine or cosine function is \(y = A \sin(B(x - C)) + D\) or \(y = A \cos(B(x - C)) + D\text{,}\) where:
  • \(A\) is the amplitude, which determines the height of the wave.
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  • if \(A\) is negative, the graph is reflected across the x-axis.
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  • \(B\) affects the period of the function. The period is \(\frac{2\pi}{|B|}\text{.}\)
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  • \(C\) is the phase shift, which moves the graph horizontally.
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  • \(D\) is the vertical shift, which moves the graph vertically.
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  • The key points of the sine and cosine functions can be found by evaluating the function at specific x-values, such as \(0\text{,}\) \(\frac{\pi}{2}\text{,}\) \(\pi\text{,}\) \(\frac{3\pi}{2}\text{,}\) and \(2\pi\) for one period.
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  • Sine is an odd function, meaning it is symmetric about the origin, while Cosine is an even function, meaning it is symmetric about the y-axis.
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Graphs of Secant and Cosecant
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The general form of a secant or cosecant function is \(y = A \sec(B(x - C)) + D\) or \(y = A \csc(B(x - C)) + D\text{,}\) where:
  • \(A\) is the amplitude, which determines the height of the wave.
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  • if \(A\) is negative, the graph is reflected across the x-axis.
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  • \(B\) affects the period of the function. The period is \(\frac{2\pi}{|B|}\text{.}\)
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  • \(C\) is the phase shift, which moves the graph horizontally.
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  • \(D\) is the vertical shift, which moves the graph vertically.
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  • Secant and cosecant functions have vertical asymptotes where the function is undefined (i.e., where the Sine or Cosine function is zero).
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  • The key points/asymptotes of the secant and cosecant functions can be found by evaluating the function at specific x-values, such as \(0\text{,}\) \(\frac{\pi}{2}\text{,}\) \(\pi\text{,}\) \(\frac{3\pi}{2}\text{,}\) and \(2\pi\) for one period.
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Graphs of Tangent and Cotangent
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The general form of a tangent or cotangent function is \(y = A \tan(B(x - C)) + D\) or \(y = A \cot(B(x - C)) + D\text{,}\) where:
  • \(A\) is the amplitude, which determines the height of the first and third quartiles of the wave.
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  • if \(A\) is negative, the graph is reflected across the x-axis.
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  • \(B\) affects the period of the function. The period is \(\frac{\pi}{|B|}\text{.}\)
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  • \(C\) is the phase shift, which moves the graph horizontally.
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  • \(D\) is the vertical shift, which moves the graph vertically.
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  • Tangent and cotangent functions have vertical asymptotes where the function is undefined.
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  • The key points/asymptotes of the tangent and cotangent functions can be found by evaluating the function at specific x-values, such as \(0\text{,}\) \(\frac{\pi}{4}\text{,}\) \(\frac{\pi}{2}\text{,}\) \(\frac{3\pi}{4}\text{,}\) and \(\pi\) for one period.
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Inverse Trigonometric Functions
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For a function to have an inverse, it must be one-to-one. The inverse trigonometric functions are defined by restricting the domains of the original trigonometric functions to make them one-to-one.
  • The inverse sine function, denoted as \(\sin^{-1}(x)\) or \(\arcsin(x)\text{,}\) is the inverse of the sine function restricted to the domain \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\text{.}\)
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  • The inverse cosine function, denoted as \(\cos^{-1}(x)\) or \(\arccos(x)\text{,}\) is the inverse of the cosine function restricted to the domain \([0, \pi]\text{.}\)
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  • The inverse tangent function, denoted as \(\tan^{-1}(x)\) or \(\arctan(x)\text{,}\) is the inverse of the tangent function restricted to the domain \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\text{.}\)
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While the original trigonometric functions are periodic, their inverses are not periodic and are defined only on specific intervals. We will use trigonometric inverse functions to solve trigonometric equations and find angles in right triangles.
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Worksheet TF-2 Trigonometric Graphs

Periods, Domains, and Symmetry
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Basic Trigonometric Graphs-part 1
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Basic Trigonometric Graphs-part 2
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Basic Trigonometric Graphs-part 3
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Transformed Sine and Cosine Graphs
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Transformed Secant and Cosecant Graphs
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Transformed Tangent and Cotangent Graphs
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Section TF-2 Rubric

Worksheet Worksheet

Before attempting the TF-2 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.
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Table 23. TF-2 Rubric
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Criteria M P R N
Simplifying Algebraic Expressions All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner. No algebraic errors but some expressions are not simplified. There are some algebraic errors and some expressions were not simplified. There are some algebraic errors and some expressions were not simplified.
Basics of Graphing Graph is labeled with units for each axis.  The graph includes all of the important features (intercepts, asymptotes, deleted points, etc).  The graph is appropriately drawn to accommodate all features. Graph is labeled and has most of the required features. Graph has most of the required features. No graph was given.
Problem Structure and Mathematical Communication Problems have a clear beginning, middle and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and provides a clear solution. Problems have a clear beginning, middle and end. Work progresses clearly from one step to the next. The work provided uses algebraic methods and provides a solution. Parts of the mathematical structure are missing or steps in the progress of the solution are missing. The work provided uses algebraic methods and provides a solution. There is not a clear structure to the solution and it is unclear what methods are being used.
Trigonometric attributes Correctly identify the domain, range, period, amplitude, phase shift, vertical shift and reflections of a trigonometric function. Correctly identify the domain, range, period, phase shift, and reflections of a trigonometric function. Correctly identify the domain and range of a trigonometric function. No evidence shown of understanding trigonometric attributes.
Key features and the principle period Given a transformed trigonometric function be able to find the principle period of the function and identify the 5 key features using correct notation. Given a transformed trigonometric function be able to find the principle period of the function and identify the 5 key features using correct notation. Given a transformed trigonometric function can identify either the principle period of the function or at least 2 of the key features. No evidence shown of understanding the principle period or the key features of that period.
Trigonometric Graph Correctly graphed two periods of the function or the function on the designated interval. Correctly graphed one period of the function on part of the designated interval. Graphed one period of the function but not on the correct interval or period. No evidence shown of understanding the graph of the trigonometric function.
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Worksheet TF-2 Graphs of Trigonometric Functions

This outcome covers graphing the six trigonometric functions and transformation of them. You should be able to graph the sine, cosine, tangent, cosecant, secant and cotangent functions by hand. You should be able to identify the period, amplitude, reflections, phase shift and vertical shift of a trigonometric function from its equation and use this information to graph the function. You should also be able to find the range, intercepts, and asymptotes of the trigonometric functions.
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For the quiz on this outcome you would be given 30 minutes to complete 1 problem. Here are samples of the types of problems you will encounter:
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1.

Given the function: \(y=-3\cos\left(\frac{3\pi x}{4}+\frac{\pi}{2}\right)+2\)
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(a)
State the five points that give the main features of the shifted principal period.
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Hint.
Use the principal period and the transformations to find the key points.
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Answer.
\((-\frac{2}{3}, -1), (0, 2), (\frac{2}{3}, 5), (\frac{4}{3}, 2), (2, -1)\)
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(b)
State the range of the function
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Hint.
To find the range, consider the maximum and minimum values of the cosine function along with the amplitude and any vertical shift.
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Answer.
The range is \([-1, 5]\text{.}\)
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Solution.
The cosine function oscillates between -1 and 1. With an amplitude of 3, it oscillates between -3 and 3. Adding the vertical shift of 2, the range becomes \([-1, 5]\text{.}\)
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(c)
State the amplitude of the function
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Hint.
The amplitude is the absolute value of the coefficient of the cosine function.
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Answer.
\(Amp=3\)
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(d)
State the period of the function
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Hint.
The period is given by \(\frac{2\pi}{|b|}\) where \(b\) is the coefficient of \(x\) in the argument of the cosine function.
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Answer.
\(Period = \frac{8}{3}\)
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Solution.
\(p=\frac{2\pi}{|\frac{3\pi}{4}|}=\frac{8}{3}\)
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(e)
State the phase shift of the function
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Hint.
The phase shift is given by \(-\frac{c}{b}\) where \(c\) is the constant term in the argument of the cosine function and \(b\) is the coefficient of \(x\text{.}\)
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Answer.
\(Phase Shift = left \frac{2}{3}\)
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Solution.
\(ps=-\frac{\frac{\pi}{2}}{\frac{3\pi}{4}}=-\frac{2}{3}\)
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(f)
State the vertical shift of the function
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Hint.
The vertical shift is given by the constant term added to the cosine function.
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Answer.
\(Vertical Shift = 2\)
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Solution.
\(vs=2\)
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(g)
State the intercepts of the function.
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Hint.
To find the intercepts, set the function equal to zero and solve for x.
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Answer.
\((0,2),(\frac{4}{3\pi}\arccos\left(\frac{2}{3}\right)-\frac{2}{3}+\frac{8n}{3},0))\)
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Solution.
\begin{align*} y &=-3\cos\left(\frac{3\pi(0)}{4}+\frac{\pi}{2}\right)+2\\ y &=-3\cos\left(\frac{\pi}{2}\right)+2\\ y &=2 \end{align*}
\begin{align*} -3\cos\left(\frac{3\pi x}{4}+\frac{\pi}{2}\right)+2 &=0\\ \cos\left(\frac{3\pi x}{4}+\frac{\pi}{2}\right) &=\frac{2}{3}\\ \frac{3\pi x}{4}+\frac{\pi}{2} &=\pm \arccos\left(\frac{2}{3}\right)+2n\pi\\ \frac{3\pi x}{4} &=\pm\arccos\left(\frac{2}{3}\right)+2n\pi-\frac{\pi}{2}\\ x=\pm\frac{4}{3\pi}\arccos\left(\frac{2}{3}\right)+\frac{2}{3}+\frac{8n}{3} \end{align*}
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(h)
Graph the function on the interval \([0, 2\pi]\text{.}\)
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
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Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
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Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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2.

Given the function: \(y=\tan\frac{2\pi}{3}\left(x+\frac{1}{3}\right)+2\)
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(a)
State the five points/asymptotes that give the main features of the shifted principal period.
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Hint.
Use the principal period and the transformations to find the key points.
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Answer.
\(x=-\frac{13}{12}, \left(-\frac{17}{24},1\right), \left(-\frac{1}{3}, 2\right), \left(\frac{1}{3}, 2\right), \left(\frac{1}{24}, 3\right), x=\frac{5}{12}\)
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(b)
State the range of the function
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Hint.
To find the range, consider the maximum and minimum values of the cosine function along with the amplitude and any vertical shift.
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Answer.
The range is \((-\infty, \infty)\text{.}\)
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Solution.
The tangent function has a range of all real numbers, so the range is \((-\infty, \infty)\text{.}\)
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(c)
State the amplitude of the function
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Hint.
The amplitude is the absolute value of the coefficient of the tangent function.
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Answer.
\(Amp=1\)
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(d)
State the period of the function
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Hint.
The period is given by \(\frac{\pi}{|b|}\) where \(b\) is the coefficient of \(x\) in the argument of the tangent function.
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Answer.
\(Period = \frac{3}{2}\)
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Solution.
\(p=\frac{\pi}{|\frac{2\pi}{3}|}=\frac{3}{2}\)
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(e)
State the phase shift of the function
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Hint.
The phase shift is given by \(c\) where \(c\) is the constant term in the argument inside of the tangent function.
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Answer.
\(Phase Shift = left \frac{1}{3}\)
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(f)
State the vertical shift of the function
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Hint.
The vertical shift is given by the constant term added to the tangent function.
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Answer.
\(Vertical Shift = 2\)
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Solution.
\(vs=2\)
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(g)
State the intercepts of the function.
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Hint.
To find the intercepts, set the function equal to zero and solve for x.
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Answer.
\(\left(0,\tan\left(\frac{2\pi}{9}\right)+2\right), \left(-\frac{1}{3}+\frac{3}{2\pi}\arctan(-2)+\frac{3n}{2},0\right)\)
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Solution.
\begin{align*} y&=\tan\frac{2\pi}{3}\left(0+\frac{1}{3}\right)+2\\ y&=\tan\left(\frac{2\pi}{9}\right)+2 \end{align*}
\begin{align*} \tan\frac{2\pi}{3}\left(x+\frac{1}{3}\right)+2 &=0\\ \tan\frac{2\pi}{3}\left(x+\frac{1}{3}\right) &=-2\\ \frac{2\pi}{3}\left(x+\frac{1}{3}\right) &=\arctan(-2)+n\pi\\ x+\frac{1}{3} &=\frac{3}{2\pi}\arctan(-2)+\frac{3n}{2}\\ x &=-\frac{1}{3}+\frac{3}{2\pi}\arctan(-2)+\frac{3n}{2} \end{align*}
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(h)
Graph the function on the interval \([0, 2\pi]\text{.}\)
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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