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**Graphing the Trigonometric Functions**

The graphs of trigonometric functions are usually easily recognizable — after you become familiar with the basic graph for each function and the possibilities for transformations of the basic graphs. Trig functions are periodic. That is, they repeat the same function values over and over, so their graphs repeat the same curve over and over. The trick is to recognize how often this curve repeats and where one of the basic graphs starts for a particular function.

An interesting feature of four of the trig functions is that they have asymptotes — those not-really-there lines used as guides to the shape of a curve. The other four functions have vertical asymptotes to mark where their domains have gaps. Locating and drawing in vertical asymptotes for the tangent, cotangent, secant, and cosecant functions. Adjusting the amplitude of the sine or cosine when the basic curve has a multiplier.

Not misreading the period of the trig function when a transformation involves a fraction. Give a rule for the equations of the asymptotes. Letting k be an integer, the general rule for the equations of the asymptotes is. She taught at Bradley University in Peoria, Illinois, for more than 35 years. Graphing Trig Functions in Pre-Calculus.

Credit: Illustration by Thomson Digital.Intro Amp. Shift Phase Shift. You've already learned the basic trig graphs.

We can transform and translate trig functions, just like you transformed and translated other functions in algebra. This function has an amplitude of 1 because the graph goes one unit up and one unit down from the midline of the graph. Do you see that this second graph is three times as tall as was the first graph?

Rsync systemd timerThe amplitude has changed from 1 in the first graph to 3 in the second, just as the multiplier in front of the sine changed from 1 to 3. This relationship is always true: Whatever number A is multiplied on the trig function gives you the amplitude that is, the "tallness" or "shortness" of the graph ; in this case, that amplitude number was 3.

For this function, the value of the amplitude multiplier A is given by 0.

Supernatural fanfiction dean protects sam from johnFor this function, the value of the amplitude multiplier A is —2so the amplitude is:. Technically, the amplitude is the absolute value of whatever is multiplied on the trig function. The amplitude just says how "tall" or "short" the curve is; it's up to you to notice whether there's a "minus" on that multiplier, and thus whether or not the function is in the usual orientation, or upside-down.

Do you see how this third graph is squished in from the sides, as compared with the first graph? Do you see that the sine wave is cycling twice as fast, so its period is only half as long? In the sine wave graphed above, the value of the period multiplier B was 2.

Sometimes the value of B inside the function will be negative, which is why there are absolute-value bars on the denominator.

X plane addonsThis right- or left-shifting is called "phase shift". Because this value is added to the variable, then the shift is to the left.

### Questions on Graphs of Trigonometric Functions

Then the phase shift is:. This number is subtracted from the variable, so the shift will be to the right. Do you see how the graph was shifted up by three units? This relationship is always true: If a number D is added outside the function, then the graph is shifted up by that number of units; if a number D is subtracted, then the graph is shifted down by that number of units. There's nothing else going on inside of the function, nor multiplied in front of it, so this is the regular cosine wave, but it's:.

So this is the regular tangent curve, but:. Putting it all together in terms of the sine wave, we have the general sine function:. Remember that the phase shift comes from what is added or subtracted directly to the variable.High school Trigonometry classes introduce students to various trigonometric identities, properties, and functions in detail.

Students typically take Trigonometry after completing previous coursework in Algebra and Geometry, but before taking Pre-Calculus and Calculus.

## Graphs of Trig Functions

Information students learn in Trigonometry helps them succeed in later higher-level mathematics courses, as well as in science courses like Physics, where trigonometric functions are used to model certain physical phenomena. Trigonometry in particular investigates trigonometric functions, and in the process teaches students how to graph sine, cosine, secant, cosecant, tangent, cotangent, arcsin, arccos, and arctan functions, as well as how to perform phase shifts and calculate their periods and amplitudes.

Trigonometric operations are also discussed, and students also learn about trigonometric equations, including how to understand, set up, and factor trig equations, how to solve individual trigonometric equations, as well as systems of trigonometric equations, how to find trig roots, and how to use the quadratic formula on trigonometric equations.

Trigonometric identities are also discussed in Trigonometry classes; students learn about the sum and product identities, as well as identities of inverse operations, squared trigonometric functions, halved angles, and doubled angles.

Students also learn to work with identities with angle sums, complementary and supplementary identities, pythagorean identities, and basic and definitional identities. Another major part of Trigonometry is learning to analyze specific kinds of special triangles. Students learn to determine angles and side lengths in and right triangles using the law of sines and the law of cosines, as well as how to identify similar triangles and determine proportions using proportionality. Trigonometry also teaches students about the unit circles and radians, focusing on how to convert degrees into radians and vice versa.

Complementary, supplementary, and coterminal angles are all discussed. This focus on angles in the unit circle is also applied to the coordinate plane when angles in different quadrants are examined.

As may now be apparent, many students find themselves very apprehensive about taking, and keeping up with, a Trigonometry course.

Each Trigonometry Practice Test features a dozen multiple-choice Trigonometry questions, and each question comes with a full step-by-step explanation to help students who miss it learn the concepts being tested. Questions are organized in Practice Tests, which draw from various topics taught in Trigonometry; questions are also organized by concept. So, if a student wants to focus on only answering questions about using the law of sines, questions organized by concept makes this possible.

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Cga connectorEmail address: Your name:. Take the Varsity Learning Tools free diagnostic test for Trigonometry to determine which academic concepts you understand and which ones require your ongoing attention. Each Trigonometry problem is tagged down to the core, underlying concept that is being tested. The Trigonometry diagnostic test results highlight how you performed on each area of the test.

You can then utilize the results to create a personalized study plan that is based on your particular area of need. Test Difficulty :. Average Time Spent : 4 hrs 8 mins. Average Time Spent : 5 hrs 30 mins. Average Time Spent : 4 hrs 28 mins. Average Time Spent : 3 hrs 25 mins.Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials.

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Social Studies - History. History World History. For All Subject Areas. See All Resource Types. This mini-unit includes 16 pages of practice on Graphing Trigonometric Functions, a match and sort activity, and a general review of transformations with sine and cosine curves. The answer keys are included. There is an extension activity at the end of the Sort and Match Activity. The handouts can be.

PreCalculusTrigonometryAlgebra 2. WorksheetsLesson Plans BundledHomework. Add to cart. Wish List. Focused notes help students to take clear and effective notes during math lessons, can be used for introducing new topics or for review, and can be used with any curricul.

Scaffolded NotesInteractive Notebooks. Vertical and phase shifts are included. Students will also identify the amplitude, period, phase shift, vertical shift, and midline of each g.

Diagram based comcast wireless diagram completedMathTrigonometryAlgebra 2. ActivitiesGraphic Organizers. Graphs of Trigonometric Functions Art Project. I really love how this activity allows students to show their creative sides.Try this with the Unit Circle.

A complete repetition of the pattern of the function is called a cycle and the period is the horizontal length of one complete cycle. Because the trig functions are cyclical in nature, they are called periodic functions. Before we go into more detail of each of the trig functions, here are some tables that might help. The second table shows more details for each of the trig functions. Starting and stopping points may be changed, as long the graph covers one complete cycle period.

Here are more detailed graphs of the six trig Functions. This is one complete revolution around the unit circle. This graph repeats itself one-half of a revolution of the unit circle.

### Math 150 Precalculus Worksheets

Vertical asymptotes were discussed here in the Graphing Rational Functions, including Asymptotes section. The tan function is an odd function ; we learned about Even and Odd Functions here. The cscsecand cot are sometimes called the reciprocal functionssince they are the reciprocal of sincosand tanrespectively.

Vertical asymptotes were discussed here in the Graphing Rational Functions, including Asymptotes. The sec function is an even function ; we learned about Even and Odd Functions here. The graph repeats itself one-half of a revolution of the unit circle. You can use the graphing calculator to graph trig functions, as follows:. We can check a complete revolution of the graph in a graphing calculator — looks good!

There are also examples of using the calculator to solve trig equations here in the Solving Trigonometric Equations section. Understand these problems, and practice, practice, practice! Skip to content. Table of Trigonometric Parent Functions Before we go into more detail of each of the trig functions, here are some tables that might help.Notice the wave shape of the graph. This is called the period.

Notice the similar wave shape of the graph. Notice that the two graphs look very similar. The distances between the peaks for each graph is the same.

The height of the peaks and the depths of the troughs are also the same. This means that:. There is an easy way to visualise the tangent graph. For this graph we see that this point has not been shifted. For this graph we see that the graph has not been shifted. We are given a table with values and so we plot each of these points and join them with a smooth curve. We note that in this case the graph is not shifted.

We note that in this case the graph is shifted upwards by 1 unit. We also note the graph is not stretched. Plot the points and join with a smooth curve Notice the wave shape of the graph. Plot the points and join with a smooth curve Notice the similar wave shape of the graph. Plot the points and join with a smooth curve There is an easy way to visualise the tangent graph. Don't get left behind Join thousands of learners improving their maths marks online with Siyavula Practice.

Sign up here. Exercise 6. Given the following graph, identify a function that matches each of the following equations:. With the assistance of the table below sketch the three functions on the same set of axes.

Give the equations for each of the following graphs:. Given the following graph. Given the following graph:. Previous Exponential functions. Next Interpretation of graphs.

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