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# ZingPath: Graphs of Trigonometric Functions

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## Graphs of Trigonometric Functions

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Explore the full path to learning Graphs of Trigonometric Functions

### Lesson Focus

#### Graphing Sine Functions

Algebra-2

Identify characteristics of the sine function, then sketch its graph in y = a . sin (bx+c) + d form.

### Now You Know

After completing this tutorial, you will be able to complete the following:

• Identify characteristics of the sine function, including specific values and period.
• Sketch the graph of transformations of the sine function.

### Everything You'll Have Covered

This Activity Object is designed to introduce the function y = sinx, the concept of periodicity as it applies to trigonometric functions, and calculating specific values allowing the learner to determine the x-intercepts and absolute maximum and minimum values necessary to graph translation of the sine function.

The unit circle is used to define the sine function.

The correlation between the unit circle and the graph of the sine function (from 0 to ), visually establishes the relationship between the specific values of t=0, corresponding ordered pairs on the graph of y =sin t, i.e., the sin t = 0 when t = 0, , the sin t = 1 when , and sin t = -1 when . Underscoring this relationship, for the learner, is critical for understanding and mastery of the guided practice problems provided in Section 3 of the Activity Object.

The period of y = sin x is .

A 5 step process can be used to help the learner sketch the graph of y = a.sin(bx + c) + d.

Step 1: Using the formula , the learner is asked to calculate the period of the given function, e.g., for the given function y = 3sin(2x) the period is . A fraction and Pi symbol tool are provided to help learners correctly notate results.

Step 2: The learner is asked to calculate the specific values for the argument . It should be noted here, that the calculated values in this step will be the same for all given problems, which are the values for the function sin x, i.e., the sin x = 0

when x = 0,, the sin x = 1 when , and sin x = -1 when ; moreover, is the absolute maximum, is the absolute minimum, and sin x , where are the zeros of the function.

Step 3: The learner, in this step, is asked to calculate by multiplying the results of Step

Step 4: This step, the most difficult, requires that the learner sets the function (asin(bx + c) + d) equal to the specific values and solve for x. For example:

Step 5: The final step, asks the learner to take the calculations in Steps 3 and 4 and format the results as ordered pairs, as a prelude to sketching the graph of the given function.

The final piece of the guided practice activity, found in Section 3, is for the learner to sketch the graph (by clicking on the button), where they see the relationship between the graph and table of specific values.

The following key vocabulary terms will be used throughout this Activity Object:

• co-terminal angles - angles which, drawn in standard position, share a terminal side, e.g., 60°, -360°, and 420° are co-terminal angles.
• specific values - input values, from the domain, that produce local maximums or minimums, or zero, e.g.,<.
• domain - the set of input values for which a function is defined, for example, the domain of y=sinx is the set of all real numbers.
• function - a special type of relationship in which each element of the domain is paired with exactly one element of the range, for example, .
• period - the fixed interval that a graph, with independent variables, repeats, e.g., the period of y=sinx is .
• periodic - recurring or repeating at regular intervals.
• Pi - a mathematical constant whose value is the ratio of a circle's circumference to its diameter.
• radians - a unit of angular measure equal to .
• range - set of all output values produced by a function, e.g., the range of the sine function y=sinx is .
• sine - the sine function is one of the basic functions encountered in trigonometry and is defined as in a right triangle.
• terminal sides - in an angle, the ray which rotates and determines the measure of the angle.
• trigonometric - refers to the basic functions used in trigonometry, which used to relate the angles to the lengths of the sides of a right triangle.
• unit circle - used to define the trigonometric ratios of angles 0 to 360 degrees. The unit circle has a radius of 1 unit; sine is the y-coordinate of the intersection point of the intersection point of the terminal arm of the angle; cosine is the x-coordinate of the same point.

### Tutorial Details

 Approximate Time 35 Minutes Pre-requisite Concepts basic trigonometric ratios, solving functions using , solving functions using specific values Course Algebra-2 Type of Tutorial Procedural Development Key Vocabulary graphing, period, sine function