Identify characteristics of the sine function, then sketch its graph in y = a . sin (bx+c) + d form.
After completing this tutorial, you will be able to complete the following:
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:
|Approximate Time||35 Minutes|
|Pre-requisite Concepts||basic trigonometric ratios, solving functions using <img src../../../tutorial
|Type of Tutorial||Procedural Development|
|Key Vocabulary||graphing, period, sine function|