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Algebra-2

Graph the cosine function, and variations of the graph that involve the cosine expression.

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

- Sketch the graph of the cosine function.
- Sketch the graph of a function that involves cosine expression.

The cosine function is periodic.

The cosine function is an example of a periodic function; this means that the function repeats its range values at regular intervals. This is useful in graphing in that, once the learner finds the period of the function and the shape of the function, it will be repeated as the values of x increase without bound.

A cosine function can have several transformations in a single function.

The general form of the cosine function, y = a • cos (bx + c) + d, has several transformations. First, the 'a' in the function stands for the maximum absolute value of a periodic curve measured along the vertical axis (or, in other words, the highest value on the y-axis that the curve reaches). The 'b' helps determine the period of the function (period = ).

The 'c' is the phase shift (whether the graph is shifted to the left or right of its original point), and the'd' is the vertical shift (whether the graph is shifted up or down from the original point.

There can also be a reflection of the cosine function.

If there is a negative sign in front of the cosine function, y = -cos x, the graph will be reflected about the x-axis. The graph will appear to be flipped upside down from its original position.

The unit circle is used to define the cosine function.

The correlation between the unit circle and the graph of the cosine function (from to ), visually establishes the relationship between the specific values of t = 0, , , and corresponding

ordered pairs on the graph of x = cos t, i.e., the cos t = 0 when t = 0, ?, and 2?, the cost = 1 when x = , and cos t = -1 when x = . Underscoring this relationship, for the learner, is specific for understanding and mastery of the guided practice problems provided in Section 3 of the Activity Object.

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

For the given function y = 2 cos (4x):

Step 1: Using the formula when the coefficient of x is b, calculate the period of the given function. A fraction and pi button are provided in the Activity Object to help correctly notate results.

Step 2: Calculate cosine of specific values, (bx + c) or in this example, 4x. It should be noted here, that the calculated vales in this step will be the same for all given problems, which are the values for the function cos x ( i.e., the cos x = 0 when

is the absolute maximum, = -1 is the absolute minimum, and cos x, where x = 0, ?, and 2? the zeros of the function.

Step 3: Find the result of entire function according to the cosines of specific values. Calculate a • cos (bx + c) by multiplying the results of Step 2 by a, in this example a = 2.

Step 4: Calculate the values of x for the cosine function.

Step 5: Write the x and y values as ordered pairs to sketch the graph. Take the calculations in Steps 3 (y-coordinate) and 4 (x-coordinate) and format the results as ordered pairs.

To see the graph using the Activity Object, click on the button

These key vocabulary terms will be used throughout this Activity Object:

- cosine - the x-coordinate of an angle on the unit circle.
- co-terminal angles - angles that share a terminal side.
- 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.
- origin - the intersection of the x- and y-axes; the ordered pair of the origin is (0,0).
- period - The length of the fixed interval that a function repeats itself, e.g., the period of
- periodic function - A function that repeats all of its range values at regular intervals.
- phase shift - A graph's horizontal movement to the left or right from the origin.
- pi or ? - a mathematical irrational constant whose value is the ratio of a circle's circumference to its diameter.
- radian - a unit of angular measure equal to .
- range - the set of all output values produced by a function; also, the y-coordinates.
- terminal sides - in an angle, the ray which rotates and determines the measure of the angle in standard position.
- trigonometric - Refers to the basic functions used in trigonometry, which are used to relate the angles to the lengths of the sides of a right triangle.
- unit circle - Used to define the trigonometric ratios of the angles. It has a radius of 1 unit and its center is at the origin.
- vertical shift - A graph's movement up or down from the origin.

Approximate Time | 35 Minutes |

Pre-requisite Concepts | cosine, domain and range, graphs of functions, period |

Course | Algebra-2 |

Type of Tutorial | Procedural Development |

Key Vocabulary | cosine, graph, graphing cosine functions |