Graph y=-3cos(2x)

Problem

y equals negative 3 cosine 2 x

Solution

1. Identify the parent function and the general form of the trigonometric equation. The equation follows the form y equals A cosine B x where A is the amplitude and B affects the period.

2. Determine the amplitude by taking the absolute value of the coefficient A Here, the absolute value of A equals the absolute value of negative 3 equals 3 The graph will oscillate between y equals 3 and y equals negative 3

3. Identify the vertical reflection caused by the negative sign in front of the amplitude. Since A equals negative 3 the graph of the cosine function is reflected across the xaxis, meaning it starts at a minimum point open paren 0 comma negative 3 close paren instead of a maximum.

4. Calculate the period using the formula P equals 2 pi over B With B equals 2 the period is P equals 2 pi over 2 equals pi One full cycle occurs over the interval open bracket 0 comma pi close bracket

5. Determine the key points by dividing the period into four equal increments. The increment size is pi over 4 The xcoordinates for the key points are 0 pi over 4 pi over 2 3 pi over 4 and pi

6. Evaluate the function at these key points to find the ycoordinates:

• At x equals 0 y equals negative 3 cosine 0 equals negative 3

• At x equals pi over 4 y equals negative 3 the cosine of pi over 2 equals 0

• At x equals pi over 2 y equals negative 3 cosine pi equals 3

• At x equals 3 pi over 4 y equals negative 3 the cosine of 3 pi over 2 equals 0

• At x equals pi y equals negative 3 cosine 2 pi equals negative 3

7. Plot the points and draw a smooth wave connecting open paren 0 comma negative 3 close paren open paren pi over 4 comma 0 close paren open paren pi over 2 comma 3 close paren open paren 3 pi over 4 comma 0 close paren and open paren pi comma negative 3 close paren

Final Answer

y equals negative 3 cosine 2 x

(The graph is a cosine wave with amplitude 3 period pi reflected over the xaxis, passing through open paren 0 comma negative 3 close paren open paren pi over 2 comma 3 close paren and open paren pi comma negative 3 close paren)

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