Graph y=-3cos(x)

Problem

y=−3*cos(x)

Solution

1. Identify the parent function and its properties. The parent function is y=cos(x) which has an amplitude of 1 a period of 2*π and starts at a maximum point (0,1)

2. Determine the amplitude of the given function. The coefficient a=−3 indicates that the amplitude is |−3|=3 This means the graph will oscillate between y=3 and y=−3

3. Account for the reflection across the x-axis. Because the coefficient a is negative, the graph of cos(x) is reflected. Instead of starting at a maximum, the graph starts at a minimum point (0,−3)

4. Identify the period and key points. The period remains 2*π since the coefficient of x is 1 Key points for one cycle occur at x=0,π/2,π,(3*π)/2,2*π

5. Calculate the coordinates for one full cycle:

• At x=0 y=−3*cos(0)=−3

• At x=π/2 y=−3*cos(π/2)=0

• At x=π y=−3*cos(π)=3

• At x=(3*π)/2 y=−3*cos((3*π)/2)=0

• At x=2*π y=−3*cos(2*π)=−3

6. Sketch the curve by plotting these points and connecting them with a smooth, wave-like shape that repeats every 2*π units.

Final Answer

y=−3*cos(x)

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