Graph y=5cos(x)

Problem

y=5*cos(x)

Solution

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

2. Determine the amplitude of the given function. The coefficient of the cosine function is 5 so the amplitude is |5|=5 This means the graph oscillates between y=5 and y=−5

3. Determine the period of the function. Since the coefficient of x is 1 the period remains 2*π

4. Identify key points over one period [0,2*π] The cosine function starts at its maximum, crosses the x-axis, reaches its minimum, crosses the x-axis again, and returns to its maximum.

5. Calculate coordinates for the key points:

• At x=0 y=5*cos(0)=5*(1)=5 Point: (0,5)

• At x=π/2 y=5*cos(π/2)=5*(0)=0 Point: (π/2,0)

• At x=π y=5*cos(π)=5*(−1)=−5 Point: (π,−5)

• At x=(3*π)/2 y=5*cos((3*π)/2)=5*(0)=0 Point: ((3*π)/2,0)

• At x=2*π y=5*cos(2*π)=5*(1)=5 Point: (2*π,5)

6. Sketch the curve by connecting these points with a smooth, wave-like shape and extending the pattern in both directions.

Final Answer

y=5*cos(x)

The graph is a cosine wave with amplitude 5 period 2*π and key points at (0,5) (π/2,0) (π,−5) ((3*π)/2,0) and (2*π,5)

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