Graph y=cos(1/2x)

Problem

y=cos(1/2*x)

Solution

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

2. Determine the amplitude. The coefficient of the cosine function is 1 so the amplitude is |1|=1 This means the graph oscillates between y=1 and y=−1

3. Calculate the period. The period P is found using the formula P=(2*π)/b where b is the coefficient of x Here, b=1/2

P=(2*π)/(1/2)

P=4*π

4. Find the key points for one cycle. Divide the period into four equal intervals of length (4*π)/4=π The xcoordinates are 0,π,2*π,3*π,4*π

5. Evaluate the function at the key xvalues:

• At x=0 y=cos(0)=1

• At x=π y=cos(π/2)=0

• At x=2*π y=cos(π)=−1

• At x=3*π y=cos(3*π/2)=0

• At x=4*π y=cos(2*π)=1

6. Sketch the curve by plotting the points (0,1) (π,0) (2*π,−1) (3*π,0) and (4*π,1) and connecting them with a smooth wave.

Final Answer

y=cos(1/2*x)* has amplitude *1* and period *4*π

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