Graph y=-1/2*cos(x)

Problem

y=−1/2*cos(x)

Solution

1. Identify the parent function, which is y=cos(x) The standard form of the cosine function is y=A*cos(B*(x−C))+D

2. Determine the amplitude |A| Here, A=−1/2 so the amplitude is |−1/2|=1/2 This means the graph oscillates between y=1/2 and y=−1/2

3. Identify the reflection. Since A is negative (A=−1/2, the graph is reflected across the xaxis. Instead of starting at a maximum, the graph starts at a minimum.

4. Calculate the period. The coefficient of x is B=1 The period is calculated as (2*π)/|B|=(2*π)/1=2*π

5. Find the key points for one cycle. Divide the period into four intervals of length (2*π)/4=π/2

• At x=0 y=−1/2*cos(0)=−1/2 (Minimum)

• At x=π/2 y=−1/2*cos(π/2)=0 (Intercept)

• At x=π y=−1/2*cos(π)=1/2 (Maximum)

• At x=(3*π)/2 y=−1/2*cos((3*π)/2)=0 (Intercept)

• At x=2*π y=−1/2*cos(2*π)=−1/2 (Minimum)

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

Final Answer

y=−1/2*cos(x)

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