Graph y=cos(5x)

Problem

y=cos(5*x)

Solution

1. Identify the parent function and its properties. The function is of the form y=A*cos(B*x) where the parent function is y=cos(x)

2. Determine the amplitude, which is the absolute value of the coefficient A Here, A=1 so the amplitude is |1|=1 This means the graph oscillates between y=1 and y=−1

3. Calculate the period using the formula P=(2*π)/B In this function, B=5

P=(2*π)/5

4. Find the horizontal scale by dividing the period into four equal intervals. This helps identify the x-coordinates for the maximums, minimums, and intercepts.

Interval=P/4=(2*π)/5⋅1/4=π/10

5. Identify key points over one period starting from x=0

• At x=0 y=cos(0)=1 (Maximum)

• At x=π/10 y=cos(π/2)=0 (x-intercept)

• At x=(2*π)/10=π/5 y=cos(π)=−1 (Minimum)

• At x=(3*π)/10 y=cos((3*π)/2)=0 (x-intercept)

• At x=(4*π)/10=(2*π)/5 y=cos(2*π)=1 (Maximum)

6. Sketch the curve by plotting these key points and drawing a smooth, wave-like shape that repeats every (2*π)/5 units along the x-axis.

Final Answer

y=cos(5*x)

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