Graph y=-5cos(x)

Problem

y=−5*cos(x)

Solution

1. Identify the parent function and its key characteristics. 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 in front of the cosine is a=−5 The amplitude is the absolute value |a|=|−5|=5

3. Identify the reflection across the x-axis. Because the coefficient a is negative, the graph of cos(x) is reflected vertically. Instead of starting at a maximum, the graph starts at a minimum point (0,−5)

4. Determine the period of the function. The coefficient of x is 1 so the period remains T=(2*π)/1=2*π

5. Locate key points over one period [0,2*π] Divide the period into four equal intervals to find the x-coordinates: 0 π/2 π (3*π)/2 and 2*π

6. Calculate the y-coordinates for the key points:

• At x=0 y=−5*cos(0)=−5*(1)=−5

• At x=π/2 y=−5*cos(π/2)=−5*(0)=0

• At x=π y=−5*cos(π)=−5*(−1)=5

• At x=(3*π)/2 y=−5*cos((3*π)/2)=−5*(0)=0

• At x=2*π y=−5*cos(2*π)=−5*(1)=−5

7. Sketch the curve by plotting these points and connecting them with a smooth wave.

Final Answer

To graph y=−5*cos(x) plot a cosine wave with amplitude 5 period 2*π and a vertical reflection, passing through key points (0,−5) (π/2,0) (π,5) ((3*π)/2,0) and (2*π,−5)

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