Graph y=-4cos(x)

Problem

y=−4*cos(x)

Solution

1. Identify the parent function and its properties. The base function is y=cos(x) which has a period of 2*π an amplitude of 1 and starts at a maximum point (0,1)

2. Determine the amplitude of the given function. The coefficient in front of the cosine function is −4 The amplitude is the absolute value |−4|=4

3. Identify the vertical reflection caused by the negative sign. Since the coefficient is negative, the graph of y=cos(x) is reflected across the xaxis. Instead of starting at a maximum, the graph starts at a minimum point (0,−4)

4. Determine the period and key points. The coefficient of x is 1 so the period remains 2*π Key points occur every quarter-period (π/2 units).

5. Calculate the coordinates for one full cycle:

• At x=0 y=−4*cos(0)=−4 (Minimum)

• At x=π/2 y=−4*cos(π/2)=0 (xintercept)

• At x=π y=−4*cos(π)=4 (Maximum)

• At x=(3*π)/2 y=−4*cos((3*π)/2)=0 (xintercept)

• At x=2*π y=−4*cos(2*π)=−4 (Minimum)

6. Sketch the curve by plotting these points and connecting them with a smooth, wave-like shape that repeats every 2*π units.

Final Answer

y=−4*cos(x)

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