Questions: Find the amplitude and period of the function, and sketch its graph. y=-5+cos(3πx) The cosine curve y=a cos(kx) has amplitude and period

Transcript text: Find the amplitude and period of the function, and sketch its graph. \[ y=-5+\cos (3 \pi x) \] The cosine curve $y=a \cos (k x)$ has amplitude $\square$ and period $\square$

Solution

Solution Steps

Step 1: Identify the amplitude

The amplitude of the function $ y = -5 + \cos(3 \pi x) $ is determined by the coefficient of the cosine function. Since the coefficient is 1, the amplitude is: \[ \text{Amplitude} = 1 \]

Step 2: Identify the period

The period of the function $ y = -5 + \cos(3 \pi x) $ is determined by the coefficient of $ x $ inside the cosine function. The period $ T $ is given by: \[ T = \frac{2\pi}{3\pi} = \frac{2}{3} \]

Step 3: Sketch the graph

The function $ y = -5 + \cos(3 \pi x) $ is a cosine wave with an amplitude of 1, a period of $\frac{2}{3}$, and a vertical shift of -5.

Final Answer

Amplitude: 1
Period: $\frac{2}{3}$

{"axisType": 3, "coordSystem": {"xmin": -1, "xmax": 1, "ymin": -6, "ymax": -4}, "commands": ["y = -5 + cos(3_pi_x)"], "latex_expressions": ["$y = -5 + \\cos(3 \\pi x)$"]}

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