Problem 58 - Algebra and Trigonometry

Question

Use a vertical shift to graph one period of the function. $$y=-3 \cos 2 \pi x+2$$

Step-by-Step Solution

Answer

The plot should consist of a single, negatively oriented wave that starts at (0, 2), drops to its lowest point at (1/4, -6), and comes back to finish the period at (1/2, 2).

Step 1: Identify the Components

The given function is \(y=-3 \cos 2 \pi x+2\). It's a cosine function where: Amplitude is given by |-3|=3, Vertical shift is 2 units upward, Frequency is given by \(2\pi x\), which means the Period is \(\frac{1}{2}\).

Step 2: Plotting Points

Begin by plotting points for one period of the standard \(\cos x\) function. Normally, these would be at \((0, 1)\), \((\frac{1}{4}, 0)\), \((\frac{1}{2}, -1)\), \((\frac{3}{4}, 0)\), and \((1, 1)\). However, due to the factors affecting the function, they will be at \((0, 2)\), \((\frac{1}{8}, -1)\), \((\frac{1}{4}, -6)\), \((\frac{3}{8}, -1)\), and \(\frac{1}{2}, 2)\). This is due to the combination of the vertical shift 2 units up, and the function's amplitude being 3.

Step 3: Drawing the Plot

Use these points to sketch the function graph. Draw a wave with negative amplitude, start at \((0, 2)\), down to its lowest point at \((\frac{1}{4}, -6)\), then back up to finish at \((\frac{1}{2}, 2)\). This represents one period of the function \(y=-3 \cos 2 \pi x+2\).