Question

graph the following function. show at least two cycles. use the graph to determine the domain and range of the function. y = 4 cos x - 1 use the graphing tool to graph the equation. type pi to insert π as needed. click to enlarge graph

Explanation:

Step1: Recall properties of cosine function

The general form of a cosine - function is \(y = A\cos(Bx - C)+D\). For the function \(y = 4\cos x-1\), we have \(A = 4\), \(B = 1\), \(C = 0\), and \(D=-1\). The amplitude \(|A|\) is \(|4| = 4\), the period \(T=\frac{2\pi}{|B|}=\frac{2\pi}{1}=2\pi\).

Step2: Find key - points for one cycle

For the basic cosine function \(y=\cos x\), key - points in one cycle (\([0,2\pi]\)) are \((0,1)\), \((\frac{\pi}{2},0)\), \((\pi, - 1)\), \((\frac{3\pi}{2},0)\), \((2\pi,1)\). For \(y = 4\cos x-1\), we transform these points.

• When \(x = 0\), \(y=4\cos(0)-1=4\times1 - 1=3\).
• When \(x=\frac{\pi}{2}\), \(y = 4\cos(\frac{\pi}{2})-1=4\times0 - 1=-1\).
• When \(x=\pi\), \(y = 4\cos(\pi)-1=4\times(-1)-1=-5\).
• When \(x=\frac{3\pi}{2}\), \(y = 4\cos(\frac{3\pi}{2})-1=4\times0 - 1=-1\).
• When \(x = 2\pi\), \(y=4\cos(2\pi)-1=4\times1 - 1=3\).

Step3: Determine domain and range

The domain of the cosine function \(y = 4\cos x-1\) is all real numbers, i.e., \((-\infty,\infty)\) since we can substitute any real - number value for \(x\) into the function.
The range of \(y=\cos x\) is \([-1,1]\). For \(y = 4\cos x-1\), we multiply the range of \(\cos x\) by \(4\) and then subtract \(1\). The maximum value of \(y = 4\cos x-1\) occurs when \(\cos x = 1\), \(y_{max}=4\times1 - 1=3\). The minimum value occurs when \(\cos x=-1\), \(y_{min}=4\times(-1)-1=-5\). So the range is \([-5,3]\).

Step4: Graph the function

Plot the key - points for at least two cycles (\([0,4\pi]\)) and connect them with a smooth curve. The graph of \(y = 4\cos x-1\) is a cosine wave with an amplitude of \(4\), shifted down \(1\) unit.

Answer:

Domain: \((-\infty,\infty)\), Range: \([-5,3]\)