Question

graph the function and state the amplitude, period and midline. then give the maximum and minimum y-values and the corresponding x-values from the first full period starting at x = 0.
$f(x)=5cos(x)$

Explanation:

Step1: Find the amplitude

For the function \(y = A\cos(x)\), the amplitude is \(|A|\).
For \(f(x)=5\cos(x)\), \(A = 5\), so the amplitude \(a=|5| = 5\).

Step2: Find the period

The period of the function \(y=\cos(x)\) is \(2\pi\). For the function \(y = A\cos(Bx)\), the period is \(T=\frac{2\pi}{|B|}\). Here \(B = 1\), so the period \(T=\frac{2\pi}{|1|}=2\pi\).

Step3: Find the mid - line

The mid - line of the function \(y = A\cos(x)+k\) is \(y = k\). For \(f(x)=5\cos(x)+0\), the mid - line is \(y = 0\).

Step4: Find the maximum and minimum values

The maximum value of \(y=\cos(x)\) is \(1\) and the minimum value is \(- 1\).
For \(y = 5\cos(x)\), the maximum value \(y_{max}=5\times1 = 5\). When \(\cos(x)=1\), \(x = 2n\pi\), for the first full - period starting at \(x = 0\), \(x = 0\).
The minimum value \(y_{min}=5\times(-1)=-5\). When \(\cos(x)=-1\), \(x=(2n + 1)\pi\), for the first full - period starting at \(x = 0\), \(x=\pi\).

Answer:

• Amplitude: \(5\)
• Period: \(2\pi\)
• Midline: \(y = 0\)
• Maximum \(y\) - value: \(y = 5\) at \(x = 0\)
• Minimum \(y\) - value: \(y=-5\) at \(x=\pi\)