a) Draw the graph of the functions of y= (1)/(2)x and y=3 ((1)/(2)x+(pi)/(3)) on same axes for which 0 x 5pi.

Mathematics
a) Draw the graph of the functions of y= (1)/(2)x and y=3 ((1)/(2)x+(pi)/(3)) on same axes for which 0 x 5pi.

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Answer

\sin\left(\frac{1}{2}x\right)$.

Here's how to approach this problem:

a) Draw the graph of the functions of y=sin⁡12xy=\sin \frac{1}{2}x and y=3sin⁡(12x+π3)y=3\sin \left(\frac{1}{2}x+\frac{\pi}{3}\right) on same axes for which 0≤x≤5π0 \le x \le 5\pi.

Step 1: Analyze the first function, y=sin⁡(12x)y = \sin\left(\frac{1}{2}x\right). This is a sine wave with: • Amplitude: A=1A = 1 • Period: T=2πB=2π1/2=4πT = \frac{2\pi}{B} = \frac{2\pi}{1/2} = 4\pi • Phase shift: None The graph oscillates between −1-1 and 11.

Key points for y=sin⁡(12x)y = \sin\left(\frac{1}{2}x\right) in the interval 0≤x≤5π0 \le x \le 5\pi:

x12xy=sin⁡(12x)000ππ212ππ03π3π2−14π2π05π5π21\begin{array}{|c|c|c|} \hline x & \frac{1}{2}x & y = \sin\left(\frac{1}{2}x\right) \\ \hline 0 & 0 & 0 \\ \pi & \frac{\pi}{2} & 1 \\ 2\pi & \pi & 0 \\ 3\pi & \frac{3\pi}{2} & -1 \\ 4\pi & 2\pi & 0 \\ 5\pi & \frac{5\pi}{2} & 1 \\ \hline \end{array}

Step 2: Analyze the second function, y=3sin⁡(12x+π3)y = 3\sin\left(\frac{1}{2}x + \frac{\pi}{3}\right). This is a sine wave with: • Amplitude: A=3A = 3 • Period: T=2πB=2π1/2=4πT = \frac{2\pi}{B} = \frac{2\pi}{1/2} = 4\pi • Phase shift: To find the phase shift, factor out the coefficient of xx from the argument: 12x+π3=12(x+2π3)\frac{1}{2}x + \frac{\pi}{3} = \frac{1}{2}\left(x + \frac{2\pi}{3}\right). The phase shift is 2π3\frac{2\pi}{3} units to the left. The graph oscillates between −3-3 and 33.

Key points for y=3sin⁡(12x+π3)y = 3\sin\left(\frac{1}{2}x + \frac{\pi}{3}\right) in the interval 0≤x≤5π0 \le x \le 5\pi:

\begin{array}{|c|c|c|c|} \hline x & \frac{1}{2}x + \frac{\pi}{3} & \sin\left(\frac{1}{2}x + \frac{\pi}{3}\right) & y = 3\sin\left(\frac{1}{2}x + \frac{\pi}{3}\right) \\ \hline
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Quick Answer

Here's how to approach this problem: a) Draw the graph of the functions of y= (1)/(2)x and y=3 ((1)/(2)x+()/(3)) on same axes for which 0 x 5.

a) Draw the graph of the functions of y= (1)/(2)x and y=3 ((1)/(2)x+(pi)/(3)) on same axes for which 0 x 5pi.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Here's how to approach this problem: a) Draw the graph of the functions of y= (1)/(2)x and y=3 ((1)/(2)x+()/(3)) on same axes for which 0 x 5. Step 1: Analyze the first function, y = ((1)/(2)x). This is a sine wave with: • Amplitude: A = 1 • Period: T = (2)/(B) = (2)/(1/2) = 4 • Phase shift: None The graph oscillates between -1 and 1. Key points for y = ((1)/(2)x) in the interval 0 x 5: |c|c|c| x & (1)/(2)x & y = ((1)/(2)x) \\ 0 & 0 & 0 \\ & ()/(2) & 1 \\ 2 & & 0 \\ 3 & (3)/(2) & -1 \\ 4 & 2 & 0 \\ 5 & (5)/(2) & 1 \\ Step 2: Analyze the second function, y = 3((1)/(2)x + ()/(3)). This is a sine wave with: • Amplitude: A = 3 • Period: T = (2)/(B) = (2)/(1/2) = 4 • Phase shift: To find the phase shift, factor out the coefficient of x from the argument: (1)/(2)x + ()/(3) = (1)/(2)(x + (2)/(3)). The phase shift is (2)/(3) units to the left. The graph oscillates between -3 and 3. Key points for y = 3((1)/(2)x + ()/(3)) in the interval 0 x 5: |c|c|c|c| x & (1)/(2)x + ()/(3) & ((1)/(2)x + ()/(3)) & y = 3((1)/(2)x + ()/(3)) \\