Question

question:
consider the function $y = 2\sin(x)$ for $0^\circ \leq x \leq 360^\circ$.

1. graph the function:

plot the function $y = 2\sin(x)$ on a coordinate plane. label the x - axis as \angle (degrees)\ and the y - axis as \y - value\. mark the coordinates of all key points where the graph intersects the x - axis, reaches its maximum, or minimum within the given interval.

Explanation:

Step1: Find x - axis intersection points

Set \(y = 0\), so \(2\sin(x)=0\), then \(\sin(x)=0\).
For \(0^{\circ}\leq x\leq360^{\circ}\), \(x = 0^{\circ},180^{\circ},360^{\circ}\). The coordinates are \((0,0)\), \((180,0)\), \((360,0)\).

Step2: Find maximum points

The general form of \(y = A\sin(x)\) has maximum when \(\sin(x)=1\). Here \(A = 2\), when \(\sin(x)=1\), \(x = 90^{\circ}\), and \(y=2\times1 = 2\). The coordinate is \((90,2)\).

Step3: Find minimum points

The general form of \(y = A\sin(x)\) has minimum when \(\sin(x)= - 1\). Here \(A = 2\), when \(\sin(x)=-1\), \(x = 270^{\circ}\), and \(y=2\times(-1)=-2\). The coordinate is \((270, - 2)\).

To graph the function \(y = 2\sin(x)\):

• Start at the origin \((0,0)\).
• As \(x\) increases from \(0^{\circ}\) to \(90^{\circ}\), \(y\) increases from \(0\) to \(2\).
• As \(x\) increases from \(90^{\circ}\) to \(180^{\circ}\), \(y\) decreases from \(2\) to \(0\).
• As \(x\) increases from \(180^{\circ}\) to \(270^{\circ}\), \(y\) decreases from \(0\) to \(-2\).
• As \(x\) increases from \(270^{\circ}\) to \(360^{\circ}\), \(y\) increases from \(-2\) to \(0\).

Answer:

Key points: \((0,0)\), \((90,2)\), \((180,0)\), \((270,-2)\), \((360,0)\)