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: Determine the amplitude

The general form of a sine function is \(y = A\sin(x)\), where \(A\) is the amplitude. For \(y = 2\sin(x)\), \(A = 2\). This means the maximum value of \(y\) is \(2\) and the minimum value is \(- 2\).

Step2: Find the x - intercepts

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 of the x - intercepts are \((0^{\circ},0)\), \((180^{\circ},0)\), \((360^{\circ},0)\)

Step3: Find the maximum and minimum points

The sine function \(y=\sin(x)\) has a maximum at \(x = 90^{\circ}\) (where \(\sin(90^{\circ})=1\)) and a minimum at \(x = 270^{\circ}\) (where \(\sin(270^{\circ})=-1\)). For \(y = 2\sin(x)\), when \(x = 90^{\circ}\), \(y=2\times1 = 2\) (coordinate \((90^{\circ},2)\)); when \(x = 270^{\circ}\), \(y=2\times(-1)=-2\) (coordinate \((270^{\circ},-2)\))

Answer:

• X - intercepts: \((0^{\circ},0)\), \((180^{\circ},0)\), \((360^{\circ},0)\)
• Maximum point: \((90^{\circ},2)\)
• Minimum point: \((270^{\circ},-2)\)

To graph the function, plot these key points and draw a smooth sine - wave curve passing through them within the interval \(0^{\circ}\leq x\leq360^{\circ}\). Label the x - axis as "Angle (degrees)" and the y - axis as "y - value".