Question

graph the following function.
y = - 4 + 6 sin x
use the graphing tool to graph
the function.
(for any answer boxes shown
with the grapher, type an exact
answer. type the word pi to
insert the symbol π as needed.)

Explanation:

Step1: Identify the general form of the sine function

The general form of a sine function is \(y = A\sin(x)+B\). In the given function \(y=-4 + 6\sin(x)\), \(A = 6\) (amplitude) and \(B=-4\) (vertical shift).

Step2: Determine the key points of the basic sine function \(y=\sin(x)\)

The key points of \(y = \sin(x)\) over the interval \([0,2\pi]\) are \((0,0)\), \((\frac{\pi}{2},1)\), \((\pi,0)\), \((\frac{3\pi}{2},-1)\), \((2\pi,0)\).

Step3: Apply the amplitude and vertical shift to the key points

For amplitude \(A = 6\), the \(y\)-values of the key points of \(y=\sin(x)\) are multiplied by \(6\). For vertical shift \(B=-4\), \( - 4\) is added to the \(y\)-values.

• For \((0,0)\): \(y=-4+6\times0=-4\), so the point is \((0, - 4)\)
• For \((\frac{\pi}{2},1)\): \(y=-4 + 6\times1=2\), so the point is \((\frac{\pi}{2},2)\)
• For \((\pi,0)\): \(y=-4+6\times0=-4\), so the point is \((\pi,-4)\)
• For \((\frac{3\pi}{2},-1)\): \(y=-4+6\times(-1)=-10\), so the point is \((\frac{3\pi}{2},-10)\)
• For \((2\pi,0)\): \(y=-4+6\times0=-4\), so the point is \((2\pi,-4)\)

Step4: Plot the key points and draw the sine - wave

Plot the points \((0, - 4)\), \((\frac{\pi}{2},2)\), \((\pi,-4)\), \((\frac{3\pi}{2},-10)\), \((2\pi,-4)\) and then connect them with a smooth curve. Since the sine function is periodic with period \(2\pi\), the pattern repeats every \(2\pi\) units.

Answer:

Using the key points \((0, - 4)\), \((\frac{\pi}{2},2)\), \((\pi,-4)\), \((\frac{3\pi}{2},-10)\), \((2\pi,-4)\) and the periodicity of the sine function (\(T = 2\pi\)), the graph of \(y=-4 + 6\sin(x)\) can be drawn.