Question

Graph y = -5 sin ( 1/7 x) - 3

Answer 1

Here is the graph of y = -5 sin(1/7 x) - 3:

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0 14 28

In this graph, the x-axis represents values from 0 to 28, and the y-axis represents values from -8 to 2. The graph shows a sine wave with a period of 2π(7) = 14π, an amplitude of 5, and a vertical shift downward by 3 units. The graph crosses the x-axis at x = 7π, 21π, and has its minimum value of y = -8 at x = 14π.

Answer 2

See attachments.

Explanation:

The sine function is periodic, meaning it repeats forever.

Standard form of a sine function:

`f(x) = A \sin (B(x+C))+D`

where:

• |A| = amplitude (height from the mid-line to the peak).
• 2π/B = period (horizontal length of one cycle of the curve).
• C = phase shift (horizontal shift - positive is to the left).
• D = vertical shift.

Given function:

`f(x)=-5 \sin \left((1)/(7)x\right)-3`

Therefore:

• |A| = 5
• B = 1/7
• C = 0
• D = -3

The given function has no horizontal shift and its period is:

`\sf Period=(2 \pi)/(B)=(2 \pi)/((1)/(7))=14\pi`

As "A" is negative, the curve is reflected in the x-axis.

Therefore, the x-values of the minimum and maximum points are:

`\implies x_(\sf min)=(1)/(4) \cdot \text{period}+\text{period} \cdot n=(7)/(2)\pi+14\pi n`

`\implies x_(\sf max)=(3)/(4) \cdot \text{period}+\text{period} \cdot n=(21)/(2)\pi+14\pi n`

The mid-line of the function is y = D, therefore the mid-line of the given function is y = -3.

As the amplitude is 5, the maximum and minimum points of the curve are 5 more and 5 less than the mid-line:

`\implies y=-3+5=2`

`\implies y=-3-5=-8`

Therefore, the minimum points of the graph are:

`\left((7)/(2)\pi + 14 \pi n, -8\right)`

Therefore, the maximum points of the graph are:

`\left((21)/(2)\pi + 14 \pi n, 5\right)`

As the mid-line of the function y = -3, there is no horizontal shift and its period is 14π, the function crosses the mid-line at:

`\left(7\pi n, -3\right)`

To graph the given function on the given small coordinate grid (attachment 1):

• Maximum point at (-3π/2, 2).
• Minimum point at (3π/2, -8).
• Point intersecting the mid-line: (0, -3)

To graph the given function on a larger coordinate grid, see attachment 2.