identifying features from a graph\nto analyze the given graph, complete the following sentences.\nthe graph…

Explanation:

Step1: Identify sine - function pattern

The sine function (y = a\sin(x)+k) starting at the origin has no phase - shift. The general form of a sine function is (y = a\sin(x)+k). Since it starts at the origin and has the typical sine - wave shape, it is of the form (y = a\sin(x)) with no vertical or horizontal shift.

Step2: Determine the value of (a)

The maximum value of the graph is (4) and the minimum value is (- 4). For the sine function (y = a\sin(x)), the amplitude (|a|) is the distance from the mid - line to the maximum or minimum value. The mid - line is (y = 0). Since the maximum value is (4) and the minimum is (-4), (a = 4) (because the graph goes up to (y = 4) first, so (a>0)).

Step3: Find the maximum and minimum values

By observing the graph, the maximum value is (4) and the minimum value is (-4).

Step4: Calculate the amplitude

The amplitude (|a|) of the function (y=a\sin(x)) is given by the formula (|a|=\frac{\text{max}-\text{min}}{2}). Substituting (\text{max}=4) and (\text{min}=-4), we get (|a|=\frac{4 - (-4)}{2}=\frac{8}{2}=4).

Step5: Determine the vertical translation (k)

The mid - line of the graph is (y = 0). For the function (y=a\sin(x)+k), the equation of the mid - line is (y = k). So, (k = 0).

Step6: Write the equation of the graph

The equation of the graph is (y = 4\sin(x)) as (a = 4) and (k = 0).

Answer:

The graph follows the pattern for the sine function: (y=a\sin(x)) starting at the origin. So, (a) is (4). The maximum value is (4) and the minimum value is (-4). So, the amplitude of the function, (|a|), is (4). The vertical translation is (k = 0). This is the graph of (y = 4\sin(x)).