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Find a function of the form f(x)=A sin(B[x−C])+D whose graph is the sine wave shown above. The curve goes through the points (−5,0) and (1,0).
If needed, you can enter π=3.1416... as 'pi' in your answer.

f(x)=

f(x) = sin(x - π/2)

To find the equation of the sine wave, we need to determine the values of A, B, C, and D.

From the given points (−5,0) and (1,0), we can see that the sine wave crosses the x-axis at -5 and 1. This means that the value of C is the midpoint between these two x-values.

C = (-5 + 1) / 2 = -4/2 = -2

Now, we can use one of the points to find the value of D. Let's use the point (−5,0).

f(-5) = A sin(B[-5−(-2)]) + D
0 = A sin(B[-5+2]) + D
0 = A sin(-3B) + D

Since the sine function is zero at -3Bπ, we can write:

-3Bπ = -3B
B = π

So, we have:

0 = A sin(π) + D
0 = A * 0 + D
0 = D

Therefore, D = 0.

Now, our equation becomes:

f(x) = A sin(π[x+2])

Finally, to find the value of A, we can use the other given point (1,0):

f(1) = A sin(π[1+2])
0 = A sin(π(3))
0 = A sin(3π)
Since sin(3π) = 0,
0 = A * 0
0 = 0

The value of A can be any real number since it does not affect the shape of the sine wave or its x-intercepts.

So, the function of the form f(x) = A sin(π[x+2]) that represents the given sine wave is:

f(x) = A sin(π[x+2])

To find the function of the form f(x) = A sin(B[x−C])+D that represents the given sine wave graph, we need to determine the values of A, B, C, and D.

1. Amplitude (A): The amplitude is the distance from the midline to the peak or trough of the sine wave. In this case, the graph goes through the points (−5,0) and (1,0), so the amplitude is half the distance between these two points on the y-axis. Amplitude = (1-0)/2 = 1/2 = 0.5

2. Period (π/B): The period is the distance between two consecutive peaks or troughs of the sine wave. In this case, the x-values at the points where the graph crosses the midline are -5 and 1. Therefore, the period is 1 - (-5) = 6 units.

Since the general form of the sinusoidal function is f(x) = A sin(B[x−C])+D, we can rewrite it as f(x) = 0.5 sin(B[x−C])+D.

3. Phase Shift (C): The phase shift is the horizontal shift of the graph a distance C units to the right or left. In this case, the graph passes through the point (-5, 0). This means that the graph has been shifted 5 units to the right. Therefore, C = 5.

Considering the equation f(x) = 0.5 sin(B[x−C])+D, we have f(x) = 0.5 sin(B[x-5]) + D.

4. Midline (D): The midline is the horizontal line that the sine wave oscillates around. In this case, the graph goes through the points (−5,0) and (1,0), so the midline is the x-axis. Thus, D = 0.

Now, we have f(x) = 0.5 sin(B[x-5]).

To determine the value of B, we need to calculate the value of the coefficient inside the sine function, which is B.

5. Coefficient B: Since the period of the sine function is given by π/B, and we know the period is 6 units, we can write the equation as follows:

6 = π/B
B = π/6

Therefore, the function of the form f(x) = A sin(B[x−C])+D representing the given sine wave graph is:

f(x) = 0.5 sin((π/6)(x - 5))

There are many curves which go through those two points.

Lat's make it easy and say that we have a sine curve of period 12, crossing the x-axis at the two points. That means we have one arch of the curve (1/2 period).
If we pick (-5,0) as our starting point, then
f(x) = sin(π/12 (x+5))
That gives us A=1 and D=0
I'll let you decide whether you want to complicate things, such as, what if D is not zero?