Find a Sinusoidal Function Given its Graph

Step 1: Find Vertical Shift and Amplitude
The scaling along the y-axis is one unit for one large division. The maximum value is \( y_{\text{max}} = 1 \) and the minimum is \( y_{\text{min}} = -7 \).
Vertical shift (\(c\)): \[ c = \dfrac{y_{\text{max}} + y_{\text{min}}}{2} = \dfrac{1 + (-7)}{2} = -3 \] Amplitude (\(a\)): \[ |a| = \dfrac{y_{\text{max}} - y_{\text{min}}}{2} = \dfrac{1 - (-7)}{2} = 4 \] Because we will track a maximum point to determine the shift, we can use a positive cosine function, so \( a = 4 \).

Step 2: Find the Period and \(b\)
The scaling along the x-axis is \( \pi \) for one large division and \( \dfrac{\pi}{5} \) for one small division.
Points A and B are consecutive maximums. The horizontal distance between them is the Period \( P = 5\pi \).
\[ P = \dfrac{2\pi}{|b|} = 5\pi \implies |b| = \dfrac{2}{5} \]

Step 3: Find the Phase Shift (\(d\))
We use the cosine form: \( y = 4 \cos\left[ \dfrac{2}{5}(x - d) \right] - 3 \).
The unshifted cosine function has a maximum at \( x = 0 \). Our graph has a maximum at point A, which is one small division to the left of the y-axis: \( x = -\dfrac{\pi}{5} \). Thus, \( d = -\dfrac{\pi}{5} \).

Conclusion:
Substituting all values, the equation is: \[ y = 4 \cos\left[ \dfrac{2}{5}\left(x - \left(-\dfrac{\pi}{5}\right)\right) \right] - 3 \implies \mathbf{y = 4 \cos\left[ \dfrac{2}{5}\left(x + \dfrac{\pi}{5} \right) \right] - 3} \]

Step 1: Find Vertical Shift and Amplitude
One large division on the y-axis is 1 unit, so a small division is \( 0.2 \). From the graph, \( y_{\text{max}} = 0.2 \) and \( y_{\text{min}} = -1.4 \).
\[ c = \dfrac{0.2 + (-1.4)}{2} = -0.6 \] \[ |a| = \dfrac{0.2 - (-1.4)}{2} = 0.8 \] We will use a positive cosine function, so \( a = 0.8 \).

Step 2: Find the Period and \(b\)
The x-axis has large divisions of \( \pi \). Points A and B are consecutive maximums at \( A\left(\dfrac{\pi}{2}, 0.2\right) \) and \( B\left(\dfrac{9\pi}{2}, 0.2\right) \).
The period is the distance between A and B: \( P = \dfrac{9\pi}{2} - \dfrac{\pi}{2} = 4\pi \).
\[ \dfrac{2\pi}{|b|} = 4\pi \implies |b| = \dfrac{1}{2} \]

Step 3: Find the Phase Shift (\(d\))
Using the cosine function, the maximum occurs at point A, where \( x = \dfrac{\pi}{2} \). This means the graph is shifted to the right by \( \dfrac{\pi}{2} \), so \( d = \dfrac{\pi}{2} \).

Conclusion:
The equation for the graph is: \[ \mathbf{y = 0.8 \cos\left[ \dfrac{1}{2}\left(x - \dfrac{\pi}{2}\right) \right] - 0.6} \]

Step 1: Find Vertical Shift and Amplitude
From the graph, \( y_{\text{max}} = 0 \) and \( y_{\text{min}} = -2 \).
\[ c = \dfrac{0 + (-2)}{2} = -1 \] \[ |a| = \dfrac{0 - (-2)}{2} = 1 \] The vertical center line is at \( y = -1 \).

Step 2: Find the Period and \(b\)
Points A and B mark a full wave cycle along the center line. Point A is at \( x = 0.6 \) and Point B is at \( x = 2.6 \).
Period \( P = 2.6 - 0.6 = 2 \).
\[ \dfrac{2\pi}{|b|} = 2 \implies |b| = \pi \]

Step 3: Find the Phase Shift (\(d\))
We are using the sine function: \( y = a \sin[ b(x - d) ] + c \).
The unshifted sine function starts at its center line and heads upwards. Our graph does exactly this at Point A, where \( x = 0.6 \) (or \( \dfrac{3}{5} \)). Therefore, the shift to the right is \( d = 0.6 \). Because the graph goes up from here, \( a \) remains positive \( 1 \).

Conclusion:
The equation of the graph is: \[ \mathbf{y = \sin\left[ \pi\left(x - \dfrac{3}{5}\right) \right] - 1} \]

Step 1: Find Vertical Shift and Amplitude
From the y-axis, \( y_{\text{max}} = -1 \) and \( y_{\text{min}} = -3 \).
\[ c = \dfrac{-1 + (-3)}{2} = -2 \] \[ |a| = \dfrac{-1 - (-3)}{2} = 1 \] Using a positive cosine function, \( a = 1 \).

Step 2: Find the Period and \(b\)
Points A and B are consecutive maximums located at \( A\left(\dfrac{3\pi}{5}, -1\right) \) and \( B\left(\dfrac{8\pi}{5}, -1\right) \).
Period \( P = \dfrac{8\pi}{5} - \dfrac{3\pi}{5} = \dfrac{5\pi}{5} = \pi \).
\[ \dfrac{2\pi}{|b|} = \pi \implies |b| = 2 \]

Step 3: Find the Phase Shift (\(d\))
The maximum occurs at Point A, where \( x = \dfrac{3\pi}{5} \). This is our horizontal shift to the right, so \( d = \dfrac{3\pi}{5} \).

Conclusion:
The equation of the graph is: \[ \mathbf{y = \cos\left[ 2\left(x - \dfrac{3\pi}{5} \right) \right] - 2} \]

Step 1: Find Amplitude and Vertical Shift
The maximum is \( 1 \) and the minimum is \( -5 \).
\[ c = \dfrac{1 + (-5)}{2} = -2 \]
The standard amplitude is \( \dfrac{1 - (-5)}{2} = 3 \). Because the problem requires \( a < 0 \), we must use a reflected cosine curve, so \( a = -3 \).

Step 2: Find the Period and \( b \)
The horizontal distance between a minimum and the *very next* maximum represents exactly half of a period.
Half-Period = \( 4 - 0 = 4 \). Therefore, the full Period \( P = 8 \).
\[ \dfrac{2\pi}{|b|} = 8 \implies b = \dfrac{2\pi}{8} = \dfrac{\pi}{4} \]

Step 3: Find the Phase Shift
A standard negative cosine curve \( (- \cos) \) starts at its minimum on the y-axis (where \( x=0 \)). Since our graph's minimum is also at \( x = 0 \), there is no horizontal phase shift required. Thus, \( d = 0 \).

Conclusion:
\[ \mathbf{y = -3\cos\left(\dfrac{\pi}{4}x\right) - 2} \]

Step 1: Find Amplitude and Vertical Shift
Max \( = 24 \), Min \( = 18 \).
Vertical shift (average temp): \( c = \dfrac{24 + 18}{2} = 21 \)
Amplitude: \( a = \dfrac{24 - 18}{2} = 3 \)

Step 2: Find the Period and \( b \)
The time between the minimum (\( t=2 \)) and maximum (\( t=14 \)) is 12 hours. This represents half a cycle.
Period \( P = 24 \) hours.
\[ \dfrac{2\pi}{|b|} = 24 \implies b = \dfrac{2\pi}{24} = \dfrac{\pi}{12} \]

Step 3: Find the Phase Shift for Sine
A positive sine function starts at its center line (\( y=21 \)) and goes *up*. We need to find the time \( t \) when this occurs. The center point going up happens exactly halfway between the minimum and the maximum.
Midpoint in time: \( t = \dfrac{2 + 14}{2} = 8 \) (which is 8:00 AM).
Therefore, the sine wave starts its cycle at \( t = 8 \), meaning the shift is \( d = 8 \).

Conclusion:
\[ \mathbf{T(t) = 3\sin\left(\dfrac{\pi}{12}(t - 8)\right) + 21} \]

Step 1: Substitute the Point
Plug in \( x = \dfrac{\pi}{9} \) and \( y = 6.5 \) into the equation:
\[ 6.5 = 5\sin\left(3\left(\dfrac{\pi}{9}\right) - C\right) + 4 \]

Step 2: Isolate the Sine Term
Subtract 4 from both sides:
\[ 2.5 = 5\sin\left(\dfrac{\pi}{3} - C\right) \]
Divide by 5:
\[ 0.5 = \sin\left(\dfrac{\pi}{3} - C\right) \]

Step 3: Solve the Trigonometric Equation
We know that the sine function equals \( 0.5 \) at angles like \( \dfrac{\pi}{6} \) and \( \dfrac{5\pi}{6} \). Let's set the inner expression equal to these principal values.

Case 1:
\[ \dfrac{\pi}{3} - C = \dfrac{\pi}{6} \implies C = \dfrac{\pi}{3} - \dfrac{\pi}{6} \implies \mathbf{C = \dfrac{\pi}{6}} \]

Case 2:
\[ \dfrac{\pi}{3} - C = \dfrac{5\pi}{6} \implies C = \dfrac{\pi}{3} - \dfrac{5\pi}{6} \implies C = -\dfrac{3\pi}{6} = -\dfrac{\pi}{2} \]
Since the problem asks for the *smallest positive* value, we select the result from Case 1.

Conclusion:
The smallest positive value is \( C = \dfrac{\pi}{6} \).