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Find an equation for the sine graph f(x) in the form f(x) = A sin(Bx + C) + D.

Check the final answer first, then review the worked steps.

Problem

Find an equation for the sine graph f(x) in the form f(x) = A sin(Bx + C) + D.

Answer

\(f(x) = \sin(x) + 5\)

Step-by-step solution

  1. Determine the Amplitude (A): The amplitude is half the difference between the maximum and minimum values of the function. The maximum value is 6 and the minimum value is 4. So, $A = \frac{6 - 4}{2} = \frac{2}{2} = 1$.
  2. Determine the Vertical Shift (D): The vertical shift is the average of the maximum and minimum values. So, $D = \frac{6 + 4}{2} = \frac{10}{2} = 5$.
  3. Determine the Period: The period is the horizontal distance over which the function completes one full cycle. By observing the graph, we can see that a full cycle occurs between $x = 0$ and $x = 2\pi$. Therefore, the period is $2\pi$.
  4. Determine the Angular Frequency (B): The angular frequency is related to the period by the formula: Period $= \frac{2\pi}{|B|}$. Since the period is $2\pi$, we have $2\pi = \frac{2\pi}{|B|}$. This implies $|B| = 1$. We will assume $B = 1$ for simplicity, as the graph is not reflected across the x-axis.
  5. Determine the Phase Shift (C): The standard sine function $y = \sin(x)$ starts at $(0,0)$ and increases. Our graph has a midline at $y=5$. The maximum value is at $y=6$ and the minimum value is at $y=4$. Let's find a point where the graph crosses its midline and is increasing. Looking at the graph, the function crosses the midline $y=5$ at $x = \frac{\pi}{2}$ and is increasing. For a standard sine function $y = \sin(x)$, it crosses the x-axis at $x=0$ and is increasing. Therefore, the phase shift is $C=0$. The equation is of the form $f(x) = A\sin(Bx + C) + D$. Substituting the values we found: $f(x) = 1\sin(1x + 0) + 5$. Simplifying this gives $f(x) = \sin(x) + 5$.
  6. Verify the equation: Let's check a few points. At $x=0$, $f(0) = \sin(0) + 5 = 0 + 5 = 5$. This matches the graph. At $x = \frac{\pi}{2}$, $f(\frac{\pi}{2}) = \sin(\frac{\pi}{2}) + 5 = 1 + 5 = 6$. This matches the graph (maximum value). At $x = \frac{3\pi}{2}$, $f(\frac{3\pi}{2}) = \sin(\frac{3\pi}{2}) + 5 = -1 + 5 = 4$. This matches the graph (minimum value).