To graph the function $f(x) = 5\sin\left(\frac{\pi}{2} x - \pi\right) + 3$, let's break down the key components of the sine function and how they affect the graph:

1. Amplitude:

• The amplitude is the coefficient in front of the sine function, which is $5$. This means the graph oscillates 5 units above and below the midline.

2. Period:

• The period of the sine function is determined by the coefficient of $x$ inside the sine function. The period is calculated as: $\text{Period} = \frac{2\pi}{\frac{\pi}{2}} = 4$
• This means the sine wave completes one full cycle every 4 units along the x-axis.

3. Phase Shift:

• The phase shift is determined by the constant added or subtracted inside the sine function. The phase shift is calculated by setting the inside of the sine function equal to zero: $\frac{\pi}{2}x - \pi = 0 \quad \Rightarrow \quad x = 2$
• This means the graph is shifted 2 units to the right.

4. Vertical Shift:

• The entire graph is shifted up by 3 units because of the "+3" at the end of the function.

5. Midline:

• The midline of the graph is at $y = 3$, which is the vertical shift.

Graphing Steps:

1. Plot the Midline: Draw a horizontal line at $y = 3$.
2. Determine Key Points:
   • The sine function starts at $x = 2$ (due to the phase shift), reaching the maximum at $x = 2 + 1 = 3$, crossing the midline at $x = 4$, reaching the minimum at $x = 5$, and completing the cycle at $x = 6$.
3. Sketch the Sine Curve:
   • Start from the midline, go up to the maximum of $y = 8$ (since $3 + 5 = 8$), back down to the midline, then to the minimum of $y = -2$ (since $3 - 5 = -2$), and return to the midline.

Would you like the graph or more detailed explanations on specific aspects?

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Here are some related questions you might find useful:

1. How does changing the amplitude affect the graph of a sine function?
2. What is the effect of a horizontal phase shift on trigonometric functions?
3. How do you determine the period of any sinusoidal function?
4. What role does the vertical shift play in trigonometric graphs?
5. Can you describe how to find the maximum and minimum points on the graph of a sine function?

Tip: When graphing sine and cosine functions, always start by identifying the amplitude, period, phase shift, and vertical shift to plot the key points accurately.