Question

Precalc B: Unit 2 Test Part 2 Modified 5 points S25

Find the function represented by the graph below. Be sure to show your work step by step or you will not earn any credit. If any of the parameters in y = af(bx +c) +d are zero, state that and justify why they are zero. Do not just leave the parameter out (5 pts)

Answer 1

To find the function represented by a given graph, you will typically need to analyze its key features, which may include:

1. **Amplitude** (a)
2. **Period** (b)
3. **Phase shift** (c)
4. **Vertical shift** (d)

### Steps to Determine the Function

1. **Identify the Type of Function**: Determine if the graph represents a sine, cosine, or other type of periodic function. Let's assume it is a sine function for this example: \(y = a \sin(bx + c) + d\).

2. **Determine the Amplitude (a)**:
- The amplitude is the distance from the midline of the graph to the maximum or minimum value.
- Calculate the peak height and trough depth.
- Use the formula \(a = \frac{\text{Max} - \text{Min}}{2}\).

3. **Determine the Vertical Shift (d)**:
- The vertical shift is the average of the maximum and minimum values, which gives the midline.
- Use the formula \(d = \frac{\text{Max} + \text{Min}}{2}\).

4. **Determine the Period (b)**:
- The period is the distance between consecutive peaks (or troughs) of the graph.
- Use the formula for the sine function, which has a period of \(2\pi\), and relate it to the parameter \(b\):
\[ \text{Period} = \frac{2\pi}{|b|} \]

5. **Determine the Phase Shift (c)**:
- If necessary, observe where the function starts relative to the typical sine function.
- If the graph appears shifted horizontally, relate the starting point to the sine function's properties, where \( \sin(0) \) starts at \( 0 \) or a cosine function starts at its maximum.

6. **Construct the Equation**:
- Combine all determined parameters into the function form \(y = a \sin(bx + c) + d\).

7. **Special Cases**:
- If any of the parameters (a, b, c, d) are zero, explain why and how it affects the graph:
- \(a = 0\): The function becomes a constant.
- \(b = 0\): The function does not oscillate (the output is constant).
- \(c = 0\): No horizontal shift (starts at the origin).
- \(d = 0\): The midline is at \(y = 0\).

### Example Calculation (Hypothetical Graph)
Assuming that the graph's maximum is 3, minimum is -1:
- Min = -1, Max = 3
- Amplitude \(a = \frac{3 - (-1)}{2} = \frac{4}{2} = 2\)
- Vertical Shift \(d = \frac{3 + (-1)}{2} = \frac{2}{2} = 1\)

If the period is found to be 4:
- Set \( \frac{2\pi}{|b|} = 4 \Rightarrow |b| = \frac{2\pi}{4} = \frac{\pi}{2}\)

If the phase shift is 0 (starts at origin):
- \(c = 0\)

Putting it all together:
- The function is \(y = 2\sin\left(\frac{\pi}{2}x\right) + 1\)

### Conclusion
In conclusion, analyze the graph thoroughly and apply these steps. If you provide the actual characteristics from the graph you're working with, I can help further in constructing the specific function!