AP Precalculus Practice Test: Unit 3 FRQ #1 Periodic, Sinusoidal, Graphs, and Coordinates

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

This is a Free Response Question!

**AP Precalculus Practice Test: Unit 3, FRQ #1** involves periodic, sinusoidal graphs and coordinates.

---

### **Key Concepts**:
1. **Periodic Functions**: Functions that repeat in regular intervals. Trigonometric functions like sine and cosine are periodic.
2. **Sinusoidal Functions**: The general form is:
\[
y = a \sin(bx + c) + d \quad \text{or} \quad y = a \cos(bx + c) + d
\]
- \( a \): Amplitude (vertical stretch/compression)
- \( b \): Affects the period, \( \text{Period} = \frac{2\pi}{|b|} \)
- \( c \): Phase shift, \( \text{Phase shift} = \frac{-c}{b} \)
- \( d \): Vertical shift
3. **Key Points**: For \( y = \sin(x) \), important points are \( (0, 0) \), \( (\frac{\pi}{2}, 1) \), \( (\pi, 0) \), \( (\frac{3\pi}{2}, -1) \), and \( (2\pi, 0) \), repeating every period.

---

### **Steps to Solve**:

1. **Identify Parameters**: Extract \( a \), \( b \), \( c \), and \( d \) from the function.
2. **Calculate Period, Amplitude, Phase Shift, and Vertical Shift**:
- Period = \( \frac{2\pi}{|b|} \)
- Amplitude = \( |a| \)
- Phase shift = \( \frac{-c}{b} \)
- Vertical shift = \( d \)
3. **Graph the Function**: Plot key points based on these parameters, considering symmetry of sine or cosine graphs.

---

### **Example**:

For \( f(x) = 2 \sin(3x - \frac{\pi}{2}) + 1 \):

1. **Parameters**:
- \( a = 2 \), Amplitude = 2
- \( b = 3 \), Period = \( \frac{2\pi}{3} \)
- \( c = -\frac{\pi}{2} \), Phase shift = \( \frac{\pi}{6} \) right
- \( d = 1 \), Vertical shift = 1

2. **Graph**: The midline is \( y = 1 \), with the graph oscillating between -1 and 3.

3. **Key Points**: At \( x = 0 \), \( f(0) = -1 \); use the period and phase shift to find other points.

---

### **Summary**:
- Sinusoidal graphs have amplitude, period, phase shift, and vertical shift. Use these features to graph the function and find key coordinates.

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study