AP Precalculus Section 3.6 Example: Graph y = 4sin(x/2) and Find Amplitude, Period, Midline, etc.

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

To graph the function \(y = 4 \sin\left(\frac{x}{2}\right)\) and find its amplitude, period, midline, and other characteristics, follow these steps:

### Graphing \(y = 4 \sin\left(\frac{x}{2}\right)\):

1. **Amplitude (\(A\)):**
- The amplitude is the coefficient of the sine function. In this case, \(A = 4\), so the amplitude is \(4\). This means the graph will oscillate between \(4\) and \(-4\).

2. **Period (\(T\)):**
- The period of the sine function is given by \(T = \frac{2\pi}{\text{coefficient of } x}\). Here, \(T = \frac{2\pi}{\frac{1}{2}} = 4\pi\). So, the period is \(4\pi\).

3. **Midline:**
- The midline is the horizontal line that the sine function oscillates around. For \(y = 4 \sin\left(\frac{x}{2}\right)\), the midline is \(y = 0\).

4. **Vertical Stretch/Compression:**
- The coefficient of \(x\) inside the sine function affects the vertical stretch or compression. Here, \(\frac{1}{2}\) indicates a stretch by a factor of \(2\) compared to the standard sine function.

5. **Graph the Function:**
- Plot key points based on the amplitude, period, and midline. Use the fact that the sine function starts at the midline, reaches its maximum at \(\text{midline} + \text{amplitude}\), returns to the midline, reaches its minimum at \(\text{midline} - \text{amplitude}\), and repeats.

### Summary:

- **Amplitude (\(A\)):** \(4\)
- **Period (\(T\)):** \(4\pi\)
- **Midline:** \(y = 0\)
- **Vertical Stretch/Compression:** Stretch by a factor of \(2\)
- **Key Points:** Use the midline, amplitude, and period to graph the sine function.

The graph of \(y = 4 \sin\left(\frac{x}{2}\right)\) will exhibit a stretched oscillation with an amplitude of \(4\) and a period of \(4\pi\), oscillating around the midline \(y = 0\).

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

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

#APPrecalculus
#PreCalcProblems
#MathMinds
#AdvancedPreCalc
#TrigTales
#PrecalcPuzzles
#FunctionFiesta
#GraphGoals
#CalcReady
#PreCalcLife
#AlgebraicAdventures
#DerivativeDreams
#IntegrationInsights
#MathematicsMagic
#PreCalcReview
#PrecalcConcepts
#LogarithmLove
#TrigonometryTips
#MathMastermind
#APCalcPrep
#Mathematics
#MathMinds
#Math
#Maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study