AP Precalculus Section 3.7 Example: Identify Parts of a Sinusoidal Function

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

The given function is \(h(t) = 30 \sin\left(\frac{\pi t}{15}\right) + 60\). Let's break down each part of the function:

1. **\(h(t)\):**
- This represents the height of a sinusoidal pattern at time \(t\). In this context, it could represent the height of an object, the temperature, or any other quantity that varies periodically with time.

2. **\(30\):**
- This is the amplitude (\(A\)) of the sinusoidal function. The amplitude is the half vertical distance between the maximum and minimum values of the function. Here, \(A = 30\), so the function oscillates between \(60 + 30 = 90\) and \(60 - 30 = 30\).

3. **\(\sin\left(\frac{\pi t}{15}\right)\):**
- This is the sine function with a period of \(T = 15\) units. The period (\(T\)) is the time it takes for one complete cycle of the sinusoidal pattern. In this case, it takes \(15\) units of time for the function to complete one cycle.

4. **\(\frac{\pi t}{15}\):**
- This part inside the sine function represents the argument of the sine function. It determines the rate at which the sinusoidal function oscillates. In this case, the function completes one cycle as \(t\) ranges from \(0\) to \(15\), so it has a frequency of \(1/T = \frac{1}{15}\).

5. **\(60\):**
- This is the vertical shift (\(D\)) of the sinusoidal function. It represents a vertical translation of the entire function. Here, the function is shifted upward by \(60\) units.

In summary, the function \(h(t) = 30 \sin\left(\frac{\pi t}{15}\right) + 60\) models a sinusoidal pattern with an amplitude of \(30\), a period of \(15\), a frequency of \(\frac{1}{15}\), and a vertical shift upward by \(60\) units. The variable \(t\) represents time, and \(h(t)\) represents the height or value of the periodic quantity at time \(t\).

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

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

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