AP Precalculus Section 2.3 Example: Exponential Function from a Table

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

If you have a table of reasonable values and you want to write an exponential equation in the form \(Y = a \cdot b^x\), you can follow these steps:

1. **Select Data Points:**
- Look at your table and select a pair of corresponding values for \(x\) and \(Y\).

2. **Substitute Values:**
- Plug these values into the equation. This gives you one equation to solve for the product \(ab^x\).

3. **Repeat with Another Pair:**
- Choose another pair of values and substitute them into the equation. This will give you a second equation.

4. **Set Up a System of Equations:**
- You now have two equations with the unknowns \(a\) and \(b\). Set up a system of equations with these two equations.

5. **Solve the System:**
- Use algebraic methods to solve the system of equations and find the values of \(a\) and \(b\).

6. **Write the Exponential Equation:**
- Once you have \(a\) and \(b\), substitute them back into the original equation \(Y = a \cdot b^x\).

For example, if your selected pairs are (1, 5) and (2, 10), you would set up the following system of equations:

Equation 1: \(5 = a \cdot b^1\)
Equation 2: \(10 = a \cdot b^2\)

Solve this system to find \(a\) and \(b\), and then substitute them back into \(Y = a \cdot b^x\) to get your exponential equation.

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:

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

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