AP Precalculus Section 2.2 Example: Write an Exponential Equation from a Graph

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 write an equation in the form \(y = a \cdot b^x\) using a graph and no calculator, you'll need to identify key information from the graph, particularly the initial value (\(a\)) and the growth or decay factor (\(b\)).

Here are the steps:

1. **Identify the Initial Value (\(a\)):**
- Look for the point where the graph intersects the y-axis. This is the y-intercept, and it corresponds to the initial value (\(a\)) in the exponential equation.

2. **Determine the Growth or Decay Factor (\(b\)):**
- Examine the behavior of the graph as it moves along the x-axis.
- If the graph rises as \(x\) increases, \(b\) is greater than 1 (exponential growth).
- If the graph falls as \(x\) increases, \(b\) is between 0 and 1 (exponential decay).
- If the graph oscillates with alternating signs, \(b\) is negative.

3. **Write the Equation:**
- Plug the identified values of \(a\) and \(b\) into the exponential form:
\[ y = a \cdot b^x \]

4. **Example:**
- Suppose the graph intersects the y-axis at \(y = 2\) and exhibits exponential growth. Your equation could be:
\[ y = 2 \cdot 3^x \]
where \(a = 2\) (initial value) and \(b = 3\) (growth factor).

5. **Check with Points on the Graph:**
- If there are other points on the graph that you can easily read, check to see if they satisfy your equation.

Remember that this method provides a rough estimate, and the accuracy depends on the clarity of the graph and your ability to visually identify key points. If you have access to a calculator, it can be beneficial for more precise calculations.

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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