Exponential and Logarithmic Functions

Graphing Exponential Functions

Exponential functions are a key topic in precalculus, often written in the form , where is a constant and is the base. Understanding their graphs is essential for analyzing growth and decay in various contexts.

• Exponential Growth: Occurs when . The function increases rapidly as increases.

• Exponential Decay: Occurs when . The function decreases as increases.

• Key Features:

  • Domain:

  • Range: for

  • Y-intercept: At

  • Horizontal Asymptote:

Example: For , the graph shifts the basic function down by 3 units. The horizontal asymptote is .

Identifying Domain and Range

The domain of an exponential function is all real numbers, while the range depends on vertical shifts and the sign of .

• Domain:

• Range: For , if , range is ; if , range is .

Matching Graphs to Equations

To match a graph to its equation, observe:

• Where the graph crosses the y-axis (y-intercept)

• The direction of growth or decay

• Any vertical shifts (asymptote location)

Exponential vs. Linear, Quadratic, and Other Functions

Recognizing the type of function is important for correct graphing and analysis.

Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. For example, .

• Used for very large or very small numbers.

• Important for expressing exponential growth or decay in real-world contexts.

Example Problem

Question: Sketch the graph of . State the domain, range, and asymptote.

• Solution:

  • Domain:

  • Range:

  • Horizontal Asymptote:

  • Y-intercept:

Practice Matching

Given a table of equations and graphs, match each equation to its corresponding graph by analyzing the features above.