Parent Graphs and Transformations of Logarithmic Functions

This section introduces the parent graphs for logarithmic functions and explores how to graph transformed logarithmic functions.

The lesson presents two cases:

1. Graph of f(x) = log_b x for b > 1
2. Graph of f(x) = log_b x for 0 < b < 1

Highlight: The graph of f(x) = log_b x is the reflection of g(x) = b^x in the line y = x.

The lesson then moves on to graphing transformed logarithmic functions. It provides an example of graphing f(x) = log_3(x-1) and explains the steps involved.

Example: Given y = a log(x-h) + k, the vertical asymptote is at x = h

Students are then given an opportunity to practice graphing a transformed logarithmic function:

Example: Graph f(x) = -log_2 x - 1

The lesson concludes with a formative assessment, asking students to rewrite and graph y = log_2 x - 1.

Vocabulary: Transformation - a change in the shape, size, or position of a graph

This comprehensive lesson provides students with a solid foundation in logarithmic function graphs, their relationship to exponential functions, and how to apply transformations to these graphs.

6.3 Day 2 Lesson: Logarithmic Functions

This lesson begins with a review of logarithmic expressions and their simplification. Students are reminded of an upcoming graded assessment and are instructed to check their answers for previous practice activities.

Highlight: The lesson emphasizes the inverse relationship between exponential and logarithmic functions.

The lesson introduces the concept that logarithmic functions are the inverse of exponential functions. This is expressed mathematically as:

g(f(x)) = log b^x = x f(g(x)) = b^(log_b x) = x

Definition: Logarithmic functions and exponential functions "undo" each other, meaning they are inverse functions.

Students are then given practice problems to simplify logarithmic expressions and evaluate logarithms.

Example: Simplify log_5(5^2) = 2

The lesson also covers finding the inverse of functions, including exponential and logarithmic functions.

Vocabulary: The inverse function is denoted as f^(-1)(x) and represents the "undoing" of the original function.