Problem 2 - College Algebra with Corequisite Support

Question

What type(s) of translation(s), if any, affect the range of a logarithmic function?

Step-by-Step Solution

Answer

Vertical translations affect the range of a logarithmic function.

Step 1: Understand the Function

A logarithmic function of the form \( f(x) = \log_b(x) \) has a domain of \( x > 0 \) and a range of all real numbers.

Step 2: Translation Definitions

Translations can occur in two primary ways: vertical translations (up or down) and horizontal translations (left or right). A function \( g(x) = f(x) + c \) represents a vertical shift, while \( h(x) = f(x - d) \) represents a horizontal shift.

Step 3: Determine Effects of Vertical Translations

When a logarithmic function \( f(x) \) is vertically translated by adding a constant \( c \), it becomes \( g(x) = f(x) + c \). This affects the output value (range), shifting the entire range by \( c \) units.

Step 4: Analyze Effects of Horizontal Translations

When the function is horizontally translated using \( h(x) = f(x - d) \), the graph shifts left or right but the range remains unchanged as the input values are the only aspect affected.

Step 5: Conclusion

Vertical translations shift the range of a logarithmic function, whereas horizontal translations do not affect the range.

Key Concepts

Translations in Functions

When we talk about translations in functions, we are referring to the shifting of the graph of a function from one place to another on a coordinate plane. These shifts do not alter the shape of the graph, only its position. Translations are used to modify functions and adjust their intersection points along the axes. In the context of a logarithmic function, the translation can either be vertical or horizontal, which will adjust where the graph appears on the coordinate plane but will follow the same logarithmic curve.

Vertical Translation

A vertical translation involves moving the entire graph of a function up or down along the y-axis. With a logarithmic function denoted by \( f(x) = \log_b(x) \), if we vertically translate it, it changes to \( g(x) = f(x) + c \), where \( c \) is a constant.

This translation impacts the output values, meaning the range of the function is affected. For instance:

• If \( c > 0 \), the graph shifts upwards, and each y-value increases by \( c \), moving the entire range up.
• If \( c < 0 \), the graph shifts downwards, and each y-value decreases by \( c \), moving the entire range down.

So, vertical translations directly alter where the graph cuts along the y-axis, hence changing the range.

Horizontal Translation

Horizontal translations move the graph of a function left or right along the x-axis. For a function \( f(x) = \log_b(x) \), a horizontal translation is represented as \( h(x) = f(x-d) \). Here, \( d \) is the amount of horizontal shift.

It is important to note:

• If \( d > 0 \), the graph shifts to the right by \( d \) units.
• If \( d < 0 \), the graph shifts to the left by \( d \) units.

However, unlike vertical translations, horizontal translations do not affect the range of a logarithmic function. They only change the domain, as the log function moves along the x-axis while preserving the set of y-values.

Domain and Range

Understanding domain and range is crucial when working with any function, including logarithms. The domain of a function refers to all possible input values (x-values), whereas the range refers to all possible output values (y-values).

For a basic logarithmic function \( f(x) = \log_b(x) \):

• The domain is all positive real numbers, \( x > 0 \). This is because logarithms of negative numbers and zero are undefined in real numbers.
• The range is all real numbers, \( y \in \mathbb{R} \), since a logarithm can produce any real number output as \( x \) varies within its domain.

Once a vertical translation is applied, it affects the range by shifting all output values up or down. In contrast, a horizontal translation modifies the domain but keeps the range untouched.