Problem 81 - Algebra and Trigonometry

Question

A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .(\text { b) How are the graphs in part (a) }\) related? $$ f(x)=\log (c x) $$

Step-by-Step Solution

Answer

The graphs for different \( c \) values are vertical shifts of \( \log(x) \) with shifts equal to \( \log(c) \).

Step 1: Understand the Family of Functions

The family of functions given is \( f(x) = \log(cx) \). This means that for different values of \( c \), the shape of the graph remains the same, but its position changes. The logarithmic function \( \log(x) \) is normally defined for \(x > 0\), and \( \log(cx) \) implies the graph will be shifted when \( c \) changes.

Step 2: Graph for \( c=1 \)

For \( c = 1 \), the function is \( f(x) = \log(x) \). This is the standard logarithmic function, which passes through the point (1,0), has a vertical asymptote at \( x = 0 \), and increases without bound as \( x \) increases.

Step 3: Graph for \( c=2 \)

For \( c = 2 \), the function becomes \( f(x) = \log(2x) \). Since \( \log(2x) = \log(2) + \log(x) \), the graph of \( \log(2x) \) is just the graph of \( \log(x) \) shifted upwards by \( \log(2) \) units.

Step 4: Graph for \( c=3 \)

For \( c = 3 \), the function is \( f(x) = \log(3x) \). Similarly, \( \log(3x) = \log(3) + \log(x) \), so the graph is shifted upwards by \( \log(3) \) units compared to \( \log(x) \).

Step 5: Graph for \( c=4 \)

For \( c = 4 \), we have \( f(x) = \log(4x) \). Here, \( \log(4x) = \log(4) + \log(x) \), indicating the graph is shifted upwards by \( \log(4) \) compared to the standard logarithm.

Step 6: Analyze Relationship Between Graphs

Each graph for \( c=1,2,3,4 \) is a vertical shift of the logarithmic function \( \log(x) \) by \( \log(c) \). All graphs share a common vertical asymptote at \( x=0 \) but differ in the height of their points based on the value of \( c \). The larger \( c \), the higher the graph is shifted.

Key Concepts

Graph Transformations

Graph transformations are an essential concept in mathematics, enabling us to alter graphs in various ways while maintaining their fundamental shape. For logarithmic functions such as \( f(x) = \log(cx) \), changing the parameter \( c \) affects the graph's positioning along the axes. Each \( c \) value results in a specific transformation of the base function \( \log(x) \).
\(\bullet\) Horizontal Stretch/Compression: Although this particular function doesn't show visible stretching or compressing, in general cases where \( f(x) = \log(kx) \), if \( k > 1 \), the graph compresses horizontally; if \( 0 < k < 1 \), it stretches.
\(\bullet\) Vertical Shift: Every transformation associated with \( f(x) = \log(cx) \) for different values of \( c \) is a vertical shift, discussed further in the next section. Understanding these shifts helps predict the graph's overall behavior and appearance.

Vertical Shifts

A vertical shift in a logarithmic function occurs when we add or subtract a constant to/from the function. In our function family \( f(x) = \log(cx) \), each change in \( c \) equates to an additional constant \( \log(c) \) being added to \( \log(x) \).
This shift is calculated as the function transforms as follows: \( \log(cx) = \log(c) + \log(x) \). Here, each graph is essentially the graph of \( \log(x) \) moved upwards by an amount equivalent to \( \log(c) \).
\(\bullet\) When \( c = 1 \), the graph remains unchanged at its base position, since \( \log(1) = 0 \).
\(\bullet\) For \( c > 1 \), the graph moves up. For example, when \( c = 2, 3, \) and \( 4 \), the shifts are \( \log(2), \log(3), \) and \( \log(4) \) units up the vertical axis, respectively.
Recognizing this shift helps one quickly determine how the graph will look relative to the standard \( \log(x) \) graph.

Asymptotes

In logarithmic functions, an asymptote is a line that the graph approaches but never actually reaches. For the family \( f(x) = \log(cx) \), the vertical asymptote is at \( x = 0 \). This asymptotic behavior appears because the logarithm is undefined for zero or negative values, meaning the function shoots towards negative infinity as \( x \) approaches zero.
This characteristic remains consistent across all values of \( c \) in \( \log(cx) \). No matter how much the function vertically shifts due to different \( c \) values, the asymptote does not move. Instead, it continually serves as a boundary that describes the graph's behavior close to \( x = 0 \).
Understanding this concept is crucial, as it dictates the set of possible \( x \)-values (domain) for the function. It reinforces how the mathematical structure of logarithmic functions controls their unique graphical properties.