Problem 73 - Algebra and Trigonometry

Question

\(73-74\) A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .\) (b) How are the graphs in part (a) related? $$ f(x)=\log (c x) $$

Step-by-Step Solution

Answer

Graphs shift right as \( c \) increases, compressing horizontally.

Step 1: Understand the Function

The given function is \( f(x) = \log(c x) \). \( c \) is a parameter that affects the logarithm's input. We'll evaluate how changing \( c \) affects the graph of the function.

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

Substitute \( c = 1 \) into the function: \( f(x) = \log(1 \cdot x) = \log(x) \). This is the standard logarithmic function graph, which starts at negative infinity as \( x \) approaches zero from the right and increases indefinitely without a bound.

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

Substitute \( c = 2 \) into the function: \( f(x) = \log(2x) \). The graph of \( f(x) = \log(2x) \) shifts the \( \log(x) \) graph to the right, meaning it also starts at negative infinity for \( x \to 0^+ \) but at a slower increasing rate.

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

Substitute \( c = 3 \) into the function: \( f(x) = \log(3x) \). This graph shifts further compared to \( \log(2x) \). The graph's progression from left to right happens at a more gradual pace.

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

Substitute \( c = 4 \) into the function: \( f(x) = \log(4x) \). The logarithm's base shifts even further right, following the pattern observed with previous values of \( c \). The graphs become less steep as they shift further to the right.

Step 6: Compare the Graphs

Observe that as \( c \) increases, the graph of \( f(x) = \log(cx) \) shifts to the right. Specifically, the vertical position at any given \( x > 0 \) increases, but the slope of increase with \( x \) becomes less steep as \( c \) increases.

Step 7: Relative Transformation

Consider the transformation effect, which is a horizontal compression. With each increasing \( c \), the same functional shape is preserved. The basic shift of the logarithmic graph \( \log(x) \) is horizontally compressed by a factor \( 1/c \).

Key Concepts

Effect of Parameter Change on Functions

When we discuss the effect of parameter change on functions, we focus on how altering a parameter, like \( c \) in the function \( f(x) = \log(c x) \), impacts the graph's shape and position. Here, \( c \) acts as a scaling factor. The parameter \( c \) directly influences where and how the graph of the logarithmic function behaves.

• For \( c = 1 \): The function \( f(x) = \log(x) \) represents the standard logarithmic curve starting from negative infinity and gradually increasing without bounds as \( x \) increases.
  
• Increasing \( c \): As you increase \( c \) to 2, 3, and 4, each consecutive graph doesn’t just shift to the right but also changes its rate of rise, becoming less steep. The graph begins at the same vertical point close to the y-axis for \( x \to 0^+ \), but the pace of increase slows.
  
• Real-World Applications: Understanding these effects is crucial in fields like physics or economics, where functions model real-world data and changes represent varying conditions.

Horizontal Compression

The concept of horizontal compression refers to how the graph of a function tightens or compresses towards the y-axis. In the function \( f(x) = \log(cx) \), the horizontal compression happens as \( c \) increases. This effect reveals itself in how the x-values at which certain y-values occur shrink closer to the origin. When the parameter \( c \) is greater than one, the function \( \log(cx) \) compresses horizontally:

• Mathematical Observation: For every increase in \( c \), the x-value corresponding to a particular y-value is \( 1/c \) of what it would be in the base function \( \log(x) \). This means the graph compresses towards the y-axis instead of stretching away from it.
• Visualizing Compression: If you compare the graphs for \( c = 1 \) and \( c = 4 \), you will notice that for the same y-value, the x they correspond to is significantly smaller for \( c = 4 \).

The results show how the higher the value of \( c \), the more the horizontal compression, affecting how and where the graph peaks and levels off.

Graph Transformations

Graph transformations are changes applied to the standard graph of a function through operations such as shifting, stretching, or reflecting. For the function \( f(x) = \log(c x) \), we see transformations in its graphical representation when changing \( c \). These transformations help us understand how functions can be adjusted to fit different scenarios or datasets. Some key transformations observed:

• Horizontal Shifting: Increasing \( c \) shifts the graph rightwards. Though the fundamental shape of the graph remains unchanged, the x-values now represent a compressed version relative to original value points.
• Transformation Analysis: As \( c \) increases, the vertical status of any particular value on the graph that was shifted also translates higher on the x-axis, making every segment cover a larger base distance.
• Application: Graph transformations, such as these, are vital in tailoring mathematical models to better fit empirical data by modifying parameters to reflect real data trends.

Understanding these transformations equips students and professionals with the knowledge to manipulate functions accurately for designing, predicting, and interpreting data.