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Magazine Chapter 4: Problem 73
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? $$f(x)=\log (c x)$$
Short Answer
Expert verified
As \( c \) increases, the graph of \( \log(cx) \) stretches horizontally, rising higher for any given \( x \).
Step by step solution
01
Understand the Function
The function given is \( f(x) = \log(cx) \). Understanding that \( c \) is a parameter, not a variable, is key. It affects the graph of the logarithmic function by stretching it horizontally based on the value of \( c \). The function is defined for positive values of \( x \).
02
Plot Graph for c=1
When \( c = 1 \), the function becomes \( f(x) = \log(x) \). This is the basic logarithmic function graph, starting at \( (1,0) \), with a vertical asymptote at \( x = 0 \), and increasing slowly as \( x \) increases.
03
Plot Graph for c=2
For \( c = 2 \), the function is \( f(x) = \log(2x) \). This graph is a horizontal stretch of the basic logarithm graph. Every \( x \) value on the basic graph now corresponds to half that \( x \) value, causing the graph to shift slightly higher at any given \( x \) compared to \( f(x) = \log(x) \).
04
Plot Graph for c=3
With \( c = 3 \), the function becomes \( f(x) = \log(3x) \). This further stretches the graph horizontally compared to \( f(x) = \log(2x) \), moving it even higher at any point \( x \) since increasing \( c \) increases the horizontal stretch.
05
Plot Graph for c=4
Finally, for \( c = 4 \), the function is \( f(x) = \log(4x) \). This graph continues the trend of stretching the basic logarithmic graph horizontally, rising faster than the previous graphs.
06
Analyze the Relationship Between Graphs
All these graphs, \( f(x)=\log(cx) \) for \( c=1, 2, 3, 4 \), show horizontal stretches of the basic \( \log(x) \) graph as \( c \) increases. The larger the \( c \), the more pronounced the shift to the right and the higher the graph appears at any given \( x \). The vertical asymptote remains at \( x = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Horizontal Stretching
Logarithmic functions like \( f(x) = \log(cx) \) can be horizontally stretched by changing the value of \( c \). When we say horizontal stretching, we mean that the graph of the function spreads out horizontally along the x-axis.
Here's how it works:
Here's how it works:
- When \( c = 1 \), the graph is the standard logarithmic graph, \( f(x) = \log(x) \).
- If \( c > 1 \), each point on the graph moves to the right, which makes the graph stretch horizontally.
- The x-value that was originally 1 will become \( \frac{1}{c} \) instead, because \( \log(c \cdot 1) = \log(c) \).
Graph Transformation
Graph transformation involves shifts and stretches that alter the appearance of a function's graph without changing its basic shape. In the case of logarithmic functions like \( f(x) = \log(cx) \), transformation happens mainly due to horizontal stretching.
When analyzing the function
When analyzing the function
- The core shape of a logarithmic function remains the same: starting at a certain point and moving upwards albeit slowly.
- The transformation, primarily controlled by \( c \), changes the nature of how quickly it rises and at which point on the x-axis it begins to increase significantly.
- Unlike vertical shifts that add or subtract whole values, these transformations are about rescaling or compressing.
Remember that the y-values follow the same pattern of increase; yet, they are spread out, or 'stretched', across different x-positions!
Asymptotic Behavior
Asymptotic behavior relates to how a function behaves as it approaches a particular line, either vertically or horizontally. For logarithmic graphs like \( f(x) = \log(cx) \), the asymptotic property is vertical as the graph nears the y-axis.
Consider these key points:
Consider these key points:
- Irrespective of the value of \( c \), the vertical asymptote for \( \log(cx) \) always remains at \( x = 0 \). This is because logarithmic functions are undefined for zero and negative x-values.
- As \( x \) gets closer to zero, the value of \( \log(cx) \) decreases without bound, depicting how the graph hugs the y-axis.
- The importance of this behavior is that it offers insight into the boundary and limitations of logarithmic functions, essentially acting as a reminder that these graphs can never cross their respective asymptote.
