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Magazine Chapter 4: Problem 35
sketch the graph of the equation without using a graphing utility. (a) \(y=\left(\frac{1}{2}\right)^{x-1}-1\) (b) \(y=\ln |x|\)
Short Answer
Expert verified
Both graph sketches involve understanding transformation effects on the function. (a) A rightward shift decaying curve; (b) a mirrored log curve around the origin.
Step by step solution
01
Understanding the Equation - Part (a)
The equation given is in the form of an exponential function: \( y = \left( \frac{1}{2} \right)^{x-1} - 1 \). This is an exponential decay function translated horizontally and vertically. We need to understand how these translations affect the graph.
02
Identify Translations - Part (a)
The function \( y = \left( \frac{1}{2} \right)^{x-1} - 1 \) is derived from \( y = \left( \frac{1}{2} \right)^x \). The term \( x - 1 \) indicates a horizontal translation to the right by 1 unit, and the \( -1 \) outside the exponential part indicates a vertical translation downward by 1 unit.
03
Locate Asymptote - Part (a)
The horizontal asymptote of the function, originally \( y = 0 \) for \( \left( \frac{1}{2} \right)^x \), is moved down to \( y = -1 \) due to the vertical translation, forming the new asymptote.
04
Plot Key Points - Part (a)
Choose key \( x \) values to calculate \( y \): for \( x = 1 \), \( y = 0 \); for \( x = 0 \), \( y = \frac{1}{2} - 1 = -\frac{1}{2} \); for \( x = 2 \), \( y = \frac{1}{4} - 1 = -\frac{3}{4} \). These points help sketch the graph.
05
Understanding the Equation - Part (b)
The given equation is \( y = \ln |x| \). This function represents the natural logarithm of the absolute value of \( x \). It includes both parts for \( x > 0 \) and \( x < 0 \). It is undefined at \( x = 0 \).
06
Identify Domain - Part (b)
The function \( y = \ln |x| \) is only defined for \( x eq 0 \). Its domain is \( x \in (-\infty, 0) \cup (0, \infty) \). It approaches negative infinity as \( x \to 0^+ \) or \( x \to 0^- \).
07
Determine Key Characteristics - Part (b)
For \( x > 0 \), the graph follows the natural logarithm shape, starting from negative infinity and increasing. For \( x < 0 \), the graph mirrors about the \( y \)-axis. As \( |x| \to \infty \), \( y \to \infty \).
08
Sketching the Graph
For (a), sketch the exponential decay reflecting horizontal and vertical shifts: a graph declining to the right from a value of \( y = 0 \) at \( x = 1 \), with an asymptote at \( y = -1 \). For (b), sketch the traditional logarithmic graph for \( x > 0 \) and reflect the shape for \( x < 0 \), avoiding crossing the \( y \)-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Decay
Exponential decay describes a process where quantities decrease rapidly at first and then taper off. It is characterized by a base between 0 and 1 raised to the power of x, such as in the function \( y = \left(\frac{1}{2}\right)^{x-1} - 1 \).
This base value of \( \frac{1}{2} \) indicates that as \( x \) increases, each successive value of \( y \) becomes half of the previous one.
Exponential decay is evident in processes like radioactive decay, cooling of objects, or depreciation of assets.
When overlaying translations, such as \( -1 \) in \( x-1 \), it communicates how the graph moves position on the axes.
This base value of \( \frac{1}{2} \) indicates that as \( x \) increases, each successive value of \( y \) becomes half of the previous one.
Exponential decay is evident in processes like radioactive decay, cooling of objects, or depreciation of assets.
When overlaying translations, such as \( -1 \) in \( x-1 \), it communicates how the graph moves position on the axes.
- The overall decay trend doesn鈥檛 change; it merely shifts according to specified horizontal and vertical translations.
Natural Logarithm
The natural logarithm, often denoted as \( \ln \), is a fundamental mathematical concept. The function is represented as \( y = \ln |x| \), which uses \( e \) (approximately 2.718) as its base.
Unlike regular logarithms that you may be familiar with, the natural logarithm calculates the power to which \( e \) must be raised to reach a given number.
In the context of \( \ln |x| \), this function considers both negative and positive values of \( x \), by using the absolute value.
Key characteristics include:
Unlike regular logarithms that you may be familiar with, the natural logarithm calculates the power to which \( e \) must be raised to reach a given number.
In the context of \( \ln |x| \), this function considers both negative and positive values of \( x \), by using the absolute value.
Key characteristics include:
- The function is undefined at \( x = 0 \). This is because dividing by zero is undefined.
- As \( x \) approaches zero from either direction, \( y \) trends towards negative infinity.
- For positive \( x \), it increases gradually, depicting growth.
Function Translation
Function translation involves shifting the entire graph horizontally or vertically.
For the function \( y = \left(\frac{1}{2}\right)^{x-1} - 1 \), we see a horizontal and a vertical translation.
These translations don't alter the fundamental nature of the graph; rather, they reposition it along the axes, allowing for more complex graph orientation and positioning in mathematical problems.
For the function \( y = \left(\frac{1}{2}\right)^{x-1} - 1 \), we see a horizontal and a vertical translation.
- The term \( x-1 \) indicates a move 1 unit to the right. This horizontal translation is directed by whatever operation is inside the exponent with \( x \).
- Meanwhile, the \( -1 \) outside the base function signifies a vertical translation downward by 1 unit.
These translations don't alter the fundamental nature of the graph; rather, they reposition it along the axes, allowing for more complex graph orientation and positioning in mathematical problems.
Horizontal Asymptote
A horizontal asymptote represents a line that a graph approaches as \( x \) heads towards positive or negative infinity.
It captures the behavior of a function at extreme values of \( x \). For \( y = \left(\frac{1}{2}\right)^x \), the original horizontal asymptote is \( y = 0 \).
But due to transformation in \( y = \left( \frac{1}{2} \right)^{x-1} - 1 \), this asymptote shifts to \( y = -1 \).
Understanding asymptotes is essential in graph analysis and helps in predicting long-term trends of functions in real-life applications.
It captures the behavior of a function at extreme values of \( x \). For \( y = \left(\frac{1}{2}\right)^x \), the original horizontal asymptote is \( y = 0 \).
But due to transformation in \( y = \left( \frac{1}{2} \right)^{x-1} - 1 \), this asymptote shifts to \( y = -1 \).
- Despite transformations, the nature of the asymptote remains; it outlines how the function behaves over its domain, describing how, as \( x \) increases infinitely or decreases infinitely, the function will not surpass that boundary line.
Understanding asymptotes is essential in graph analysis and helps in predicting long-term trends of functions in real-life applications.
