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Chapter 1: Problem 77

Start with the graph of \(y=\ln x .\) Find an equation of the graph that results from a. shifting down 3 units. b. shifting right 1 unit. c. shifting left \(1,\) up 3 units. d. shifting down \(4,\) right 2 units. e. reflecting about the \(y\) -axis. f. reflecting about the line \(y=x\)

Short Answer

Expert verified
a. \(y = \ln x - 3\), b. \(y = \ln(x - 1)\), c. \(y = \ln(x + 1) + 3\), d. \(y = \ln(x - 2) - 4\), e. \(y = \ln(-x)\), f. \(y = e^x\).

Step by step solution

01

Shifting Down 3 Units

To shift the graph of a function down, we subtract a number from the function. The original function is \(y = \ln x\). By shifting it down 3 units, we have \(y = \ln x - 3\).
02

Shifting Right 1 Unit

To shift the graph of a function to the right, we replace \(x\) with \(x-c\) where \(c\) is the number of units moved. Shifting the graph of \(y= \ln x\) right by 1 unit results in \(y = \ln(x - 1)\).
03

Shifting Left 1, Up 3 Units

To shift the graph left 1 unit, replace \(x\) with \(x + 1\), and to shift up by 3 units, add 3 to the entire function. Therefore, \(y = \ln(x + 1) + 3\).
04

Shifting Down 4, Right 2 Units

First, shift the graph right 2 units by replacing \(x\) with \(x - 2\). Then, shift the graph down by subtracting 4. This results in the function \(y = \ln(x - 2) - 4\).
05

Reflecting About the y-axis

Reflecting a graph about the y-axis involves replacing \(x\) with \(-x\) in the function. For the original function \(y = \ln x\), the reflection is \(y = \ln(-x)\). Note that this results in a transformation that changes the domain of the function.
06

Reflecting About the Line y=x

Reflecting a graph about the line \(y=x\) involves swapping \(x\) and \(y\). The original equation \(y = \ln x\) becomes \(x = \ln y\). Solving for \(y\), we use exponentiation to get \(y = e^x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Shifts
Graph shifts are a simple yet crucial concept in understanding function transformations. Whenever you move a graph from its original position on a coordinate plane, you are performing a graph shift. There are two primary types of shifts: vertical and horizontal.
Vertical shifts involve moving the graph up or down:
  • To shift a graph upward, you add a constant to the function. For instance, for the function \( y = \ln x \), shifting up by 3 units becomes \( y = \ln x + 3 \).
  • Conversely, to move downward, you subtract a constant. Thus, \( y = \ln x - 3 \) shifts the graph down by 3 units.
Horizontal shifts involve moving the graph left or right:
  • To shift right, subtract a constant from \( x \). For example, \( y = \ln(x - 1) \) shifts the graph right by 1 unit.
  • To shift left, add a constant to \( x \). Therefore, \( y = \ln(x + 1) \) indicates a shift left by 1 unit.
Logarithmic Functions
Logarithmic functions are a specific type of function important in both mathematics and real-world applications like measuring sound intensity or earthquake magnitude. The base-\( e \) logarithmic function, also known as the natural logarithm, has a specific form:
Given as \( y = \ln x \), this function has notable characteristics.
  • It is only defined for positive \( x \)-values because you cannot take the logarithm of zero or a negative number in real numbers.
  • The graph of \( y = \ln x \) is steadily increasing but at a decreasing rate.
Understanding these properties helps when transforming the graph, as different transformations will modify these features while still retaining some essence of the original function's nature.
Reflection of Graphs
Reflections are a transformation that "flips" a graph over a specific line, producing a mirror image. In terms of functions, you can reflect over the y-axis or the line \( y = x \).
Reflecting over the y-axis involves changing \( x \) to \( -x \) in the function:
  • For \( y = \ln x \), reflecting over the y-axis produces \( y = \ln(-x) \). This transformation affects the domain, as the logarithmic function is conventionally defined for positive values of \( x \).
Reflecting over the line \( y = x \) requires swapping \( x \) and \( y \) in the function:
  • So \( y = \ln x \) becomes \( x = \ln y \), and when solved for \( y \), you get \( y = e^x \), introducing exponential functions into the reflection picture.
These reflections provide an excellent visual understanding of how functions can be manipulated and how their properties change with different transformations.
Translation of Functions
Translation of functions refers to moving a graph from one location to another on the plane without changing its shape, size, or orientation, functioning as a basis for understanding other complex transformations.
A straightforward translation combines both horizontal and vertical shifts. For instance:
  • Shifting the graph to the right by 2 units and down by 4 units results in \( y = \ln(x - 2) - 4 \).
  • If translated left by 1 unit and up by 3 units, the function becomes \( y = \ln(x + 1) + 3 \).
By mastering translations, we gain valuable insight into the mechanics of graph transformations, making other transformations like rotations and reflections easier to grasp.

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