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Magazine Chapter 9: Problem 68
Consider the functions \(y=\log _{2} x+3, y=\log _{2} x-4, y=\log _{2}(x-1),\) and \(y=\log _{2}(x+2) .\) Use a graphing calculator to sketch the graphs on the same screen. Describe this family of graphs in terms of its parent graph \(y=\log _{2} x\)
Short Answer
Expert verified
The functions are vertical and horizontal shifts of the parent graph \(y = \log_{2} x\).
Step by step solution
01
Identify the Parent Function
The parent graph of the functions provided is the logarithmic function \(y = \log_{2} x\). This is a basic logarithmic function with base 2.
02
Analyze Each Function's Transformation
Each given function is a variation of the parent graph \(y = \log_{2} x\): - \(y = \log_{2} x + 3\) is a vertical shift up by 3 units.- \(y = \log_{2} x - 4\) is a vertical shift down by 4 units.- \(y = \log_{2} (x - 1)\) is a horizontal shift to the right by 1 unit.- \(y = \log_{2} (x + 2)\) is a horizontal shift to the left by 2 units.
03
Describe the Graphs on a Coordinate System
When all these functions are graphed together:
- The transformations involve shifts that do not change the shape of the graph, only its position on the coordinate plane.
- Vertical shifts move the graph up or down along the y-axis.
- Horizontal shifts move the graph left or right along the x-axis.
04
Sketching the Graphs Using a Calculator
Use the graphing feature of a calculator:
- Enter each function into the graphing calculator.
- Adjust the window settings to ensure all transformations can be visibly compared.
- Look for the effects of the shifts: for instance, note where asymptotes have moved (horizontal shifts affect vertical asymptotes but not the vertical shift functions).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parent Graph of Logarithmic Functions
The parent graph of a logarithmic function serves as the foundation for its variations. In the exercise provided, the parent function is the basic form: \( y = \log_{2}{x} \). This graph represents a logarithmic curve with base 2. It originates from the point (1, 0) on the coordinate plane, where the function value is zero. The curve gradually increases as x values increase, approaching the y-axis but never actually touching it. This behavior near the y-axis is known as having a vertical asymptote at \( x = 0 \). Understanding this parent graph is crucial because transformations of this graph maintain its core shape, modifying only its position or orientation on a graph.
Function Transformation in Logarithmic Functions
Function transformations allow us to adapt the parent graph into new forms. For logarithmic functions based on \( y = \log_{2}{x} \), transformations include shifts, reflections, or stretches. In the given functions:
- Vertical shifts adjust the graph upwards or downwards. For instance, \( y = \log_{2}{x} + 3 \) moves the graph up by 3 units, while \( y = \log_{2}{x} - 4 \) moves it down by 4 units.
- Horizontal shifts move the graph left or right along the x-axis. \( y = \log_{2}{(x - 1)} \) shifts it 1 unit to the right, whereas \( y = \log_{2}{(x + 2)} \) shifts it 2 units to the left.
Using a Graphing Calculator for Logarithmic Graphs
A graphing calculator is an invaluable tool for visualizing the behavior of complex functions. To explore the transformations of a parent logarithmic function:
- Input each function into your calculator's graphing mode.
- Ensure the window settings include a broad view of the x and y axes to accommodate shifts.
- Carefully observe how the graphs superimpose and how their paths change.
Vertical Shift in Logarithmic Functions
A vertical shift in a logarithmic function results in moving its graph either upwards or downwards without changing its shape. This shift is controlled by adding or subtracting a constant outside the logarithm expression. In the examples given:
- For \( y = \log_{2}{x} + 3 \), the entire graph lifts up by 3 units.
- In contrast, \( y = \log_{2}{x} - 4 \) adjusts the graph downwards by 4 units.
Horizontal Shift in Logarithmic Functions
Logarithmic functions can undergo horizontal shifts, displacing the graph left or right along the x-axis. This action is achieved by adding or subtracting a value within the logarithm's parentheses. In the specified functions:
- \( y = \log_{2}{(x - 1)} \) shifts the parent graph 1 unit to the right.
- Meanwhile, \( y = \log_{2}{(x + 2)} \) shifts it 2 units to the left.
