Math Problem Statement
Let ๐ฆ=๐(๐ฅ) be the function whose graph is given in the upper-left in the figure. Find formulas, in terms of ๐(๐ฅ) , for the horizontal and vertical shifts of the graph of ๐(๐ฅ) in parts (a)-(c) and determine the equation of each asymptote. Note only vertical and horizontal shifts have been applied to the graph of ๐ (no reflections, compressions, or stretches), so your formulas should be of the form ๐ฆ=๐(๐ฅโโ)+๐ where โ and ๐ are constants you determine from each graph. (c) Using shifts of ๐(๐ฅ), graph C can be written as: y=? with an asymptote at y=?
Solution
To determine the transformation and equations associated with each graph, let's analyze the shifts applied to the base function .
Analysis of Graph C
The original graph of has a vertical asymptote at and a horizontal asymptote at . The second image, representing graph C, shows a function that looks horizontally and vertically shifted.
Let's denote the transformations applied to in graph C as:
where:
- represents the horizontal shift.
- represents the vertical shift.
Step-by-Step Solution
-
Identify Horizontal Shift: Compare the location of the vertical asymptote in both graphs. For , the vertical asymptote is at . In graph C, it is at . This indicates a horizontal shift of units to the right.
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Identify Vertical Shift: Compare the location of the horizontal asymptote in both graphs. For , the horizontal asymptote is at . In graph C, the horizontal asymptote appears to be at . This indicates a vertical shift of .
Thus, the function for graph C, using shifts of , can be written as:
Asymptotes for Graph C
- Vertical Asymptote: Since there was a horizontal shift of 4, the vertical asymptote moves from to .
- Horizontal Asymptote: The horizontal asymptote for was . After shifting down by 5 units, it becomes .
Final Answer
For graph C: with an asymptote at .
Would you like more details on function transformations or asymptote shifts?
Related Questions
- How do you determine shifts of a function given an asymptote change?
- What effect does a horizontal shift have on a functionโs asymptotes?
- How do vertical shifts affect the horizontal asymptote of a function?
- How can you graphically identify horizontal and vertical shifts in a transformed function?
- What other types of transformations could be applied to a base function?
Tip
Remember, for a transformation , shifts the graph horizontally, while shifts it vertically.
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Math Problem Analysis
Mathematical Concepts
Function Transformation
Graph Shifting
Asymptotes
Formulas
y = f(x - h) + k
Theorems
Shifting Theorem
Suitable Grade Level
Grades 10-12