Math Problem Statement

Let ๐‘ฆ=๐‘“(๐‘ฅ) be the function whose graph is given in the upper-left in the figure. Find formulas, in terms of ๐‘“(๐‘ฅ) , for the horizontal and vertical shifts of the graph of ๐‘“(๐‘ฅ) 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 ๐‘ฆ=๐‘“(๐‘ฅโˆ’โ„Ž)+๐‘˜ where โ„Ž and ๐‘˜ are constants you determine from each graph. (c) Using shifts of ๐‘“(๐‘ฅ), graph C can be written as: y=? with an asymptote at y=?

Let y = f(x) be the function whose graph is given in the upper-left in the figure. Find formulas, in terms of f(x), for the horizontal and vertical shifts of the graph of f(x) in parts (a)-(c) and determine the equation of each asymptote. Note only vertical and horizontal shifts have been applied to the graph of f (no reflections, compressions, or stretches), so your formulas should be of the form y = f(x โˆ’ h) + k where h and k are constants you determine from each graph. (c) Using shifts of f(x), 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 f(x)f(x).

Analysis of Graph C

The original graph of f(x)f(x) has a vertical asymptote at x=0x = 0 and a horizontal asymptote at y=5y = 5. The second image, representing graph C, shows a function that looks horizontally and vertically shifted.

Let's denote the transformations applied to f(x)f(x) in graph C as:

y=f(xโˆ’h)+ky = f(x - h) + k

where:

  • hh represents the horizontal shift.
  • kk represents the vertical shift.

Step-by-Step Solution

  1. Identify Horizontal Shift: Compare the location of the vertical asymptote in both graphs. For f(x)f(x), the vertical asymptote is at x=0x = 0. In graph C, it is at x=4x = 4. This indicates a horizontal shift of h=4h = 4 units to the right.

  2. Identify Vertical Shift: Compare the location of the horizontal asymptote in both graphs. For f(x)f(x), the horizontal asymptote is at y=5y = 5. In graph C, the horizontal asymptote appears to be at y=0y = 0. This indicates a vertical shift of k=โˆ’5k = -5.

Thus, the function for graph C, using shifts of f(x)f(x), can be written as: y=f(xโˆ’4)โˆ’5y = f(x - 4) - 5

Asymptotes for Graph C

  • Vertical Asymptote: Since there was a horizontal shift of 4, the vertical asymptote moves from x=0x = 0 to x=4x = 4.
  • Horizontal Asymptote: The horizontal asymptote for f(x)f(x) was y=5y = 5. After shifting down by 5 units, it becomes y=0y = 0.

Final Answer

For graph C: y=f(xโˆ’4)โˆ’5y = f(x - 4) - 5 with an asymptote at y=0y = 0.

Would you like more details on function transformations or asymptote shifts?

Related Questions

  1. How do you determine shifts of a function given an asymptote change?
  2. What effect does a horizontal shift have on a functionโ€™s asymptotes?
  3. How do vertical shifts affect the horizontal asymptote of a function?
  4. How can you graphically identify horizontal and vertical shifts in a transformed function?
  5. What other types of transformations could be applied to a base function?

Tip

Remember, for a transformation y=f(xโˆ’h)+ky = f(x - h) + k, hh shifts the graph horizontally, while kk 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