https://file.aiquickdraw.com/mathbot/file/1730082215130w5zt0jhq.jpg

https://file.aiquickdraw.com/mathbot/file/1730082215131eaumxkdd.jpg

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−ℎ)+𝑘 where ℎ and 𝑘 are constants you determine from each graph. (b) Using shifts of 𝑓(𝑥), graph B can be written as: y=? with an asymptote at y=?

In the problem, we are given a function \( y = f(x) \), with its original graph in the first image (upper-left) and a horizontally and/or vertically shifted version of it in the second image. The goal is to determine the equation for the graph in the second image using transformations of \( f(x) \), in the form \( y = f(x - h) + k \), where \( h \) and \( k \) are constants representing horizontal and vertical shifts, respectively. Additionally, we need to identify the asymptote for the transformed function.

### Step-by-Step Solution

1. **Identifying the Transformations**:
   - In the original graph (first image), we observe that \( f(x) \) has a horizontal asymptote at \( y = 5 \).
   - The transformed graph in the second image appears to be shifted downward and slightly to the left.

2. **Determining the Horizontal Shift**:
   - We notice that the transformed graph seems to have a starting point further to the left of the origin compared to the original graph.
   - Let \( h \) represent this horizontal shift. By inspection, we estimate that \( h \approx 1 \), as the shift aligns with a leftward movement.

3. **Determining the Vertical Shift**:
   - In the transformed graph, the horizontal asymptote is no longer at \( y = 5 \), but it appears to have moved down.
   - Let \( k \) be the vertical shift. The asymptote in the new graph appears to be at \( y = 2 \), so \( k = -3 \) (since \( 5 - 3 = 2 \)).

4. **Formulating the Transformed Function**:
   - Combining these shifts, we can write the transformed function as:
     \[
     y = f(x + 1) - 3
     \]

5. **Determining the Asymptote of the Transformed Function**:
   - Since the vertical shift is \( k = -3 \), the horizontal asymptote of the transformed function is \( y = 2 \).

### Final Answer

For graph B, the function can be written as:
\[
y = f(x + 1) - 3
\]
with an asymptote at:
\[
y = 2
\]

Would you like more details or have any questions about this solution?

### Related Questions
1. How does the vertical shift affect the position of the horizontal asymptote?
2. What other transformations (e.g., reflections) could we use to transform the graph of \( f(x) \)?
3. How would we approach finding transformations if \( f(x) \) had a slant asymptote instead of a horizontal one?
4. How does identifying key points on \( f(x) \) help in determining transformations?
5. What methods can be used to estimate the values of \( h \) and \( k \) precisely?

### Tip
When working with transformations, focusing on identifiable points and asymptotes helps in accurately determining shifts and other changes to the function.

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−h)+k where h and k are constants you determine from each graph. (b) Using shifts of f(x), graph B can be written as: y=? with an asymptote at y=?

Explore how to find the horizontal and vertical shifts of the function f(x) and determine the equations of asymptotes. This problem involves function transformations and asymptote analysis, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation