Which Statement Correctly Describes The Graph Of G ( X ) = F ( X − 9 G(x) = F(x - 9 G ( X ) = F ( X − 9 ]?A. Function G G G Has The Same Horizontal And Vertical Asymptotes As Function F F F .B. Function G G G Has The Same Horizontal Asymptote As F F F And

When dealing with functions, particularly those that involve shifts, it's essential to understand how these transformations affect the graph of the function. In this article, we'll explore the concept of shifting a function and how it impacts the graph's asymptotes.

What is a Shifted Function?

A shifted function is a transformation of a given function, where the graph of the original function is moved horizontally or vertically. This can be achieved by adding or subtracting a constant value from the input variable (x) or the output variable (y). In the case of the function $g(x) = f(x - 9)$, the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ by 9 units to the right.

Horizontal and Vertical Asymptotes

Asymptotes are lines that the graph of a function approaches as x goes to positive or negative infinity. There are two types of asymptotes: horizontal and vertical. A horizontal asymptote is a horizontal line that the graph approaches as x goes to positive or negative infinity, while a vertical asymptote is a vertical line that the graph approaches as x goes to a specific value.

Horizontal Asymptote

The horizontal asymptote of a function is determined by the degree of the numerator and denominator of the function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Vertical Asymptote

The vertical asymptote of a function is determined by the values of x that make the denominator of the function equal to zero. If the denominator is a polynomial, the vertical asymptote is the value of x that makes the polynomial equal to zero. If the denominator is a rational function, the vertical asymptote is the value of x that makes the denominator equal to zero.

Effect of Horizontal Shift on Asymptotes

When a function is shifted horizontally, its asymptotes are also shifted. The horizontal asymptote remains the same, but the vertical asymptote is shifted by the amount of the horizontal shift.

Which Statement Correctly Describes the Graph of $g(x) = f(x - 9)$?

Now that we've discussed the concept of shifting a function and its impact on asymptotes, let's examine the given function $g(x) = f(x - 9)$. The graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ by 9 units to the right.

A. Function $g$ has the same horizontal and vertical asymptotes as function $f$.

This statement is incorrect. The horizontal asymptote of $g(x)$ remains the same as the horizontal asymptote of $f(x)$, but the vertical asymptote of $g(x)$ is shifted by 9 units to the right.

B. Function $g$ has the same horizontal asymptote as $f$ and

This statement is correct. The horizontal asymptote of $g(x)$ remains the same as the horizontal asymptote of $f(x)$.

Conclusion

In conclusion, when a function is shifted horizontally, its asymptotes are also shifted. The horizontal asymptote remains the same, but the vertical asymptote is shifted by the amount of the horizontal shift. Therefore, the correct statement is that function $g$ has the same horizontal asymptote as function $f$.

References

• [1] Calculus, 6th edition, Michael Spivak
• [2] Algebra and Trigonometry, 4th edition, James Stewart
• [3] Precalculus, 2nd edition, Michael Sullivan

Additional Resources

• Khan Academy: Shifts and Reflections
• Mathway: Shifts and Reflections
• Wolfram Alpha: Shifts and Reflections
  Frequently Asked Questions: Shifted Functions and Asymptotes =============================================================

In our previous article, we discussed the concept of shifting a function and its impact on asymptotes. We also examined the given function $g(x) = f(x - 9)$ and determined that the correct statement is that function $g$ has the same horizontal asymptote as function $f$. In this article, we'll answer some frequently asked questions related to shifted functions and asymptotes.

Q: What is the effect of a horizontal shift on the graph of a function?

A: A horizontal shift of a function moves its graph to the left or right by a specified amount. If the shift is to the right, the graph is moved to the right by the specified amount. If the shift is to the left, the graph is moved to the left by the specified amount.

Q: How does a horizontal shift affect the vertical asymptote of a function?

A: A horizontal shift affects the vertical asymptote of a function by shifting it by the amount of the horizontal shift. If the shift is to the right, the vertical asymptote is shifted to the right by the specified amount. If the shift is to the left, the vertical asymptote is shifted to the left by the specified amount.

Q: What is the effect of a horizontal shift on the horizontal asymptote of a function?

A: A horizontal shift does not affect the horizontal asymptote of a function. The horizontal asymptote remains the same, regardless of the horizontal shift.

Q: Can a function have multiple horizontal shifts?

A: Yes, a function can have multiple horizontal shifts. Each horizontal shift will move the graph of the function to the left or right by a specified amount.

Q: How do I determine the horizontal and vertical asymptotes of a shifted function?

A: To determine the horizontal and vertical asymptotes of a shifted function, you need to follow these steps:

1. Determine the horizontal and vertical asymptotes of the original function.
2. Apply the horizontal shift to the original function.
3. Determine the new horizontal and vertical asymptotes of the shifted function.

Q: Can a function have a horizontal shift and a vertical shift?

A: Yes, a function can have both a horizontal shift and a vertical shift. Each shift will move the graph of the function in a different direction.

Q: How do I determine the horizontal and vertical asymptotes of a function with both horizontal and vertical shifts?

A: To determine the horizontal and vertical asymptotes of a function with both horizontal and vertical shifts, you need to follow these steps:

1. Determine the horizontal and vertical asymptotes of the original function.
2. Apply the horizontal shift to the original function.
3. Apply the vertical shift to the shifted function.
4. Determine the new horizontal and vertical asymptotes of the shifted function.

Q: Can a function have multiple vertical shifts?

A: Yes, a function can have multiple vertical shifts. Each vertical shift will move the graph of the function up or down by a specified amount.

Q: How do I determine the horizontal and vertical asymptotes of a function with multiple vertical shifts?

A: To determine the horizontal and vertical asymptotes of a function with multiple vertical shifts, you need to follow these steps:

1. Determine the horizontal and vertical asymptotes of the original function.
2. Apply the vertical shifts to the original function.
3. Determine the new horizontal and vertical asymptotes of the shifted function.

Conclusion

In conclusion, shifted functions and asymptotes are an essential concept in mathematics. Understanding how horizontal and vertical shifts affect the graph of a function and its asymptotes is crucial for solving problems in calculus, algebra, and other areas of mathematics. We hope this article has helped you understand the concept of shifted functions and asymptotes better.

References

• [1] Calculus, 6th edition, Michael Spivak
• [2] Algebra and Trigonometry, 4th edition, James Stewart
• [3] Precalculus, 2nd edition, Michael Sullivan

Additional Resources

• Khan Academy: Shifts and Reflections
• Mathway: Shifts and Reflections
• Wolfram Alpha: Shifts and Reflections