How Would The Equation $f(x)=\log (x-2)+8$ Be Shifted With Respect To The Parent Function?A. Left 2 Units And Down 8 Units B. Left 2 Units And Up 8 Units C. Right 2 Units And Down 8 Units D. Right 2 Units And Up 8 Units

Introduction

In mathematics, functions are essential tools for modeling real-world phenomena. A parent function is a basic function from which other functions can be derived by applying various transformations. Understanding how functions shift with respect to their parent functions is crucial in mathematics and its applications. In this article, we will explore how the equation $f(x)=\log (x-2)+8$ is shifted with respect to its parent function.

Parent Function and Shifts

The parent function of the given equation is the basic logarithmic function, $f(x)=\log x$. This function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=-\infty$. The graph of the parent function is a logarithmic curve that increases as $x$ increases.

To shift a function with respect to its parent function, we need to understand the types of shifts that can occur. There are three types of shifts: horizontal, vertical, and a combination of both.

Horizontal Shifts

A horizontal shift occurs when the function is shifted left or right with respect to the parent function. This type of shift is achieved by adding or subtracting a constant value from the input variable, $x$. If the constant value is added to $x$, the function is shifted to the right. If the constant value is subtracted from $x$, the function is shifted to the left.

Vertical Shifts

A vertical shift occurs when the function is shifted up or down with respect to the parent function. This type of shift is achieved by adding or subtracting a constant value from the function. If the constant value is added to the function, the function is shifted up. If the constant value is subtracted from the function, the function is shifted down.

Combination of Horizontal and Vertical Shifts

A combination of horizontal and vertical shifts occurs when both types of shifts are applied to the function. This type of shift is achieved by adding or subtracting a constant value from the input variable, $x$, and adding or subtracting another constant value from the function.

Applying Shifts to the Given Equation

Now that we have understood the types of shifts that can occur, let's apply these shifts to the given equation, $f(x)=\log (x-2)+8$. To determine the type of shift, we need to analyze the equation and identify the constant values that are added or subtracted from the input variable, $x$, and the function.

Horizontal Shift

The equation $f(x)=\log (x-2)+8$ has a horizontal shift of $2$ units to the right. This is because the input variable, $x$, is subtracted by $2$ inside the logarithmic function. This means that the function is shifted $2$ units to the right with respect to the parent function.

Vertical Shift

The equation $f(x)=\log (x-2)+8$ has a vertical shift of $8$ units up. This is because the constant value, $8$, is added to the function. This means that the function is shifted $8$ units up with respect to the parent function.

Conclusion

In conclusion, the equation $f(x)=\log (x-2)+8$ is shifted with respect to its parent function by $2$ units to the right and $8$ units up. This means that the correct answer is:

D. Right 2 units and up 8 units

This understanding of function shifts is essential in mathematics and its applications. By analyzing the types of shifts that can occur, we can determine how a function is shifted with respect to its parent function.

Final Thoughts

In this article, we have explored how the equation $f(x)=\log (x-2)+8$ is shifted with respect to its parent function. We have analyzed the types of shifts that can occur and applied these shifts to the given equation. This understanding of function shifts is crucial in mathematics and its applications. By mastering this concept, we can better understand and analyze functions in various mathematical contexts.

References

• [1] Calculus by Michael Spivak
• [2] Mathematics for Computer Science by Eric Lehman, F Thomson Leighton, and Albert R Meyer
• [3] Introduction to Mathematical Thinking by Keith Devlin

Glossary

• Parent function: A basic function from which other functions can be derived by applying various transformations.
• Horizontal shift: A shift that occurs when the function is shifted left or right with respect to the parent function.
• Vertical shift: A shift that occurs when the function is shifted up or down with respect to the parent function.
• Combination of horizontal and vertical shifts: A shift that occurs when both types of shifts are applied to the function.
  

Introduction

In our previous article, we explored how the equation $f(x)=\log (x-2)+8$ is shifted with respect to its parent function. We analyzed the types of shifts that can occur and applied these shifts to the given equation. In this article, we will answer some frequently asked questions related to function shifts.

Q1: What is a parent function?

A parent function is a basic function from which other functions can be derived by applying various transformations. It is the original function that is used as a reference point for other functions.

Q2: What are the types of shifts that can occur?

There are three types of shifts that can occur:

• Horizontal shift: A shift that occurs when the function is shifted left or right with respect to the parent function.
• Vertical shift: A shift that occurs when the function is shifted up or down with respect to the parent function.
• Combination of horizontal and vertical shifts: A shift that occurs when both types of shifts are applied to the function.

Q3: How do I determine the type of shift that occurs?

To determine the type of shift that occurs, you need to analyze the equation and identify the constant values that are added or subtracted from the input variable, $x$, and the function.

Q4: What is a horizontal shift?

A horizontal shift occurs when the function is shifted left or right with respect to the parent function. This type of shift is achieved by adding or subtracting a constant value from the input variable, $x$.

Q5: What is a vertical shift?

A vertical shift occurs when the function is shifted up or down with respect to the parent function. This type of shift is achieved by adding or subtracting a constant value from the function.

Q6: How do I apply a horizontal shift to a function?

To apply a horizontal shift to a function, you need to add or subtract a constant value from the input variable, $x$. For example, if you want to shift the function $f(x)=\log x$ 2 units to the right, you would replace $x$ with $x-2$.

Q7: How do I apply a vertical shift to a function?

To apply a vertical shift to a function, you need to add or subtract a constant value from the function. For example, if you want to shift the function $f(x)=\log x$ 2 units up, you would add 2 to the function.

Q8: Can a function have both horizontal and vertical shifts?

Yes, a function can have both horizontal and vertical shifts. This type of shift is achieved by adding or subtracting a constant value from the input variable, $x$, and adding or subtracting another constant value from the function.

Q9: How do I determine the order of shifts when a function has both horizontal and vertical shifts?

To determine the order of shifts when a function has both horizontal and vertical shifts, you need to analyze the equation and identify the constant values that are added or subtracted from the input variable, $x$, and the function. The order of shifts is determined by the order in which the constant values are added or subtracted.

Q10: Can a function have more than two shifts?

Yes, a function can have more than two shifts. This type of shift is achieved by adding or subtracting multiple constant values from the input variable, $x$, and the function.

Conclusion

In conclusion, understanding function shifts is essential in mathematics and its applications. By analyzing the types of shifts that can occur and applying these shifts to functions, we can better understand and analyze functions in various mathematical contexts.

Final Thoughts

In this article, we have answered some frequently asked questions related to function shifts. We have explored the types of shifts that can occur, how to determine the type of shift that occurs, and how to apply shifts to functions. By mastering this concept, we can better understand and analyze functions in various mathematical contexts.

References

• [1] Calculus by Michael Spivak
• [2] Mathematics for Computer Science by Eric Lehman, F Thomson Leighton, and Albert R Meyer
• [3] Introduction to Mathematical Thinking by Keith Devlin

Glossary

• Parent function: A basic function from which other functions can be derived by applying various transformations.
• Horizontal shift: A shift that occurs when the function is shifted left or right with respect to the parent function.
• Vertical shift: A shift that occurs when the function is shifted up or down with respect to the parent function.
• Combination of horizontal and vertical shifts: A shift that occurs when both types of shifts are applied to the function.