Select The Correct Answer.The Parent Function $f(x)=\sqrt[3]{x}$ Is Transformed To $g(x)=f(x+2)-4$. Which Is The Graph Of $ G G G [/tex]?

Mar 10, 2025 by ADMIN 142 views

**Transforming Parent Functions: A Step-by-Step Guide**

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Introduction

In mathematics, parent functions are the basic functions from which other functions are derived. These functions are used as a starting point to create various transformations, such as shifts, stretches, and compressions. In this article, we will explore how to transform a parent function using the given transformation equation.

Understanding the Parent Function

The parent function given is $f(x)=\sqrt[3]{x}$. This function represents a cube root function, which is a type of radical function. The cube root function is a one-to-one function, meaning that each value of x corresponds to a unique value of y.

Understanding the Transformation Equation

The transformation equation given is $g(x)=f(x+2)-4$. This equation represents a horizontal shift of 2 units to the left and a vertical shift of 4 units down from the parent function f(x).

Breaking Down the Transformation

To understand the transformation, let's break it down into two parts:

Horizontal Shift

The horizontal shift is represented by the term (x+2) inside the function f(x). This means that the graph of g(x) will be shifted 2 units to the left compared to the graph of f(x).

Vertical Shift

The vertical shift is represented by the term -4 outside the function f(x). This means that the graph of g(x) will be shifted 4 units down compared to the graph of f(x).

Combining the Shifts

When we combine the horizontal and vertical shifts, we get the final transformation equation: $g(x)=f(x+2)-4$. This equation represents a horizontal shift of 2 units to the left and a vertical shift of 4 units down from the parent function f(x).

Graphing the Transformation

To graph the transformation, we need to start with the graph of the parent function f(x) and then apply the horizontal and vertical shifts.

Step 1: Graph the Parent Function

The graph of the parent function f(x) is a cube root function, which is a type of radical function.

Step 2: Apply the Horizontal Shift

To apply the horizontal shift, we need to shift the graph of f(x) 2 units to the left. This means that we need to replace x with (x+2) in the equation of f(x).

Step 3: Apply the Vertical Shift

To apply the vertical shift, we need to shift the graph of f(x) 4 units down. This means that we need to subtract 4 from the equation of f(x).

The Final Graph

After applying the horizontal and vertical shifts, we get the final graph of g(x).

Conclusion

In this article, we explored how to transform a parent function using the given transformation equation. We broke down the transformation into two parts: horizontal shift and vertical shift. We then combined the shifts to get the final transformation equation and graphed the transformation. The final graph represents a horizontal shift of 2 units to the left and a vertical shift of 4 units down from the parent function f(x).

Example Problems

Problem 1

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x-3)+2$.

Solution

To find the transformation equation, we need to apply the horizontal and vertical shifts.

Horizontal shift: Replace x with (x-3) in the equation of f(x).
Vertical shift: Add 2 to the equation of f(x).

The transformation equation is $g(x)=f(x-3)+2$.

Problem 2

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x+1)-3$.

Solution

To find the transformation equation, we need to apply the horizontal and vertical shifts.

Horizontal shift: Replace x with (x+1) in the equation of f(x).
Vertical shift: Subtract 3 from the equation of f(x).

The transformation equation is $g(x)=f(x+1)-3$.

Practice Problems

Problem 1

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x-2)+1$.

Problem 2

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x+4)-2$.

Problem 3

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x-1)+3$.

Problem 4

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x+3)-1$.

Problem 5

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x-4)+2$.

Problem 6

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x+2)-1$.

Problem 7

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x-1)-2$.

Problem 8

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x+1)+2$.

Problem 9

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x-3)+1$.

Problem 10

Given the parent function $f(x)=\sqrt[3]{x}$, find the transformation equation for $g(x)=f(x+2)+3$.

Solutions

Problem 1

The transformation equation is $g(x)=f(x-2)+1$.

Problem 2

The transformation equation is $g(x)=f(x+4)-2$.

Problem 3

The transformation equation is $g(x)=f(x-1)+3$.

Problem 4

The transformation equation is $g(x)=f(x+3)-1$.

Problem 5

The transformation equation is $g(x)=f(x-4)+2$.

Problem 6

The transformation equation is $g(x)=f(x+2)-1$.

Problem 7

The transformation equation is $g(x)=f(x-1)-2$.

Problem 8

The transformation equation is $g(x)=f(x+1)+2$.

Problem 9

The transformation equation is $g(x)=f(x-3)+1$.

Problem 10

The transformation equation is $g(x)=f(x+2)+3$.

Conclusion

Final Thoughts

Transforming parent functions is an essential concept in mathematics, and it has numerous applications in various fields, such as physics, engineering, and computer science. By understanding how to transform parent functions, we can create new functions that can be used to model real-world phenomena.

References

[1] "Transforming Parent Functions" by Math Open Reference
[2] "Parent Functions" by Khan Academy
[3] "Transformations of Functions" by Purplemath

Glossary

Parent Function: A basic function from which other functions are derived.
Transformation Equation: An equation that represents a transformation of a parent function.
Horizontal Shift: A shift of a function to the left or right.
Vertical Shift: A shift of a function up or down.
Radical Function: A function that involves a root, such as a square root or cube root.
One-to-One Function: A function that maps each value of x to a unique value of y.

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Introduction

In our previous article, we explored how to transform parent functions using the given transformation equation. We broke down the transformation into two parts: horizontal shift and vertical shift. We then combined the shifts to get the final transformation equation and graphed the transformation. In this article, we will answer some frequently asked questions about transforming parent functions.

Q&A

Q1: What is a parent function?

A parent function is a basic function from which other functions are derived. It is a fundamental concept in mathematics and is used as a starting point to create various transformations.

A1:

A parent function is a basic function from which other functions are derived. It is a fundamental concept in mathematics and is used as a starting point to create various transformations.

Q2: What is a transformation equation?

A transformation equation is an equation that represents a transformation of a parent function. It is used to create new functions by applying horizontal and vertical shifts to the parent function.

A2:

A transformation equation is an equation that represents a transformation of a parent function. It is used to create new functions by applying horizontal and vertical shifts to the parent function.

Q3: What is a horizontal shift?

A horizontal shift is a shift of a function to the left or right. It is represented by the term (x-h) inside the function, where h is the amount of the shift.

A3:

A horizontal shift is a shift of a function to the left or right. It is represented by the term (x-h) inside the function, where h is the amount of the shift.

Q4: What is a vertical shift?

A vertical shift is a shift of a function up or down. It is represented by the term k outside the function, where k is the amount of the shift.

A4:

A vertical shift is a shift of a function up or down. It is represented by the term k outside the function, where k is the amount of the shift.

Q5: How do I apply a horizontal shift to a parent function?

To apply a horizontal shift to a parent function, you need to replace x with (x-h) inside the function, where h is the amount of the shift.

A5:

To apply a horizontal shift to a parent function, you need to replace x with (x-h) inside the function, where h is the amount of the shift.

Q6: How do I apply a vertical shift to a parent function?

To apply a vertical shift to a parent function, you need to add or subtract k from the function, where k is the amount of the shift.

A6:

To apply a vertical shift to a parent function, you need to add or subtract k from the function, where k is the amount of the shift.

Q7: What is the difference between a horizontal shift and a vertical shift?

A horizontal shift is a shift of a function to the left or right, while a vertical shift is a shift of a function up or down.

A7:

A horizontal shift is a shift of a function to the left or right, while a vertical shift is a shift of a function up or down.

Q8: How do I graph a transformed parent function?

To graph a transformed parent function, you need to start with the graph of the parent function and then apply the horizontal and vertical shifts.

A8:

To graph a transformed parent function, you need to start with the graph of the parent function and then apply the horizontal and vertical shifts.

Q9: What are some common transformations of parent functions?

Some common transformations of parent functions include horizontal shifts, vertical shifts, reflections, and stretches.

A9:

Some common transformations of parent functions include horizontal shifts, vertical shifts, reflections, and stretches.

Q10: How do I determine the amount of a horizontal shift?

To determine the amount of a horizontal shift, you need to look at the term (x-h) inside the function, where h is the amount of the shift.

A10:

To determine the amount of a horizontal shift, you need to look at the term (x-h) inside the function, where h is the amount of the shift.

Q11: How do I determine the amount of a vertical shift?

To determine the amount of a vertical shift, you need to look at the term k outside the function, where k is the amount of the shift.

A11:

To determine the amount of a vertical shift, you need to look at the term k outside the function, where k is the amount of the shift.

Q12: What is the importance of transforming parent functions?

Transforming parent functions is an essential concept in mathematics, and it has numerous applications in various fields, such as physics, engineering, and computer science.

A12:

Transforming parent functions is an essential concept in mathematics, and it has numerous applications in various fields, such as physics, engineering, and computer science.

Conclusion

In this article, we answered some frequently asked questions about transforming parent functions. We covered topics such as parent functions, transformation equations, horizontal shifts, vertical shifts, and graphing transformed parent functions. We hope that this article has been helpful in understanding the concept of transforming parent functions.

Final Thoughts

Transforming parent functions is an essential concept in mathematics, and it has numerous applications in various fields. By understanding how to transform parent functions, we can create new functions that can be used to model real-world phenomena.

References

[1] "Transforming Parent Functions" by Math Open Reference
[2] "Parent Functions" by Khan Academy
[3] "Transformations of Functions" by Purplemath

Glossary

Parent Function: A basic function from which other functions are derived.
Transformation Equation: An equation that represents a transformation of a parent function.
Horizontal Shift: A shift of a function to the left or right.
Vertical Shift: A shift of a function up or down.
Radical Function: A function that involves a root, such as a square root or cube root.
One-to-One Function: A function that maps each value of x to a unique value of y.