Transformations of a Function

Oct 13, 2025 by miodeya.com 3 views

When analyzing a function, it's essential to understand its parent function and how it has been transformed. In this article, we'll explore the relationship between the function $$f(x)=3|x+2|+4$$ and its parent function.

What is a Parent Function?

A parent function is the simplest form of a function, without any transformations. For example, the parent function of $$f(x)=3|x+2|+4$$ is $$f(x)=|x|$$.

Transformations of a Function

A function can be transformed in several ways, including:

Vertical Stretching or Compressing: This occurs when the function is multiplied by a constant factor. If the factor is greater than 1, the function is stretched. If the factor is less than 1, the function is compressed.
Horizontal Stretching or Compressing: This occurs when the function is multiplied by a constant factor inside the absolute value. If the factor is greater than 1, the function is stretched. If the factor is less than 1, the function is compressed.
Translation: This occurs when the function is shifted horizontally or vertically.

Analyzing the Function $$f(x)=3|x+2|+4$$

The function $$f(x)=3|x+2|+4$$ can be broken down into three parts:

Vertical Stretching: The function is multiplied by a factor of 3, which means it has been stretched vertically.
Horizontal Translation: The function has been shifted 2 units to the left.
Vertical Translation: The function has been shifted 4 units upwards.

Comparing the Function to its Parent Function

The parent function of $$f(x)=3|x+2|+4$$ is $$f(x)=|x|$$.

Vertical Stretching: The function $$f(x)=3|x+2|+4$$ has been stretched vertically by a factor of 3, whereas the parent function $$f(x)=|x|$$ has not been stretched.
Horizontal Translation: The function $$f(x)=3|x+2|+4$$ has been shifted 2 units to the left, whereas the parent function $$f(x)=|x|$$ has not been translated horizontally.
Vertical Translation: The function $$f(x)=3|x+2|+4$$ has been shifted 4 units upwards, whereas the parent function $$f(x)=|x|$$ has not been translated vertically.

Conclusion

In conclusion, the function $$f(x)=3|x+2|+4$$ has been transformed from its parent function $$f(x)=|x|$$ through vertical stretching, horizontal translation, and vertical translation.

Key Takeaways

The function $$f(x)=3|x+2|+4$$ has been stretched vertically by a factor of 3.
The function $$f(x)=3|x+2|+4$$ has been shifted 2 units to the left.
The function $$f(x)=3|x+2|+4$$ has been shifted 4 units upwards.

Final Answer

In our previous article, we explored the relationship between the function $$f(x)=3|x+2|+4$$ and its parent function $$f(x)=|x|$$. In this article, we'll answer some frequently asked questions about parent functions and transformations.

Q: What is the parent function of a given function?

A: The parent function of a given function is the simplest form of the function, without any transformations. For example, the parent function of $$f(x)=3|x+2|+4$$ is $$f(x)=|x|$$.

Q: What are the different types of transformations that can occur in a function?

A: There are several types of transformations that can occur in a function, including:

Vertical Stretching or Compressing: This occurs when the function is multiplied by a constant factor. If the factor is greater than 1, the function is stretched. If the factor is less than 1, the function is compressed.
Horizontal Stretching or Compressing: This occurs when the function is multiplied by a constant factor inside the absolute value. If the factor is greater than 1, the function is stretched. If the factor is less than 1, the function is compressed.
Translation: This occurs when the function is shifted horizontally or vertically.

Q: How can I determine if a function has been stretched or compressed?

A: To determine if a function has been stretched or compressed, look for a constant factor outside the absolute value. If the factor is greater than 1, the function has been stretched. If the factor is less than 1, the function has been compressed.

Q: How can I determine if a function has been translated horizontally or vertically?

A: To determine if a function has been translated horizontally or vertically, look for a constant value inside the absolute value. If the value is positive, the function has been shifted to the right. If the value is negative, the function has been shifted to the left. If the value is greater than 0, the function has been shifted upwards. If the value is less than 0, the function has been shifted downwards.

Q: Can a function have multiple transformations?

A: Yes, a function can have multiple transformations. For example, the function $$f(x)=3|x+2|+4$$ has been stretched vertically, shifted 2 units to the left, and shifted 4 units upwards.

Q: How can I graph a function with multiple transformations?

A: To graph a function with multiple transformations, start by graphing the parent function. Then, apply each transformation in order. For example, to graph the function $$f(x)=3|x+2|+4$$, start by graphing the parent function $$f(x)=|x|$$. Then, apply the vertical stretch by multiplying the function by 3. Next, apply the horizontal translation by shifting the function 2 units to the left. Finally, apply the vertical translation by shifting the function 4 units upwards.

Q: What are some common parent functions?

A: Some common parent functions include:

Linear Functions: $$f(x)=x$$
Quadratic Functions: $$f(x)=x^2$$
Absolute Value Functions: $$f(x)=|x|$$
Polynomial Functions: $$f(x)=x^n$$

Conclusion

In conclusion, understanding the relationship between a function and its parent function is essential for analyzing and graphing functions. By recognizing the different types of transformations that can occur in a function, you can determine how a function has been transformed from its parent function.

Key Takeaways

The parent function of a given function is the simplest form of the function, without any transformations.
There are several types of transformations that can occur in a function, including vertical stretching or compressing, horizontal stretching or compressing, and translation.
A function can have multiple transformations.
To graph a function with multiple transformations, start by graphing the parent function and then apply each transformation in order.

Final Answer

The final answer is that understanding the relationship between a function and its parent function is essential for analyzing and graphing functions.