Describe How The Graph Of The Function G ( X ) = 4 ( X − 1 ) 2 G(x) = 4(x-1)^2 G ( X ) = 4 ( X − 1 ) 2 Is Related To The Graph Of The Parent Function F ( X ) = X 2 F(x) = X^2 F ( X ) = X 2 .- Vertical Translation: $\square$ (Select Choice)- Horizontal Translation: $\square$ (Select

Mar 10, 2025 by ADMIN 287 views

Understanding the Relationship Between the Graphs of $g(x) = 4(x-1)^2$ and $f(x) = x^2$

The graph of a function is a visual representation of the relationship between the input and output values of the function. In this article, we will explore the relationship between the graph of the function $g(x) = 4(x-1)^2$ and the graph of its parent function $f(x) = x^2$ . We will examine how the graph of $g(x)$ is related to the graph of $f(x)$ in terms of vertical and horizontal translations.

Vertical and Horizontal Translations

A vertical translation is a shift in the graph of a function up or down, while a horizontal translation is a shift in the graph of a function left or right. To understand the relationship between the graphs of $g(x)$ and $f(x)$ , we need to examine the equations of these functions.

The equation of the parent function $f(x) = x^2$ is a quadratic function that opens upwards. The graph of this function is a parabola that is symmetric about the y-axis.

The equation of the function $g(x) = 4(x-1)^2$ is also a quadratic function, but it has been modified to include a horizontal translation and a vertical stretch.

Horizontal Translation

The graph of $g(x)$ is a horizontal translation of the graph of $f(x)$ by 1 unit to the right. This means that the graph of $g(x)$ is shifted 1 unit to the right compared to the graph of $f(x)$ .

To see why this is the case, let's examine the equation of $g(x)$ . The term $(x-1)$ indicates that the graph of $g(x)$ is shifted 1 unit to the right compared to the graph of $f(x)$ . This is because the value of $x$ is being subtracted by 1, which means that the graph of $g(x)$ is being shifted to the right by 1 unit.

Vertical Stretch

In addition to the horizontal translation, the graph of $g(x)$ is also a vertical stretch of the graph of $f(x)$ . This means that the graph of $g(x)$ is stretched vertically by a factor of 4 compared to the graph of $f(x)$ .

To see why this is the case, let's examine the equation of $g(x)$ . The coefficient 4 in front of the squared term $(x-1)^2$ indicates that the graph of $g(x)$ is stretched vertically by a factor of 4 compared to the graph of $f(x)$ .

Combining Horizontal and Vertical Translations

Now that we have examined the horizontal and vertical translations separately, let's combine them to understand the relationship between the graphs of $g(x)$ and $f(x)$ .

The graph of $g(x)$ is a horizontal translation of the graph of $f(x)$ by 1 unit to the right, and a vertical stretch of the graph of $f(x)$ by a factor of 4. This means that the graph of $g(x)$ is shifted 1 unit to the right compared to the graph of $f(x)$ , and stretched vertically by a factor of 4 compared to the graph of $f(x)$ .

Conclusion

In conclusion, the graph of the function $g(x) = 4(x-1)^2$ is related to the graph of the parent function $f(x) = x^2$ in terms of a horizontal translation and a vertical stretch. The graph of $g(x)$ is shifted 1 unit to the right compared to the graph of $f(x)$ , and stretched vertically by a factor of 4 compared to the graph of $f(x)$ . This understanding of the relationship between the graphs of $g(x)$ and $f(x)$ is essential for graphing and analyzing quadratic functions.

Key Takeaways

The graph of $g(x) = 4(x-1)^2$ is a horizontal translation of the graph of $f(x) = x^2$ by 1 unit to the right.
The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 4.
The graph of $g(x)$ is shifted 1 unit to the right compared to the graph of $f(x)$ , and stretched vertically by a factor of 4 compared to the graph of $f(x)$ .

Graphing Quadratic Functions

When graphing quadratic functions, it is essential to understand the relationship between the graph of the function and its parent function. By understanding the horizontal and vertical translations, you can accurately graph and analyze quadratic functions.

Real-World Applications

Quadratic functions have numerous real-world applications, including physics, engineering, and economics. Understanding the relationship between the graph of a quadratic function and its parent function is essential for analyzing and solving problems in these fields.

Final Thoughts

In conclusion, the graph of the function $g(x) = 4(x-1)^2$ is related to the graph of the parent function $f(x) = x^2$ in terms of a horizontal translation and a vertical stretch. By understanding this relationship, you can accurately graph and analyze quadratic functions, and apply this knowledge to real-world problems.
Q&A: Understanding the Relationship Between the Graphs of $g(x) = 4(x-1)^2$ and $f(x) = x^2$

In our previous article, we explored the relationship between the graph of the function $g(x) = 4(x-1)^2$ and the graph of its parent function $f(x) = x^2$ . We examined how the graph of $g(x)$ is related to the graph of $f(x)$ in terms of vertical and horizontal translations. In this article, we will answer some frequently asked questions about the relationship between the graphs of $g(x)$ and $f(x)$ .

Q: What is the difference between the graphs of $g(x)$ and $f(x)$ ?

A: The graph of $g(x)$ is a horizontal translation of the graph of $f(x)$ by 1 unit to the right, and a vertical stretch of the graph of $f(x)$ by a factor of 4.

Q: Why is the graph of $g(x)$ shifted 1 unit to the right compared to the graph of $f(x)$ ?

A: The graph of $g(x)$ is shifted 1 unit to the right because of the term $(x-1)$ in the equation of $g(x)$ . This term indicates that the graph of $g(x)$ is shifted 1 unit to the right compared to the graph of $f(x)$ .

Q: Why is the graph of $g(x)$ stretched vertically by a factor of 4 compared to the graph of $f(x)$ ?

A: The graph of $g(x)$ is stretched vertically by a factor of 4 because of the coefficient 4 in front of the squared term $(x-1)^2$ in the equation of $g(x)$ . This coefficient indicates that the graph of $g(x)$ is stretched vertically by a factor of 4 compared to the graph of $f(x)$ .

Q: How can I graph the function $g(x) = 4(x-1)^2$ ?

A: To graph the function $g(x) = 4(x-1)^2$ , you can start by graphing the parent function $f(x) = x^2$ . Then, shift the graph of $f(x)$ 1 unit to the right to obtain the graph of $g(x)$ . Finally, stretch the graph of $g(x)$ vertically by a factor of 4 to obtain the final graph.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including physics, engineering, and economics. Some examples of real-world applications of quadratic functions include:

Modeling the trajectory of a projectile
Designing the shape of a parabolic mirror
Analyzing the motion of a pendulum
Modeling the growth of a population

Q: How can I use the relationship between the graphs of $g(x)$ and $f(x)$ to solve problems?

A: You can use the relationship between the graphs of $g(x)$ and $f(x)$ to solve problems by analyzing the horizontal and vertical translations of the graph of $g(x)$ compared to the graph of $f(x)$ . This can help you to identify the key features of the graph of $g(x)$ and to make predictions about its behavior.

Q: What are some common mistakes to avoid when graphing quadratic functions?

A: Some common mistakes to avoid when graphing quadratic functions include:

Failing to account for horizontal and vertical translations
Not using the correct scale for the graph
Not labeling the axes correctly
Not including key features of the graph, such as the vertex and the x-intercepts.

Conclusion

In conclusion, the graph of the function $g(x) = 4(x-1)^2$ is related to the graph of the parent function $f(x) = x^2$ in terms of a horizontal translation and a vertical stretch. By understanding this relationship, you can accurately graph and analyze quadratic functions, and apply this knowledge to real-world problems. We hope that this Q&A article has been helpful in answering your questions about the relationship between the graphs of $g(x)$ and $f(x)$ .