Which Translation Maps The Graph Of The Function F ( X ) = X 2 F(x)=x^2 F ( X ) = X 2 Onto The Function G ( X ) = X 2 + 2 X + 6 G(x)=x^2+2x+6 G ( X ) = X 2 + 2 X + 6 ?A. Left 1 Unit, Up 5 UnitsB. Right 1 Unit, Up 5 UnitsC. Left 2 Units, Up 2 UnitsD. Right 2 Units, Up 2 Units

$$$$

Understanding Graph Translation

Graph translation is a fundamental concept in mathematics, particularly in algebra and geometry. It involves shifting or moving a graph of a function to a new position on the coordinate plane. In this article, we will explore how to map the graph of the function $f(x)=x^2$ onto the function $g(x)=x^2+2x+6$.

The Function $f(x)=x^2$

The function $f(x)=x^2$ is a quadratic function that represents a parabola opening upwards. The graph of this function is a U-shaped curve that is symmetric about the y-axis. The equation of this function can be written in the form $y=ax^2$, where $a=1$.

The Function $g(x)=x^2+2x+6$

The function $g(x)=x^2+2x+6$ is also a quadratic function, but it has a different equation than $f(x)=x^2$. The graph of this function is also a parabola, but it is shifted to the left and upwards compared to the graph of $f(x)=x^2$. The equation of this function can be written in the form $y=ax^2+bx+c$, where $a=1$, $b=2$, and $c=6$.

Graph Translation: Mapping $f(x)$ onto $g(x)$

To map the graph of $f(x)$ onto the graph of $g(x)$, we need to find the translation that takes the graph of $f(x)$ to the graph of $g(x)$. This involves finding the horizontal and vertical shifts required to move the graph of $f(x)$ to the graph of $g(x)$.

Horizontal Shift

The horizontal shift is the movement of the graph of $f(x)$ to the left or right. In this case, we need to find the horizontal shift that takes the graph of $f(x)$ to the graph of $g(x)$. To do this, we need to compare the equations of the two functions.

The equation of $f(x)$ is $y=x^2$, while the equation of $g(x)$ is $y=x^2+2x+6$. By comparing the two equations, we can see that the graph of $g(x)$ is shifted to the left by 1 unit compared to the graph of $f(x)$.

Vertical Shift

The vertical shift is the movement of the graph of $f(x)$ up or down. In this case, we need to find the vertical shift that takes the graph of $f(x)$ to the graph of $g(x)$. To do this, we need to compare the equations of the two functions.

The equation of $f(x)$ is $y=x^2$, while the equation of $g(x)$ is $y=x^2+2x+6$. By comparing the two equations, we can see that the graph of $g(x)$ is shifted up by 5 units compared to the graph of $f(x)$.

Conclusion

In conclusion, the graph of $f(x)=x^2$ can be mapped onto the graph of $g(x)=x^2+2x+6$ by shifting the graph of $f(x)$ to the left by 1 unit and up by 5 units. This is the translation that takes the graph of $f(x)$ to the graph of $g(x)$.

Answer

The correct answer is:

A. Left 1 unit, up 5 units

This is the translation that maps the graph of $f(x)=x^2$ onto the graph of $g(x)=x^2+2x+6$.

Discussion

This problem involves graph translation, which is a fundamental concept in mathematics. The problem requires the student to understand how to map the graph of one function onto the graph of another function. The student needs to compare the equations of the two functions and find the horizontal and vertical shifts required to move the graph of one function to the graph of the other function.

Example Problems

Here are some example problems that involve graph translation:

• Map the graph of $f(x)=x^2$ onto the graph of $g(x)=x^2+4x+7$.
• Map the graph of $f(x)=x^2$ onto the graph of $g(x)=x^2-3x+9$.
• Map the graph of $f(x)=x^2$ onto the graph of $g(x)=x^2+5x+11$.

Solutions

Here are the solutions to the example problems:

• Map the graph of $f(x)=x^2$ onto the graph of $g(x)=x^2+4x+7$: The graph of $g(x)$ is shifted to the left by 2 units and up by 7 units compared to the graph of $f(x)$.
• Map the graph of $f(x)=x^2$ onto the graph of $g(x)=x^2-3x+9$: The graph of $g(x)$ is shifted to the right by 3 units and up by 9 units compared to the graph of $f(x)$.
• Map the graph of $f(x)=x^2$ onto the graph of $g(x)=x^2+5x+11$: The graph of $g(x)$ is shifted to the left by 5 units and up by 11 units compared to the graph of $f(x)$.

Conclusion

$$$$

Q: What is graph translation?

A: Graph translation is a fundamental concept in mathematics that involves shifting or moving a graph of a function to a new position on the coordinate plane.

Q: What are the types of graph translation?

A: There are two types of graph translation: horizontal shift and vertical shift.

• Horizontal shift: This involves moving the graph of a function to the left or right.
• Vertical shift: This involves moving the graph of a function up or down.

Q: How do I determine the horizontal shift?

A: To determine the horizontal shift, you need to compare the equations of the two functions. If the equation of the second function has a term with a negative coefficient, it means that the graph of the second function is shifted to the left. If the equation of the second function has a term with a positive coefficient, it means that the graph of the second function is shifted to the right.

Q: How do I determine the vertical shift?

A: To determine the vertical shift, you need to compare the equations of the two functions. If the equation of the second function has a constant term that is greater than the constant term of the first function, it means that the graph of the second function is shifted up. If the equation of the second function has a constant term that is less than the constant term of the first function, it means that the graph of the second function is shifted down.

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

A: A horizontal shift involves moving the graph of a function to the left or right, while a vertical shift involves moving the graph of a function up or down.

Q: Can a graph be shifted both horizontally and vertically?

A: Yes, a graph can be shifted both horizontally and vertically. This is known as a combination of horizontal and vertical shifts.

Q: How do I find the combination of horizontal and vertical shifts?

A: To find the combination of horizontal and vertical shifts, you need to compare the equations of the two functions. You need to determine the horizontal shift by comparing the coefficients of the x-term, and the vertical shift by comparing the constant terms.

Q: What is the importance of graph translation?

A: Graph translation is an important concept in mathematics because it helps us to understand how to move a graph of a function to a new position on the coordinate plane. It is used in various fields such as physics, engineering, and computer science.

Q: Can graph translation be used to solve real-world problems?

A: Yes, graph translation can be used to solve real-world problems. For example, it can be used to model the motion of an object, or to analyze the behavior of a system.

Q: What are some common applications of graph translation?

A: Some common applications of graph translation include:

• Modeling the motion of an object
• Analyzing the behavior of a system
• Solving optimization problems
• Graphing functions

Q: How can I practice graph translation?

A: You can practice graph translation by working on problems that involve shifting graphs of functions. You can also use online resources such as graphing calculators or software to visualize the graphs and practice shifting them.

Q: What are some common mistakes to avoid when working with graph translation?

A: Some common mistakes to avoid when working with graph translation include:

• Not comparing the equations of the two functions
• Not determining the horizontal and vertical shifts correctly
• Not visualizing the graphs correctly
• Not using the correct notation

Q: How can I improve my skills in graph translation?

A: You can improve your skills in graph translation by practicing regularly, working on problems that involve shifting graphs of functions, and using online resources such as graphing calculators or software to visualize the graphs and practice shifting them.