Which Translation Maps The Graph Of The Function $f(x) = X^2$ Onto The Function $g(x) = X^2 + 2x + 6$?A. Left 1 Unit, Up 5 Units B. Right 1 Unit, Up 5 Units C. Left 2 Units, Up 2 Units D. Right 2 Units, Up 2 Units

Feb 28, 2025 by ADMIN 219 views

Which Translation Maps the Graph of the Function $f(x) = x^2$ onto the Function $g(x) = x^2 + 2x + 6$ ?

Understanding Function Transformations

In mathematics, function transformations are essential concepts that help us understand how functions change under various operations. When we talk about translating a function, we are essentially shifting its graph to a new position on the coordinate plane. This can be done horizontally, vertically, or both. In this article, we will explore which translation maps the graph of the function $f(x) = x^2$ onto the function $g(x) = x^2 + 2x + 6$ .

The Graph of $f(x) = x^2$

The graph of the function $f(x) = x^2$ is a parabola that opens upwards. It has a minimum point at the origin (0, 0) and is symmetric about the y-axis. The equation of this function can be written in the form $y = ax^2$ , where $a$ is a constant. In this case, $a = 1$ .

The Graph of $g(x) = x^2 + 2x + 6$

The graph of the function $g(x) = x^2 + 2x + 6$ is also a parabola, but it is shifted to the left and upwards compared to the graph of $f(x) = x^2$ . To understand this, let's rewrite the equation of $g(x)$ in the form $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. By completing the square, we get:

g(x) = (x + 1)^2 + 5

This tells us that the vertex of the parabola is at the point $(-1, 5)$ .

Translation of the Graph

To map the graph of $f(x) = x^2$ onto the graph of $g(x) = x^2 + 2x + 6$ , we need to find the translation that takes the graph of $f(x)$ to the graph of $g(x)$ . This translation can be represented as a combination of horizontal and vertical shifts.

Let's consider the options given:

A. Left 1 unit, up 5 units B. Right 1 unit, up 5 units C. Left 2 units, up 2 units D. Right 2 units, up 2 units

We need to determine which of these translations maps the graph of $f(x) = x^2$ onto the graph of $g(x) = x^2 + 2x + 6$ .

Analyzing the Options

Let's analyze each option one by one:

A. Left 1 unit, up 5 units

If we shift the graph of $f(x) = x^2$ to the left by 1 unit, the new equation becomes $f(x + 1) = (x + 1)^2$ . If we then shift this graph upwards by 5 units, the new equation becomes $f(x + 1) + 5 = (x + 1)^2 + 5$ . This is equivalent to the equation of $g(x)$ , which is $(x + 1)^2 + 5$ . Therefore, option A is a possible translation.

B. Right 1 unit, up 5 units

If we shift the graph of $f(x) = x^2$ to the right by 1 unit, the new equation becomes $f(x - 1) = (x - 1)^2$ . If we then shift this graph upwards by 5 units, the new equation becomes $f(x - 1) + 5 = (x - 1)^2 + 5$ . This is not equivalent to the equation of $g(x)$ , which is $(x + 1)^2 + 5$ . Therefore, option B is not a possible translation.

C. Left 2 units, up 2 units

If we shift the graph of $f(x) = x^2$ to the left by 2 units, the new equation becomes $f(x + 2) = (x + 2)^2$ . If we then shift this graph upwards by 2 units, the new equation becomes $f(x + 2) + 2 = (x + 2)^2 + 2$ . This is not equivalent to the equation of $g(x)$ , which is $(x + 1)^2 + 5$ . Therefore, option C is not a possible translation.

D. Right 2 units, up 2 units

If we shift the graph of $f(x) = x^2$ to the right by 2 units, the new equation becomes $f(x - 2) = (x - 2)^2$ . If we then shift this graph upwards by 2 units, the new equation becomes $f(x - 2) + 2 = (x - 2)^2 + 2$ . This is not equivalent to the equation of $g(x)$ , which is $(x + 1)^2 + 5$ . Therefore, option D is not a possible translation.

Conclusion

Based on our analysis, we can conclude that the translation that maps the graph of the function $f(x) = x^2$ onto the function $g(x) = x^2 + 2x + 6$ is:

A. Left 1 unit, up 5 units

This translation takes the graph of $f(x) = x^2$ to the graph of $g(x) = x^2 + 2x + 6$ .
Q&A: Function Transformations and Graph Mapping

Understanding Function Transformations

In our previous article, we explored the concept of function transformations and how to map the graph of one function onto another. We analyzed the translation that maps the graph of the function $f(x) = x^2$ onto the function $g(x) = x^2 + 2x + 6$ . In this article, we will answer some frequently asked questions related to function transformations and graph mapping.

Q: What is a function transformation?

A: A function transformation is a change in the graph of a function that results from a change in the function's equation. This can include horizontal and vertical shifts, as well as changes in the function's shape or orientation.

Q: What are the different types of function transformations?

A: There are several types of function transformations, including:

Horizontal shifts: shifting the graph of a function to the left or right
Vertical shifts: shifting the graph of a function up or down
Horizontal stretches and compressions: changing the width of the graph of a function
Vertical stretches and compressions: changing the height of the graph of a function
Reflections: flipping the graph of a function over a line or axis

Q: How do I determine the type of function transformation that has occurred?

A: To determine the type of function transformation that has occurred, you can analyze the equation of the function and look for changes in the coefficients or constants. You can also use graphing software or a calculator to visualize the graph of the function and identify the type of transformation that has occurred.

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

A: A horizontal shift is a change in the x-coordinate of the graph of a function, while a vertical shift is a change in the y-coordinate of the graph of a function. Horizontal shifts move the graph of a function to the left or right, while vertical shifts move the graph of a function up or down.

Q: How do I map the graph of one function onto another?

A: To map the graph of one function onto another, you can use the following steps:

Identify the type of function transformation that has occurred
Determine the direction and magnitude of the transformation
Apply the transformation to the graph of the original function
Verify that the resulting graph is equivalent to the graph of the target function

Q: What are some common mistakes to avoid when mapping the graph of one function onto another?

A: Some common mistakes to avoid when mapping the graph of one function onto another include:

Failing to identify the type of function transformation that has occurred
Misinterpreting the direction or magnitude of the transformation
Applying the transformation incorrectly
Failing to verify that the resulting graph is equivalent to the graph of the target function

Q: How do I use function transformations to solve real-world problems?

A: Function transformations can be used to solve a wide range of real-world problems, including:

Modeling population growth or decline
Analyzing the behavior of physical systems
Optimizing the design of a product or system
Predicting the outcome of a future event or scenario

Conclusion

In this article, we have answered some frequently asked questions related to function transformations and graph mapping. We have discussed the different types of function transformations, how to determine the type of transformation that has occurred, and how to map the graph of one function onto another. We have also highlighted some common mistakes to avoid and provided examples of how function transformations can be used to solve real-world problems.