(b) Using The Appropriate Shifts And Transformations On The Function \[$ F(x) = X^2 \$\], Sketch The Graph Of \[$ Y = |x^2 - 2x| \$\].

Feb 28, 2025 by ADMIN 135 views

**Transforming and Sketching the Graph of a Function**

Introduction

In mathematics, graphing functions is an essential skill that helps us visualize and understand the behavior of mathematical relationships. When given a function, we can use various transformations and shifts to create new functions and modify the original graph. In this article, we will explore how to use the appropriate shifts and transformations on the function $f(x) = x^2$ to sketch the graph of $y = |x^2 - 2x|$ .

Understanding the Original Function

The original function is $f(x) = x^2$ . This 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.

Transforming the Function

To transform the function $f(x) = x^2$ into $y = |x^2 - 2x|$ , we need to apply two transformations:

Horizontal Shift: The function $y = |x^2 - 2x|$ can be obtained by shifting the graph of $f(x) = x^2$ two units to the right. This is because the term $-2x$ inside the absolute value function represents a horizontal shift of 2 units to the right.
Vertical Shift: The absolute value function $|x^2 - 2x|$ also represents a vertical shift of the graph of $f(x) = x^2$ . However, since the absolute value function is always non-negative, the vertical shift is not a simple translation. Instead, it represents a reflection of the graph of $f(x) = x^2$ about the x-axis.

Applying the Transformations

To apply the transformations, we can start with the original function $f(x) = x^2$ and then apply the horizontal shift and vertical shift.

Horizontal Shift: To shift the graph of $f(x) = x^2$ two units to the right, we can replace $x$ with $(x - 2)$ in the original function. This gives us $f(x - 2) = (x - 2)^2$ .
Vertical Shift: To reflect the graph of $f(x - 2) = (x - 2)^2$ about the x-axis, we can multiply the function by $-1$ . This gives us $-f(x - 2) = -(x - 2)^2$ .

Simplifying the Function

The function $-f(x - 2) = -(x - 2)^2$ can be simplified by expanding the squared term. This gives us $-f(x - 2) = -(x^2 - 4x + 4)$ .

Simplifying Further

To simplify the function further, we can distribute the negative sign to the terms inside the parentheses. This gives us $-f(x - 2) = -x^2 + 4x - 4$ .

Absolute Value Function

The function $-f(x - 2) = -x^2 + 4x - 4$ is not the same as the original function $y = |x^2 - 2x|$ . To obtain the original function, we need to take the absolute value of the function $-x^2 + 4x - 4$ . This gives us $y = | -x^2 + 4x - 4 |$ .

Sketching the Graph

To sketch the graph of $y = |x^2 - 2x|$ , we can use the transformations and shifts that we applied to the original function $f(x) = x^2$ . The graph of $y = |x^2 - 2x|$ is a V-shaped curve that is symmetric about the y-axis.

Key Features of the Graph

The graph of $y = |x^2 - 2x|$ has the following key features:

Vertex: The vertex of the graph is at the point $(0, 2)$ .
Axis of Symmetry: The axis of symmetry of the graph is the y-axis.
Intercepts: The graph has x-intercepts at the points $(0, 0)$ and $(2, 0)$ .
Asymptotes: The graph has a horizontal asymptote at the line $y = 0$ .

Conclusion

In this article, we explored how to use the appropriate shifts and transformations on the function $f(x) = x^2$ to sketch the graph of $y = |x^2 - 2x|$ . We applied a horizontal shift of 2 units to the right and a vertical shift of the graph about the x-axis to obtain the original function. We then simplified the function and took the absolute value to obtain the final graph. The graph of $y = |x^2 - 2x|$ is a V-shaped curve that is symmetric about the y-axis.

References

[1] "Graphing Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/graphing.html
[2] "Transformations of Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f1f4f/x2f1f5f/x2f1f5a

Glossary

Horizontal Shift: A transformation that shifts the graph of a function to the left or right.
Vertical Shift: A transformation that shifts the graph of a function up or down.
Absolute Value Function: A function that takes the absolute value of an expression.
Vertex: The point on a graph where the function changes from increasing to decreasing or vice versa.
Axis of Symmetry: A line that passes through the vertex of a graph and is perpendicular to the x-axis or y-axis.
Intercepts: The points where a graph intersects the x-axis or y-axis.
Asymptotes: Lines that the graph of a function approaches but never touches.
Q&A: Transforming and Sketching the Graph of a Function =====================================================

Introduction

In our previous article, we explored how to use the appropriate shifts and transformations on the function $f(x) = x^2$ to sketch the graph of $y = |x^2 - 2x|$ . In this article, we will answer some frequently asked questions about transforming and sketching the graph of a function.

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

A: A horizontal shift is a transformation that shifts the graph of a function to the left or right, while a vertical shift is a transformation that shifts the graph of a function up or down.

Q: How do I apply a horizontal shift to a function?

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

Q: How do I apply a vertical shift to a function?

A: To apply a vertical shift to a function, you need to multiply the function by a constant, $k$ , where $k$ is the amount of the shift.

Q: What is the absolute value function?

A: The absolute value function is a function that takes the absolute value of an expression. It is denoted by $|x|$ and is defined as:

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Q: How do I sketch the graph of an absolute value function?

A: To sketch the graph of an absolute value function, you need to find the vertex of the graph and then sketch the graph on either side of the vertex.

Q: What are the key features of the graph of an absolute value function?

A: The key features of the graph of an absolute value function are:

Vertex: The vertex of the graph is at the point $(0, 0)$ .
Axis of Symmetry: The axis of symmetry of the graph is the y-axis.
Intercepts: The graph has x-intercepts at the points $(0, 0)$ and $(2, 0)$ .
Asymptotes: The graph has a horizontal asymptote at the line $y = 0$ .

Q: How do I use transformations to sketch the graph of a function?

A: To use transformations to sketch the graph of a function, you need to apply the following steps:

Horizontal Shift: Apply a horizontal shift to the function by replacing $x$ with $(x - h)$ .
Vertical Shift: Apply a vertical shift to the function by multiplying the function by a constant, $k$ .
Absolute Value Function: Take the absolute value of the function to obtain the final graph.

Q: What are some common transformations that can be applied to a function?

A: Some common transformations that can be applied to a function are:

Horizontal Shift: A transformation that shifts the graph of a function to the left or right.
Vertical Shift: A transformation that shifts the graph of a function up or down.
Absolute Value Function: A function that takes the absolute value of an expression.
Reflection: A transformation that reflects the graph of a function about the x-axis or y-axis.

Conclusion

In this article, we answered some frequently asked questions about transforming and sketching the graph of a function. We discussed the difference between a horizontal shift and a vertical shift, how to apply these transformations, and how to sketch the graph of an absolute value function. We also discussed the key features of the graph of an absolute value function and how to use transformations to sketch the graph of a function.

References

[1] "Graphing Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/graphing.html
[2] "Transformations of Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f1f4f/x2f1f5f/x2f1f5a

Glossary

Horizontal Shift: A transformation that shifts the graph of a function to the left or right.
Vertical Shift: A transformation that shifts the graph of a function up or down.
Absolute Value Function: A function that takes the absolute value of an expression.
Vertex: The point on a graph where the function changes from increasing to decreasing or vice versa.
Axis of Symmetry: A line that passes through the vertex of a graph and is perpendicular to the x-axis or y-axis.
Intercepts: The points where a graph intersects the x-axis or y-axis.
Asymptotes: Lines that the graph of a function approaches but never touches.