Understanding the Original Graph

Oct 13, 2025 by windailys.com 16 views

Introduction

In mathematics, graph transformation is a crucial concept that helps us understand how different functions can be obtained from a given function. In this article, we will explore how to transform the graph of $$y=|x|$$ to make it the graph of $$y=-2|x|+3$$. We will delve into the world of absolute value functions, vertical and horizontal shifts, and scaling to understand the transformation process.

Understanding the Original Graph

The graph of $$y=|x|$$ is a V-shaped graph that opens upwards. It has a minimum point at the origin (0,0) and extends upwards on both sides of the y-axis. The absolute value function is defined as:

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

This function is symmetric about the y-axis, meaning that for every point (x,y) on the graph, there is a corresponding point (-x,y) also on the graph.

Transforming the Graph

To transform the graph of $$y=|x|$$ to make it the graph of $$y=-2|x|+3$$, we need to apply three transformations:

Vertical Scaling: We need to scale the graph vertically by a factor of 2. This means that every y-coordinate on the graph will be multiplied by 2.
Reflection: We need to reflect the graph about the x-axis. This means that every y-coordinate on the graph will be negated.
Vertical Shift: We need to shift the graph upwards by 3 units. This means that every y-coordinate on the graph will be increased by 3.

Applying the Transformations

Let's apply the transformations one by one:

Vertical Scaling: To scale the graph vertically by a factor of 2, we multiply every y-coordinate by 2. This gives us:

y = 2|x|

The graph of $$y=2|x|$$ is a V-shaped graph that opens upwards, with a minimum point at the origin (0,0) and extends upwards on both sides of the y-axis. The graph is twice as tall as the original graph.

Reflection: To reflect the graph about the x-axis, we negate every y-coordinate. This gives us:

y = -2|x|

The graph of $$y=-2|x|$$ is a V-shaped graph that opens downwards, with a minimum point at the origin (0,0) and extends downwards on both sides of the y-axis. The graph is a mirror image of the original graph.

Vertical Shift: To shift the graph upwards by 3 units, we add 3 to every y-coordinate. This gives us:

y = -2|x| + 3

The graph of $$y=-2|x|+3$$ is a V-shaped graph that opens downwards, with a minimum point at the origin (0,3) and extends downwards on both sides of the y-axis. The graph is a vertical shift of the previous graph by 3 units.

Conclusion

In this article, we explored how to transform the graph of $$y=|x|$$ to make it the graph of $$y=-2|x|+3$$. We applied three transformations: vertical scaling, reflection, and vertical shift. By understanding these transformations, we can visualize how different functions can be obtained from a given function. This concept is crucial in mathematics, particularly in graph theory and function analysis.

Key Takeaways

The graph of $$y=|x|$$ is a V-shaped graph that opens upwards.
To transform the graph of $$y=|x|$$ to make it the graph of $$y=-2|x|+3$$, we need to apply three transformations: vertical scaling, reflection, and vertical shift.
Vertical scaling involves multiplying every y-coordinate by a factor.
Reflection involves negating every y-coordinate.
Vertical shift involves adding a constant to every y-coordinate.

Introduction

In our previous article, we explored how to transform the graph of $$y=|x|$$ to make it the graph of $$y=-2|x|+3$$. We applied three transformations: vertical scaling, reflection, and vertical shift. In this article, we will answer some frequently asked questions about graph transformation and function analysis.

Q&A

Q: What is graph transformation?

A: Graph transformation is the process of changing the graph of a function to obtain a new function. This can be done by applying various transformations such as vertical scaling, reflection, and vertical shift.

Q: What is the difference between vertical scaling and vertical shift?

A: Vertical scaling involves multiplying every y-coordinate by a factor, while vertical shift involves adding a constant to every y-coordinate.

Q: How do I determine the type of transformation applied to a graph?

A: To determine the type of transformation applied to a graph, look for the following:

Vertical scaling: If the graph is stretched or compressed vertically, it is a vertical scaling.
Reflection: If the graph is a mirror image of the original graph, it is a reflection.
Vertical shift: If the graph is shifted up or down, it is a vertical shift.

Q: Can I apply multiple transformations to a graph?

A: Yes, you can apply multiple transformations to a graph. However, make sure to apply the transformations in the correct order to obtain the desired result.

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

A: To graph a function with multiple transformations, follow these steps:

Apply the first transformation to the original graph.
Apply the second transformation to the resulting graph.
Continue applying transformations until you obtain the desired graph.

Q: What are some common transformations used in graphing?

A: Some common transformations used in graphing include:

Vertical scaling: Multiplying every y-coordinate by a factor.
Reflection: Negating every y-coordinate.
Vertical shift: Adding a constant to every y-coordinate.
Horizontal shift: Adding a constant to every x-coordinate.
Horizontal scaling: Multiplying every x-coordinate by a factor.

Q: How do I determine the equation of a transformed graph?

A: To determine the equation of a transformed graph, follow these steps:

Identify the original equation of the graph.
Apply the transformations to the original equation.
Simplify the resulting equation to obtain the equation of the transformed graph.

Conclusion

In this article, we answered some frequently asked questions about graph transformation and function analysis. We hope that this article has provided you with a better understanding of graph transformation and function analysis. If you have any further questions, please don't hesitate to ask.

Key Takeaways

Graph transformation is the process of changing the graph of a function to obtain a new function.
Vertical scaling involves multiplying every y-coordinate by a factor.
Reflection involves negating every y-coordinate.
Vertical shift involves adding a constant to every y-coordinate.
Multiple transformations can be applied to a graph.