Determine If The Following Statement Is True Or False:To Graph $y = (x-2)^3$, Shift The Graph Of $y = X^3$ To The Left 2 Units.Choose The Correct Answer:A. The Statement Is False; Since 2 Is Subtracted From The Input, The Graph Is
Introduction
Graphing polynomial functions can be a complex task, especially when dealing with transformations and shifts. In this article, we will explore the concept of shifting the graph of a polynomial function and determine if a given statement is true or false. We will focus on the function $y = (x-2)^3$ and compare it to the graph of $y = x^3$.
Understanding the Function
The function $y = (x-2)^3$ is a cubic function that has been transformed from the original function $y = x^3$. To understand this transformation, let's break down the function into its components.
- The term $(x-2)$ represents a horizontal shift of the graph of $y = x^3$.
- The exponent $3$ indicates that the graph will be raised to the third power.
Horizontal Shifts
A horizontal shift is a transformation that moves the graph of a function to the left or right. In the case of the function $y = (x-2)^3$, the graph of $y = x^3$ is shifted to the left by 2 units.
Determining the Truth of the Statement
The statement claims that the graph of $y = (x-2)^3$ can be obtained by shifting the graph of $y = x^3$ to the left 2 units. To determine if this statement is true or false, let's analyze the function.
- The term $(x-2)$ represents a horizontal shift of the graph of $y = x^3$.
- The exponent $3$ indicates that the graph will be raised to the third power.
Since the term $(x-2)$ represents a horizontal shift of the graph of $y = x^3$, the graph of $y = (x-2)^3$ can be obtained by shifting the graph of $y = x^3$ to the left 2 units. Therefore, the statement is true.
Conclusion
In conclusion, the statement that the graph of $y = (x-2)^3$ can be obtained by shifting the graph of $y = x^3$ to the left 2 units is true. This is because the term $(x-2)$ represents a horizontal shift of the graph of $y = x^3$, and the exponent $3$ indicates that the graph will be raised to the third power.
Understanding the Graph of $y = x^3$
The graph of $y = x^3$ is a cubic function that has a positive leading coefficient. This means that the graph will open upwards and will have a single turning point, known as the vertex.
Graphing the Function $y = (x-2)^3$
To graph the function $y = (x-2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the left by 2 units.
Shifting the Graph
To shift the graph of $y = x^3$ to the left by 2 units, we can replace the variable $x$ with $(x+2)$. This will give us the function $y = (x+2)^3$.
- The term $(x+2)$ represents a horizontal shift of the graph of $y = x^3$ to the left by 2 units.
- The exponent $3$ indicates that the graph will be raised to the third power.
Graphing the Function $y = (x-2)^3$
To graph the function $y = (x-2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the left by 2 units.
Graphing the Function $y = (x+2)^3$
To graph the function $y = (x+2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the right by 2 units.
Conclusion
In conclusion, the graph of $y = (x-2)^3$ can be obtained by shifting the graph of $y = x^3$ to the left 2 units. This is because the term $(x-2)$ represents a horizontal shift of the graph of $y = x^3$, and the exponent $3$ indicates that the graph will be raised to the third power.
Understanding the Graph of $y = (x-2)^3$
The graph of $y = (x-2)^3$ is a cubic function that has a positive leading coefficient. This means that the graph will open upwards and will have a single turning point, known as the vertex.
Graphing the Function $y = (x-2)^3$
To graph the function $y = (x-2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the left by 2 units.
Shifting the Graph
To shift the graph of $y = x^3$ to the left by 2 units, we can replace the variable $x$ with $(x+2)$. This will give us the function $y = (x+2)^3$.
- The term $(x+2)$ represents a horizontal shift of the graph of $y = x^3$ to the left by 2 units.
- The exponent $3$ indicates that the graph will be raised to the third power.
Graphing the Function $y = (x+2)^3$
To graph the function $y = (x+2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the right by 2 units.
Conclusion
In conclusion, the graph of $y = (x-2)^3$ can be obtained by shifting the graph of $y = x^3$ to the left 2 units. This is because the term $(x-2)$ represents a horizontal shift of the graph of $y = x^3$, and the exponent $3$ indicates that the graph will be raised to the third power.
Understanding the Graph of $y = (x-2)^3$
The graph of $y = (x-2)^3$ is a cubic function that has a positive leading coefficient. This means that the graph will open upwards and will have a single turning point, known as the vertex.
Graphing the Function $y = (x-2)^3$
To graph the function $y = (x-2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the left by 2 units.
Shifting the Graph
To shift the graph of $y = x^3$ to the left by 2 units, we can replace the variable $x$ with $(x+2)$. This will give us the function $y = (x+2)^3$.
- The term $(x+2)$ represents a horizontal shift of the graph of $y = x^3$ to the left by 2 units.
- The exponent $3$ indicates that the graph will be raised to the third power.
Graphing the Function $y = (x+2)^3$
To graph the function $y = (x+2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the right by 2 units.
Conclusion
In conclusion, the graph of $y = (x-2)^3$ can be obtained by shifting the graph of $y = x^3$ to the left 2 units. This is because the term $(x-2)$ represents a horizontal shift of the graph of $y = x^3$, and the exponent $3$ indicates that the graph will be raised to the third power.
Understanding the Graph of $y = (x-2)^3$
The graph of $y = (x-2)^3$ is a cubic function that has a positive leading coefficient. This means that the graph will open upwards and will have a single turning point, known as the vertex.
Graphing the Function $y = (x-2)^3$
To graph the function $y = (x-2)^3$, we can start by graphing the function $y = x^3$. Then, we can shift the graph to the left by 2 units.
Shifting the Graph
Q&A: Graphing Polynomial Functions
Q: What is a horizontal shift in a polynomial function?
A: A horizontal shift is a transformation that moves the graph of a polynomial function to the left or right. In the case of the function $y = (x-2)^3$, the graph of $y = x^3$ is shifted to the left by 2 units.
Q: How do I determine if a statement about a polynomial function is true or false?
A: To determine if a statement about a polynomial function is true or false, you need to analyze the function and understand the transformations that have been applied. In the case of the function $y = (x-2)^3$, the statement that the graph can be obtained by shifting the graph of $y = x^3$ to the left 2 units is true.
Q: What is the difference between a horizontal shift and a vertical shift in a polynomial function?
A: A horizontal shift moves the graph of a polynomial function to the left or right, while a vertical shift moves the graph up or down. In the case of the function $y = (x-2)^3$, the graph is shifted horizontally to the left by 2 units.
Q: How do I graph a polynomial function with a horizontal shift?
A: To graph a polynomial function with a horizontal shift, you need to start by graphing the original function. Then, you can shift the graph to the left or right by the specified amount.
Q: What is the vertex of a polynomial function?
A: The vertex of a polynomial function is the turning point of the graph. In the case of the function $y = (x-2)^3$, the vertex is located at the point $(2, 0)$.
Q: How do I find the vertex of a polynomial function?
A: To find the vertex of a polynomial function, you need to analyze the function and understand the transformations that have been applied. In the case of the function $y = (x-2)^3$, the vertex is located at the point $(2, 0)$.
Q: What is the difference between a cubic function and a quadratic function?
A: A cubic function is a polynomial function of degree 3, while a quadratic function is a polynomial function of degree 2. In the case of the function $y = (x-2)^3$, the graph is a cubic function.
Q: How do I graph a cubic function?
A: To graph a cubic function, you need to start by graphing the original function. Then, you can shift the graph to the left or right by the specified amount.
Q: What is the significance of the exponent in a polynomial function?
A: The exponent in a polynomial function determines the degree of the function. In the case of the function $y = (x-2)^3$, the exponent is 3, which means that the graph is a cubic function.
Q: How do I determine the degree of a polynomial function?
A: To determine the degree of a polynomial function, you need to analyze the function and understand the transformations that have been applied. In the case of the function $y = (x-2)^3$, the degree is 3.
Conclusion
In conclusion, graphing polynomial functions with shifts and transformations can be a complex task, but by understanding the concepts and techniques, you can master this skill. Remember to analyze the function, understand the transformations, and apply the techniques to graph the function.