Which Statement Describes The Graph Of F ( X ) = ⌊ X ⌋ − 2 F(x)=\lfloor X\rfloor-2 F ( X ) = ⌊ X ⌋ − 2 On {0,3 }$?A. The Steps Are At Y = − 2 Y=-2 Y = − 2 For 0 ≤ X \textless 1 0 \leq X\ \textless \ 1 0 ≤ X \textless 1 , At Y = − 1 Y=-1 Y = − 1 For 1 ≤ X \textless 2 1 \leq X\ \textless \ 2 1 ≤ X \textless 2 , And At Y = 0 Y=0 Y = 0

Introduction

In mathematics, the floor function, denoted by $\lfloor x\rfloor$, is a fundamental concept that plays a crucial role in various mathematical disciplines, including calculus, algebra, and analysis. The floor function of a real number $x$ is defined as the largest integer less than or equal to $x$. In this article, we will explore the graph of the function $f(x)=\lfloor x\rfloor-2$ on the interval $[0,3]$. We will examine the characteristics of the graph, including the location of the steps, and determine which statement accurately describes the graph.

The Floor Function

The floor function $\lfloor x\rfloor$ is a step function that takes on integer values. For any real number $x$, the floor function returns the largest integer less than or equal to $x$. For example, $\lfloor 3.7\rfloor = 3$, $\lfloor -2.3\rfloor = -3$, and $\lfloor 0\rfloor = 0$. The graph of the floor function consists of a series of horizontal line segments, each corresponding to an integer value.

The Function $f(x)=\lfloor x\rfloor-2$

The function $f(x)=\lfloor x\rfloor-2$ is a transformation of the floor function. The graph of this function will also consist of a series of horizontal line segments, but with a vertical shift of $-2$ units. This means that each step in the graph will be shifted downward by $2$ units.

**Graph of $f(x)=\lfloor x\rfloor-2$ on $[0,3]$

To determine the graph of $f(x)=\lfloor x\rfloor-2$ on the interval $[0,3]$, we need to examine the values of the function at various points within the interval.

• For $0 \leq x < 1$, the floor function takes on the value $0$, so $f(x) = 0 - 2 = -2$.
• For $1 \leq x < 2$, the floor function takes on the value $1$, so $f(x) = 1 - 2 = -1$.
• For $2 \leq x < 3$, the floor function takes on the value $2$, so $f(x) = 2 - 2 = 0$.

Which Statement Describes the Graph?

Based on the analysis above, we can determine which statement accurately describes the graph of $f(x)=\lfloor x\rfloor-2$ on the interval $[0,3]$.

A. The steps are at $y=-2$ for $0 \leq x\ \textless \ 1$, at $y=-1$ for $1 \leq x\ \textless \ 2$, and at $y=0$ for $2 \leq x\ \textless \ 3$.

This statement accurately describes the graph of $f(x)=\lfloor x\rfloor-2$ on the interval $[0,3]$. The steps in the graph are located at $y=-2$ for $0 \leq x < 1$, at $y=-1$ for $1 \leq x < 2$, and at $y=0$ for $2 \leq x < 3$.

Conclusion

Q: What is the floor function, and how does it relate to the graph of $f(x)=\lfloor x\rfloor-2$?

A: The floor function, denoted by $\lfloor x\rfloor$, is a mathematical function that returns the largest integer less than or equal to a given real number $x$. In the context of the graph of $f(x)=\lfloor x\rfloor-2$, the floor function is used to determine the value of the function at various points within the interval $[0,3]$.

Q: What is the significance of the vertical shift of $-2$ units in the graph of $f(x)=\lfloor x\rfloor-2$?

A: The vertical shift of $-2$ units in the graph of $f(x)=\lfloor x\rfloor-2$ means that each step in the graph is shifted downward by $2$ units. This is a result of the transformation of the floor function by subtracting $2$ from each value.

Q: How do the steps in the graph of $f(x)=\lfloor x\rfloor-2$ relate to the values of the floor function?

A: The steps in the graph of $f(x)=\lfloor x\rfloor-2$ correspond to the values of the floor function at various points within the interval $[0,3]$. For example, the step at $y=-2$ corresponds to the value of the floor function being $0$, the step at $y=-1$ corresponds to the value of the floor function being $1$, and the step at $y=0$ corresponds to the value of the floor function being $2$.

Q: What is the relationship between the graph of $f(x)=\lfloor x\rfloor-2$ and the graph of the floor function?

A: The graph of $f(x)=\lfloor x\rfloor-2$ is a transformation of the graph of the floor function. The graph of the floor function consists of a series of horizontal line segments, each corresponding to an integer value. The graph of $f(x)=\lfloor x\rfloor-2$ is obtained by shifting the graph of the floor function downward by $2$ units.

Q: How can the graph of $f(x)=\lfloor x\rfloor-2$ be used in real-world applications?

A: The graph of $f(x)=\lfloor x\rfloor-2$ can be used in various real-world applications, such as modeling the behavior of physical systems, analyzing data, and making predictions. For example, the graph of $f(x)=\lfloor x\rfloor-2$ can be used to model the behavior of a physical system that has a step-like response to changes in the input.

Q: What are some common mistakes to avoid when working with the graph of $f(x)=\lfloor x\rfloor-2$?

A: Some common mistakes to avoid when working with the graph of $f(x)=\lfloor x\rfloor-2$ include:

• Failing to account for the vertical shift of $-2$ units
• Misinterpreting the values of the floor function
• Failing to recognize the relationship between the graph of $f(x)=\lfloor x\rfloor-2$ and the graph of the floor function

Q: How can the graph of $f(x)=\lfloor x\rfloor-2$ be used to solve problems in mathematics and science?

A: The graph of $f(x)=\lfloor x\rfloor-2$ can be used to solve problems in mathematics and science by modeling the behavior of physical systems, analyzing data, and making predictions. For example, the graph of $f(x)=\lfloor x\rfloor-2$ can be used to model the behavior of a physical system that has a step-like response to changes in the input.

Conclusion

In conclusion, the graph of $f(x)=\lfloor x\rfloor-2$ is a transformation of the graph of the floor function, obtained by shifting the graph downward by $2$ units. The graph of $f(x)=\lfloor x\rfloor-2$ can be used in various real-world applications, such as modeling the behavior of physical systems, analyzing data, and making predictions. By understanding the graph of $f(x)=\lfloor x\rfloor-2$, we can solve problems in mathematics and science and gain a deeper understanding of the behavior of physical systems.