Graphing F(x) = Floor(x) - 2 On [0, 3) A Comprehensive Guide

Hey guys! Today, we're diving into the fascinating world of floor functions and how they transform into step graphs. Specifically, we're going to break down the graph of f(x) = ⌊x⌋ - 2 over the interval [0, 3). If you've ever wondered what those stair-like graphs are all about, you're in the right place. Let's jump in and make sure we understand every step (pun intended!) of this process.

Understanding the Floor Function: The Foundation of Our Graph

Before we can tackle the graph itself, we need to get cozy with the floor function, denoted by ⌊x⌋. Think of it as a number's personal bouncer, only allowing whole numbers (integers) onto the dance floor. But there's a catch: it only lets in the greatest integer less than or equal to x. So, if x is already an integer, like 2, then ⌊2⌋ = 2. No problem there. But what if x is a decimal, like 2.7? The floor function chops off the decimal part, giving us ⌊2.7⌋ = 2. It's like rounding down, but only to the nearest integer. Now that we understand the foundation, let’s move on to how this plays out in a graph.

Breaking Down the Floor Function in Action

Let's walk through some examples to really solidify this concept. Imagine x is somewhere between 0 and 1, but not quite reaching 1. For any value in this range, say 0.5, the floor function ⌊0.5⌋ = 0. Similarly, for any x between 1 and 2 (but less than 2), like 1.99, we have ⌊1.99⌋ = 1. See the pattern? The floor function holds steady at each integer value until we hit the next integer, then it jumps up. This is what creates the steps in our graph. Understanding this behavior of the floor function is key to understanding the graph of f(x) = ⌊x⌋ - 2. Each step corresponds to a range of x values where the floor function gives the same integer result. This makes analyzing and sketching the graph much simpler.

The Impact of the '- 2' Shift

Now that we've got the floor function down, let's throw a little curveball: the '- 2' in our function f(x) = ⌊x⌋ - 2. This is a vertical shift. It means we're taking the basic floor function graph and moving it down two units on the y-axis. So, every point on the graph gets shifted down by 2. This seemingly small change has a big impact on where our steps will be located. Instead of the steps starting at y = 0, y = 1, and so on, they'll now start at y = -2, y = -1, and so on. This vertical shift is crucial to understanding the final shape and position of the graph. To master graphing these functions, it’s essential to first comprehend the floor function and then how vertical shifts like '- 2' affect the graph's position. With these principles, deciphering the graph of f(x) = ⌊x⌋ - 2 becomes straightforward.

Graphing f(x) = ⌊x⌋ - 2 on [0, 3): Step-by-Step

Alright, let's put it all together and sketch the graph of f(x) = ⌊x⌋ - 2 on the interval [0, 3). This interval means we're only looking at the portion of the graph where x is between 0 (inclusive) and 3 (but not including 3). We'll break this down into smaller chunks based on the integer values within our interval.

The First Step: 0 ≤ x < 1

Let's focus on the first segment, where x is between 0 (inclusive) and 1 (exclusive). This is written as 0 ≤ x < 1. In this range, the floor function ⌊x⌋ will always be 0. Remember, it takes the greatest integer less than or equal to x. So, if x is 0.5, ⌊0.5⌋ = 0. If x is 0.99, ⌊0.99⌋ = 0. Now, we plug this into our function: f(x) = ⌊x⌋ - 2 = 0 - 2 = -2. This means that for all x in the interval 0 ≤ x < 1, the graph will be a horizontal line at y = -2. We draw a line segment at y = -2, starting at x = 0 (a closed circle, because x can be 0) and extending to x = 1 (an open circle, because x cannot be 1). So, the first step in our graph is a horizontal line at y = -2.

The Second Step: 1 ≤ x < 2

Moving on to the next segment, we're now looking at x values between 1 (inclusive) and 2 (exclusive), or 1 ≤ x < 2. In this range, the floor function ⌊x⌋ will always be 1. For instance, if x is 1.3, ⌊1.3⌋ = 1, and if x is 1.999, ⌊1.999⌋ = 1. Plugging this into our function, we get f(x) = ⌊x⌋ - 2 = 1 - 2 = -1. This tells us that for all x in the interval 1 ≤ x < 2, the graph will be a horizontal line at y = -1. We draw another line segment, this time at y = -1, starting at x = 1 (a closed circle) and extending to x = 2 (an open circle). So, the second step in our graph sits one unit higher than the first, at y = -1.

The Third Step: 2 ≤ x < 3

Finally, we consider the last segment of our interval, 2 ≤ x < 3. Here, the floor function ⌊x⌋ will be 2. Whether x is 2.1 or 2.99, ⌊x⌋ will equal 2. So, f(x) = ⌊x⌋ - 2 = 2 - 2 = 0. This means that for 2 ≤ x < 3, the graph is a horizontal line at y = 0. We draw our final step, a line segment at y = 0, starting at x = 2 (a closed circle) and going up to x = 3 (an open circle, since 3 is not included in our interval). Therefore, the third step is a horizontal line at y = 0, completing our graph on the interval [0, 3).

Identifying the Correct Statement: Putting It All Together

Now that we've thoroughly analyzed and graphed f(x) = ⌊x⌋ - 2 on the interval [0, 3), we can confidently identify the statement that accurately describes the graph. Let's recap what we found:

• For 0 ≤ x < 1, the graph is at y = -2. (First Step)
• For 1 ≤ x < 2, the graph is at y = -1. (Second Step)
• For 2 ≤ x < 3, the graph is at y = 0. (Third Step)

With this in mind, we can evaluate the given statements and choose the one that matches our findings. If we see a statement describing these exact steps at these specific y-values for the correct intervals of x, then that's our winner! By breaking down the problem into manageable steps, understanding the floor function, and accounting for the vertical shift, we've successfully navigated this graph and are ready to tackle similar problems with confidence.

The Correct Answer

Based on our analysis, the statement that accurately describes the graph of f(x) = ⌊x⌋ - 2 on [0, 3) is:

A. The steps are at y = -2 for 0 ≤ x < 1, at y = -1 for 1 ≤ x < 2, and at y = 0 for 2 ≤ x < 3.

This statement perfectly aligns with the steps we identified and graphed earlier. We saw that the function f(x) takes the value -2 for x in the interval [0, 1), -1 for x in the interval [1, 2), and 0 for x in the interval [2, 3). Therefore, the graph consists of three horizontal line segments at y values of -2, -1, and 0, respectively, over these intervals.

Conclusion: Mastering the Floor Function and Its Graphs

And there you have it! We've successfully navigated the graph of f(x) = ⌊x⌋ - 2 on the interval [0, 3). By understanding the floor function, recognizing the impact of vertical shifts, and breaking down the interval into manageable segments, we were able to sketch the graph and identify the correct statement. Remember, the key to mastering these types of problems is a solid grasp of the fundamentals and a systematic approach. Keep practicing, and you'll become a pro at graphing floor functions in no time! Understanding these concepts not only boosts your math skills but also helps in various fields like computer science, where floor functions are used in algorithms and data structures. So, keep exploring, keep learning, and remember to enjoy the process of unraveling mathematical challenges!