Points On The Graph Y = 1.5 + [x] Explained With Examples
Hey guys! Today, we're diving into the fascinating world of graphs and functions, specifically focusing on the equation y = 1.5 + [x]. This equation involves the floor function, denoted by [x], which might seem a bit intimidating at first, but trust me, it's super cool once you get the hang of it. We're going to figure out which points lie on the graph of this equation. So, let's put on our math hats and get started!
Understanding the Floor Function and the Equation
Before we jump into the points, let's make sure we all understand what the floor function is. The floor function, [x], simply gives you the greatest integer less than or equal to x. Think of it as rounding down to the nearest whole number. For example, [3.14] is 3, [5] is 5, and [-2.7] is -3. See? Not so scary after all!
Now, let's break down our equation: y = 1.5 + [x]. What this equation is telling us is that for any given x-value, we first find its floor (i.e., the greatest integer less than or equal to x), and then we add 1.5 to that result to get the corresponding y-value. This means that the graph of this equation will have a step-like appearance, because the floor function only changes its value at integer points.
To truly grasp this, let's consider a few examples. If x is 2.3, then [x] is 2, and y would be 1.5 + 2 = 3.5. If x is -1.8, then [x] is -2, and y would be 1.5 + (-2) = -0.5. By walking through these specific cases, we begin to see how the equation behaves and how the floor function influences the y-values based on different x-values. This understanding is crucial as we proceed to evaluate the given points and determine their location relative to the graph. Visualizing these points and their transformations through the floor function gives us a clearer picture of the overall graph and its step-like nature.
By considering more examples, such as x = 0 (where [x] = 0 and y = 1.5) and x = 4 (where [x] = 4 and y = 5.5), we reinforce our understanding of the equation's behavior. The constant addition of 1.5 shifts the floor function's typical step pattern upwards, creating a graph that hovers above the standard integer steps. This shift is a key characteristic of the graph and significantly influences which points will lie on it.
Remember, the graph of this function will consist of horizontal line segments because the y-value remains constant for any x-value within the same integer interval. For instance, between x = 2 and x = 3 (excluding 3), the floor of x is always 2, making y consistently 3.5. These constant segments contribute to the overall step-like appearance, where each step represents an integer value of [x]. Therefore, to determine if a point lies on the graph, we need to check whether its y-coordinate matches the output of the equation for its given x-coordinate.
Evaluating the Given Points
Now, let's get to the meat of the problem. We have a few points to check, and we need to figure out which ones sit on the graph of y = 1.5 + [x]. To do this, we'll simply plug in the x-coordinate of each point into the equation and see if the resulting y-value matches the y-coordinate of the point.
Point 1: (-4.5, -2.5)
Let's start with the point (-4.5, -2.5). We'll plug x = -4.5 into our equation: y = 1.5 + [-4.5]. The floor of -4.5 is -5 (remember, we round down to the nearest integer). So, y = 1.5 + (-5) = -3.5. The y-coordinate we calculated (-3.5) does not match the y-coordinate of the point (-2.5). Therefore, the point (-4.5, -2.5) is not on the graph.
This is a great example to illustrate how the floor function works with negative numbers. Many people might initially think the floor of -4.5 is -4, but it's crucial to remember that it's the greatest integer less than or equal to -4.5, which is -5. This distinction is vital for accurately evaluating points and understanding the graph's behavior in the negative domain. The discrepancy between the calculated y-value and the given y-coordinate highlights the importance of precise calculation and a solid understanding of the floor function's properties.
Point 2: (-0.8, 0.5)
Next up, we have the point (-0.8, 0.5). Plugging x = -0.8 into our equation, we get y = 1.5 + [-0.8]. The floor of -0.8 is -1. So, y = 1.5 + (-1) = 0.5. Hey, the y-coordinate we calculated (0.5) matches the y-coordinate of the point (0.5)! This means the point (-0.8, 0.5) is on the graph.
This point demonstrates a case where the fractional part of the x-coordinate plays a significant role. The value -0.8, being between -1 and 0, has a floor of -1. Adding 1.5 to this floor results in a y-value of 0.5, which precisely matches the point's y-coordinate. This consistency confirms that the point lies on the graph and reinforces our understanding of how the floor function and the constant addition of 1.5 work together to define the graph's specific characteristics.
Point 3: (7.9, 9.5)
Let's move on to the point (7.9, 9.5). We plug x = 7.9 into our equation: y = 1.5 + [7.9]. The floor of 7.9 is 7. So, y = 1.5 + 7 = 8.5. The y-coordinate we calculated (8.5) does not match the y-coordinate of the point (9.5). Therefore, the point (7.9, 9.5) is not on the graph.
This example illustrates how the floor function effectively truncates the decimal part of a number, leading to a specific integer value. The floor of 7.9 is 7, not 8, which is a critical distinction. Adding 1.5 to this value results in 8.5, a y-value that differs from the point's given y-coordinate of 9.5. This discrepancy reinforces the step-like nature of the graph, where each step is defined by the integer values produced by the floor function.
Point 4: (4.5, 6)
Now, let's check the point (4.5, 6). Plugging x = 4.5 into our equation, we get y = 1.5 + [4.5]. The floor of 4.5 is 4. So, y = 1.5 + 4 = 5.5. The y-coordinate we calculated (5.5) does not match the y-coordinate of the point (6). Therefore, the point (4.5, 6) is not on the graph.
Here, the floor of 4.5 correctly yields 4, which when added to 1.5, results in a y-value of 5.5. This value contrasts with the point's y-coordinate of 6, indicating that the point does not align with the graph defined by the equation. This discrepancy highlights the consistent relationship between the floor function and the resulting y-values, which is essential for correctly identifying points that lie on the graph.
Point 5: (1.3, 3.5)
Finally, let's evaluate the point (1.3, 3.5). Plugging x = 1.3 into our equation, we get y = 1.5 + [1.3]. The floor of 1.3 is 1. So, y = 1.5 + 1 = 2.5. The y-coordinate we calculated (2.5) does not match the y-coordinate of the point (3.5). Therefore, the point (1.3, 3.5) is not on the graph.
This instance further exemplifies how the floor function determines the integer base for the y-value calculation. The floor of 1.3 is 1, and the addition of 1.5 results in 2.5, a value that is distinct from the point's y-coordinate of 3.5. This difference underscores the importance of precise floor function calculation and its direct impact on the resulting y-values. The clear mismatch in coordinates confirms that this point does not belong to the graph of the given equation.
Conclusion: Identifying the Points on the Graph
Alright, guys, we've done the math! By plugging in the x-coordinates of each point into the equation y = 1.5 + [x] and comparing the resulting y-values with the given y-coordinates, we've determined which points lie on the graph. Let's recap our findings:
- (-4.5, -2.5): Not on the graph
- (-0.8, 0.5): On the graph
- (7.9, 9.5): Not on the graph
- (4.5, 6): Not on the graph
- (1.3, 3.5): Not on the graph
So, the only point from the given options that sits on the graph of y = 1.5 + [x] is (-0.8, 0.5). Understanding the floor function is key to tackling these types of problems. Remember, it's all about finding the greatest integer less than or equal to x, and then applying that to the equation. Keep practicing, and you'll be a pro in no time!