Graph Of F(x) = (4x^2 - 16) / (2x - 4) Explained
In this article, we will delve into a comprehensive analysis of the function f(x) = (4x^2 - 16) / (2x - 4), aiming to identify the graph that accurately represents it. This exploration involves simplifying the function, identifying any points of discontinuity, and understanding its overall behavior. By carefully examining these aspects, we can confidently determine the correct graphical representation of the function.
Simplifying the Function
The first step in understanding the graph of a rational function like f(x) = (4x^2 - 16) / (2x - 4) is to simplify it. Simplification often reveals hidden characteristics of the function, such as its true form and any potential discontinuities. To simplify this function, we begin by factoring both the numerator and the denominator.
The numerator, 4x^2 - 16, is a difference of squares and can be factored as follows:
4x^2 - 16 = 4(x^2 - 4) = 4(x - 2)(x + 2)
The denominator, 2x - 4, can be factored by taking out the common factor of 2:
2x - 4 = 2(x - 2)
Now, we can rewrite the function with the factored numerator and denominator:
f(x) = [4(x - 2)(x + 2)] / [2(x - 2)]
We can simplify further by canceling the common factor of (x - 2) from the numerator and denominator. However, it's crucial to note that canceling this factor introduces a point of discontinuity, which we will address later. After canceling the common factor, we have:
f(x) = 2(x + 2), for x ≠2
This simplified form, f(x) = 2(x + 2), represents a linear function. A linear function generally graphs as a straight line. However, the condition x ≠2 is critical. It indicates that there is a hole or a removable discontinuity in the graph at x = 2. This means the graph will look like a straight line, but there will be a gap at the point where x = 2.
In summary, simplifying the function reveals that it is essentially a linear function, but with a crucial exception: a point of discontinuity at x = 2. This understanding is essential for accurately identifying the graph of the function. We now know to look for a straight line with a hole at x = 2, which will guide us in selecting the correct graph from the given options. The simplification process not only makes the function easier to visualize but also highlights the importance of considering domain restrictions when graphing functions.
Identifying Points of Discontinuity
As we've seen, identifying points of discontinuity is a critical step in accurately graphing a rational function. In the case of f(x) = (4x^2 - 16) / (2x - 4), the simplification process revealed a removable discontinuity, often referred to as a hole, in the graph. To fully understand this discontinuity, let's delve deeper into how it arises and its implications for the graph.
The discontinuity occurs because we canceled the factor (x - 2) from both the numerator and the denominator during simplification. While canceling this factor simplifies the function algebraically, it's essential to remember that the original function is undefined when the denominator is zero. In this case, the denominator 2x - 4 equals zero when x = 2. This means that the original function f(x) = (4x^2 - 16) / (2x - 4) is undefined at x = 2.
However, after canceling the (x - 2) factor, we obtained the simplified form f(x) = 2(x + 2). This linear function is defined at x = 2. This discrepancy is what creates the removable discontinuity. The simplified form represents the function everywhere except at the point where the original function was undefined.
To find the coordinates of the hole, we substitute x = 2 into the simplified function:
f(2) = 2(2 + 2) = 2(4) = 8
Thus, the hole is located at the point (2, 8). This means that on the graph, there will be a gap or an open circle at the point (2, 8), indicating that the function is not defined at that specific point.
Understanding the concept of removable discontinuities is crucial in graphing rational functions accurately. It's not enough to simply simplify the function; we must also account for any values of x that make the original denominator zero. These values represent points where the function is undefined, and they manifest as holes or vertical asymptotes in the graph. In this case, we've identified a hole at (2, 8), which will be a key feature in distinguishing the correct graph from other options. The presence of this hole demonstrates that while the function behaves linearly for most values of x, it has a specific point where its behavior is interrupted.
Understanding the Overall Behavior
Having simplified the function f(x) = (4x^2 - 16) / (2x - 4) to f(x) = 2(x + 2) (for x ≠2) and identified the point of discontinuity at (2, 8), we now need to understand the overall behavior of the function to accurately represent its graph. This involves analyzing the simplified linear form and considering the impact of the discontinuity on the graph's appearance. The overall behavior of the function is key to visualizing it.
The simplified form, f(x) = 2(x + 2), is a linear equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, m = 2 and b = 4 (since 2(x + 2) = 2x + 4). This tells us that the graph is a straight line with a positive slope of 2, meaning it rises from left to right. The y-intercept is 4, indicating that the line crosses the y-axis at the point (0, 4).
Now, let's consider the impact of the discontinuity at (2, 8). As we established earlier, this point is a hole in the graph. The line will exist everywhere except at this specific point. To visualize this, imagine drawing a straight line with a slope of 2 and a y-intercept of 4. Then, at the point where x = 2, you would erase a tiny circle, creating a hole. This hole signifies that the function is not defined at x = 2, even though the rest of the line is continuous.
The behavior of the function as x approaches 2 from both sides is also crucial to understand. As x approaches 2 from values less than 2, the function values approach 8. Similarly, as x approaches 2 from values greater than 2, the function values also approach 8. This is consistent with the linear nature of the simplified function. However, the function never actually reaches 8 at x = 2; it gets infinitely close but remains undefined at that precise point.
In summary, the overall behavior of f(x) = (4x^2 - 16) / (2x - 4) can be described as a straight line with a slope of 2 and a y-intercept of 4, but with a hole at the point (2, 8). This understanding allows us to confidently identify the correct graph: a straight line with the specified slope and intercept, interrupted by a single point of discontinuity at (2, 8). The correct graph will visually represent this combination of linear behavior and a removable discontinuity.
Identifying the Correct Graph
Based on our comprehensive analysis, we now have a clear picture of what the graph of f(x) = (4x^2 - 16) / (2x - 4) should look like. We've established that the function, after simplification, is essentially a linear function f(x) = 2(x + 2), which represents a straight line with a slope of 2 and a y-intercept of 4. However, we've also identified a crucial detail: a removable discontinuity, or a hole, at the point (2, 8). Therefore, identifying the correct graph involves looking for these key features.
When presented with multiple graph options, we can systematically evaluate each one against our findings:
- Look for a Straight Line: The primary shape of the graph should be a straight line, reflecting the linear nature of the simplified function. Graphs that show curves, parabolas, or other non-linear shapes can be immediately ruled out.
- Check the Slope and y-intercept: The line should have a positive slope of 2, meaning it should rise as you move from left to right. Also, the line should intersect the y-axis at the point (0, 4). Verify that the graph meets these criteria.
- Identify the Hole: The most critical feature to look for is the hole at (2, 8). The graph should appear to be a continuous line, but there should be an open circle or a gap at this specific point. This indicates the removable discontinuity.
- Eliminate Incorrect Options: Graphs that do not exhibit all these characteristics can be eliminated. For instance, a graph that shows a continuous line without a hole at (2, 8) is incorrect. Similarly, a graph with a different slope or y-intercept, or with a vertical asymptote instead of a hole, is not the correct representation of the function.
By carefully comparing each graph option to these criteria, we can confidently select the one that accurately represents f(x) = (4x^2 - 16) / (2x - 4). The correct graph will be a straight line with a slope of 2, a y-intercept of 4, and a hole at the point (2, 8). This systematic approach ensures that we don't overlook any critical details and that we choose the most accurate representation of the function.
In conclusion, determining the graph of f(x) = (4x^2 - 16) / (2x - 4) involves simplifying the function, identifying points of discontinuity, understanding the overall behavior, and carefully comparing graph options. By following these steps, we can confidently select the graph that accurately represents the function.
