Understanding F(x) = 100/x - 5 Graph Transformations

Let's dive into understanding how the function f(x) = 100/x - 5 behaves and how its graph transforms. This function is super practical, especially when you're dealing with filling a tank, like a fish tank, and want to know how much time is left. So, stick around, and we'll break it down together!

The Parent Function: f(x) = 1/x

Before we get into the specifics of f(x) = 100/x - 5, let's quickly chat about its parent function, which is f(x) = 1/x. Understanding this basic reciprocal function is key to grasping the transformations. The graph of f(x) = 1/x is a hyperbola, featuring two branches that sit in the first and third quadrants. As x gets larger, f(x) gets closer to zero, creating a horizontal asymptote at y = 0. Similarly, as x approaches zero, f(x) shoots off to infinity, giving us a vertical asymptote at x = 0. This parent function is a cornerstone for understanding more complex reciprocal functions.

Key characteristics of f(x) = 1/x:

• Asymptotes: Vertical asymptote at x = 0 and horizontal asymptote at y = 0.
• Symmetry: Symmetric about the origin.
• Branches: Two branches in the first and third quadrants.

Knowing this parent function helps us understand how transformations affect the graph. When we start adding, subtracting, multiplying, or dividing, we shift, stretch, or reflect this basic shape. So, with that in mind, let's move on to how f(x) = 100/x - 5 builds upon this foundation.

Understanding f(x) = 100/x - 5

Alright, let's break down f(x) = 100/x - 5 piece by piece. This function tells us the remaining time needed to fill a 100-gallon tank, given that it's filling at a rate of x gallons per minute and has already been filling for 5 minutes. The 100/x part calculates the total time to fill the tank, and the - 5 subtracts the 5 minutes that have already passed. So, f(x) gives us the remaining time.

Vertical Stretch: The Role of 100

The first transformation to consider is the multiplication by 100. In the function f(x) = 100/x, the 100 acts as a vertical stretch factor. This means that every y-value on the graph of f(x) = 1/x is multiplied by 100. So, instead of the graph getting close to zero as x increases, it gets close to zero much more slowly. Think of it like stretching the graph vertically, making it taller. For example, when x = 1, in the parent function f(x) = 1/x, f(1) = 1. But in f(x) = 100/x, f(1) = 100. That's a huge difference! This vertical stretch significantly impacts the shape and scale of the graph.

Vertical Shift: The Impact of -5

Next up, we have the - 5 in f(x) = 100/x - 5. This is a vertical shift. It moves the entire graph down by 5 units. So, every point on the graph is shifted down 5 units. The horizontal asymptote, which was at y = 0 for f(x) = 100/x, is now at y = -5. This shift is crucial because it represents the initial 5 minutes that the tank has already been filling. It changes the context of the graph, showing the remaining time relative to this initial period. Because of the vertical shift down, all the values of f(x) are reduced by 5, which means the y-axis has been translated to the negative direction.

Asymptotes: Revisited

Let's revisit those asymptotes. For f(x) = 100/x - 5, the vertical asymptote remains at x = 0. This is because dividing by zero is still undefined. However, the horizontal asymptote shifts down to y = -5 due to the vertical shift. As x gets very large, 100/x approaches zero, and f(x) approaches -5. These asymptotes are important guides when sketching the graph.

Graphing the Function

To graph f(x) = 100/x - 5, start with the parent function f(x) = 1/x. Then, apply the vertical stretch by a factor of 100 and the vertical shift down by 5 units. Plot a few key points to get a sense of the shape. Remember that the graph will have a vertical asymptote at x = 0 and a horizontal asymptote at y = -5. The graph will exist in the first and third quadrants relative to the new origin at (0, -5).

How Changing 'x' Affects the Graph

Now, let's explore how changing the value of x affects the graph and what it practically means in our tank-filling scenario. Remember, x represents the rate at which the tank is filling in gallons per minute.

Increasing 'x'

If we increase x (the filling rate), the value of 100/x decreases. This means the total time to fill the tank decreases. Since f(x) = 100/x - 5, the remaining time f(x) also decreases. On the graph, this corresponds to moving to the right along the curve in the first quadrant, closer to the horizontal asymptote y = -5. Practically, this means that if the tank fills faster, the remaining time needed to fill it gets shorter.

Decreasing 'x'

Conversely, if we decrease x (the filling rate), the value of 100/x increases. This means the total time to fill the tank increases. Consequently, the remaining time f(x) also increases. On the graph, this corresponds to moving to the left along the curve in the first quadrant, further away from the horizontal asymptote. In practical terms, if the tank fills slower, the remaining time needed to fill it becomes longer.

Domain and Range Considerations

In the context of the problem, x must be greater than zero because we can't have a negative or zero filling rate. This restricts the domain of the function to x > 0. Also, f(x) must be greater than zero, otherwise we are dealing with negative time, which is meaningless. Therefore, 100/x - 5 > 0, which simplifies to x < 20. Since x must be both greater than 0 and less than 20, it restricts the domain of the function to 0 < x < 20. As x approaches 0, f(x) approaches infinity, meaning it would take infinitely long to fill the tank. As x approaches 20, f(x) approaches zero, meaning the tank is almost full. The horizontal asymptote at y = -5 is never reached because f(x) is always greater than zero in our context. The range of the function, given these restrictions, is f(x) > 0.

Real-World Implications

Understanding these transformations isn't just about math; it's about applying these concepts to real-world situations. Imagine you're managing a water tank for a community. Knowing how the filling rate affects the remaining time can help you plan water usage efficiently. Or, if you're setting up a complex irrigation system, this function can assist in predicting how long it will take to fill reservoirs based on pump rates.

Moreover, this type of analysis extends beyond just filling tanks. It applies to various scenarios involving rates and time, such as production lines, data processing speeds, and even project management. The key is to identify the underlying relationship between variables and use mathematical functions to model and predict outcomes.

Conclusion

So, there you have it! The function f(x) = 100/x - 5 is a fantastic example of how transformations affect the graph of a reciprocal function. By understanding the vertical stretch and vertical shift, you can easily sketch the graph and interpret its meaning in a real-world context. Whether you're filling a fish tank or managing a larger project, these mathematical principles provide valuable insights. Keep exploring, and you'll find that math is incredibly useful in everyday life! Keep up the great work, and happy graphing!