Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of \(f(x)=1 / x\) with the graph of \(g\). $$g(x)=f(x)-2=\frac{1}{x}-2$$

Step-by-Step Solution

Answer

The graph of \(g(x)=\frac{1}{x}-2\) is the same as the graph of \(f(x)=\frac{1}{x}\) but shifted 2 units downward.

Step 1: Identify base function

The base function here is \(f(x) = \frac{1}{x}\). This is a simple reciprocal function, and its graph is a hyperbola with the x and y axis as asymptotes.

Step 2: Identify transformation

The transformation in the function \(g(x)=\frac{1}{x}-2\) involves subtracting 2 from the function \(f(x)\). This means that every y-coordinate in the graph of \(f(x)\) is translated 2 units down to obtain the graph of \(g(x)\).

Step 3: Conclusion

Therefore, the graph of the function \(g(x)=\frac{1}{x}-2\) is exactly like the graph of \(f(x)=\frac{1}{x}\), but shifted down 2 units. This means that the asymptote of the function \(g(x)\) is now the line \(y=-2\), not the x-axis.

Key Concepts

Understanding Reciprocal Functions

At the heart of our discussion is a particular type of function known as a reciprocal function, represented typically as \(f(x) = \frac{1}{x}\). When you visualize this function on a graph, it creates a shape known as a hyperbola. The defining characteristic of the reciprocal function's graph is that it approaches but never touches either the x-axis or the y-axis, illustrating a relationship where as x gets larger, f(x) gets smaller, and as x approaches zero, f(x) grows without bound.

As a foundational concept in algebra and calculus, reciprocal functions are an excellent example of rational functions, ones made up of two polynomials where the denominator is not equal to zero. This non-zero denominator is vital as it avoids division by zero, which is undefined in mathematics. Understanding this function is crucial because its behavior under transformation can help with grasping more complex concepts such as limits, continuity, and asymptotic behavior in calculus.

Exploring Vertical Shifts

A vertical shift in graph transformations is a straightforward but fundamental concept. It occurs when we add or subtract a constant value to a function's output. To grasp this, consider our function \(f(x) = \frac{1}{x}\) and its transformation to become \(g(x) = f(x) - 2\). What we're effectively doing is moving the entire graph of \(f(x)\) down by 2 units. Each point (x, y) on the graph of the original function has its y-value decreased by 2, resulting in a new point (x, y-2) on the graph of \(g(x)\).

Recognizing and performing vertical shifts allows us to manipulate and control graphs to match particular constraints or conditions. This skill is particularly useful when trying to infer or illustrate the effect of changes in real-world phenomena within a mathematical context.

Asymptotes Decoded

Lastly, let's delve into the concept of asymptotes. An asymptote is a line that a graph approaches but never actually reaches or touches. In the context of reciprocal functions like \(f(x) = \frac{1}{x}\), we have two asymptotes: one horizontal (the x-axis) and one vertical (the y-axis). These asymptotes define the boundaries of the graph's behavior but are never included in the graph itself.

When we applied a transformation to create \(g(x) = \frac{1}{x} - 2\), it introduced a vertical shift, which also affected the position of the horizontal asymptote. Originally at y=0 for the function \(f(x)\), the horizontal asymptote for \(g(x)\) has moved down to y=-2. It's essential to understand that vertical shifts do not alter the nature of the relationship between the variables – they simply relocate the entire graph up or down along the y-axis, resulting in a corresponding shift in the horizontal asymptote.