Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f(x)=2 / x\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=f(x)+1$$

Step-by-Step Solution

Answer

The function \(g(x)\) is exactly the same as function \(f(x)\) but shifted vertically upward by 1 unit on the graph.

Step 1: Understand the Functions

In this case, we have two functions. The first one is \(f(x) = 2/x\), which is a rational function. This will be a hyperbola centered in the origin. The second function \(g(x) = f(x) + 1\), which is the same as \(2/x + 1\). By adding 1 to the first function, we essentially move the graph of \(f(x)\) vertically upward by one unit.

Step 2: Graph the Functions

Using a graphing utility, plot both functions \(f(x) = 2/x\) and \(g(x) = 2/x + 1\). The common graphical representation will allow visualizing the relationship between the two functions.

Step 3: Compare the Functions

The graph of the function \(g(x)\) should appear identical to the graph of the function \(f(x)\), but it will be shifted vertically upward by one unit. This is because the function \(g(x) = f(x) + 1\), meaning every output value of \(f(x)\) is increased by 1 to create \(g(x)\).

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help us visualize mathematical functions, including rational functions such as the one in this exercise. They offer a graphical representation, making it easier to understand complex relationships between different parts of functions. Graphing utilities can be software programs on computers, apps on tablets, or calculators that allow plotting of many types of functions. They are particularly helpful in visualizing changes and shifts in a graph rapidly without manual plotting.

• You can enter functions directly to see their graphs.
• These tools allow for zooming in and out to better understand different parts of the graph.
• They also provide options to shift graphs or change variables dynamically to see immediate effects.

In this exercise, using a graphing utility helps by plotting both functions, \(f(x) = \frac{2}{x}\) and \(g(x) = \frac{2}{x} + 1\), in the same window. This can help visually compare their differences, especially when functions are shifted vertically.

Vertical Shifts

Vertical shifts refer to moving a graph up or down in the coordinate plane. This occurs when we add or subtract a constant from a function. Therefore, the graph of the function itself changes its vertical position relative to the x-axis.For example, given a function \(f(x) = \frac{2}{x}\), the adjustment \(g(x) = f(x) + 1\) results in the graph being moved up by one unit.

• Adding a positive constant moves the graph upwards.
• Subtracting a constant moves the graph downwards.

Why does this happen? Because each y-value of the function is increased by the constant amount. So, if you add 1 to \(f(x)\), every corresponding y-value of \(f(x)\) is incremented by 1, resulting in an upward shift. Vertical shifts do not affect the shape of the graph; they only adjust the graph's position along the y-axis.

Hyperbolas

A rational function, such as \(f(x) = \frac{2}{x}\), often forms a hyperbola when graphed. Hyperbolas are curves that look like two opposing, mirrored arches. They have distinct characteristics such as two separate branches, and they approach the axes asymptotically.

• The center of a hyperbola is the point from which it appears to expand, not to be confused with the origin unless specified.
• Hyperbolas have vertical and horizontal asymptotes, lines that the graph gets infinitely close to but never touches.

In our example, the hyperbola stems from dividing the constant 2 by \(x\). Such functions tend to show undefined behavior at \(x = 0\) due to division by zero. Thus, graphing \(f(x) = \frac{2}{x}\) produces a curve with branches that get closer to the x-axis and y-axis but never touch them, depicting typical hyperbolic behavior.