Problem 46 - College Algebra with Modeling and Visualization

Question

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x+2} $$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is \(f(x+2)\), shifted left by 2 units, with asymptotes at \(x=-2\) and \(y=0\).

Step 1: Identify the Base Function

The given function is \(g(x) = \frac{1}{x+2}\). This resembles the function \(f(x) = \frac{1}{x}\). Therefore, we'll use \(f(x)\) as our base function.

Step 2: Determine the Transformation

The function \(g(x) = \frac{1}{x+2}\) can be rewritten in terms of \(f(x)\) by identifying the transformation as a horizontal shift. \(g(x)\) can be expressed as \(f(x+2)\). This represents a shift of the graph of \(f(x)\) left by 2 units.

Step 3: Analyze the Asymptotes

For \(f(x) = \frac{1}{x}\), the vertical asymptote is at \(x = 0\). After shifting 2 units left, for \(g(x)\), the vertical asymptote moves to \(x = -2\). There is no horizontal asymptote shifting, so it remains at \(y = 0\).

Step 4: Sketch the Graph

Draw the axes and plot the vertical asymptote at \(x = -2\) and the horizontal asymptote at \(y = 0\). Sketch the hyperbola that approaches these asymptotes as \(x\) moves away from \(-2\) and 0.

Key Concepts

Rational Functions

Rational functions are a special type of function represented by the ratio of two polynomials. The simplest form of a rational function is \( f(x) = \frac{1}{x} \), where the numerator is 1 and the denominator is the polynomial \( x \).
This type of function inherently has certain "rules" it follows, like undefined points, which are points where the function does not have a real number output.
In our example, we're considering \( g(x) = \frac{1}{x+2} \), clearly a transformation of our base rational function \( f(x) = \frac{1}{x} \).

• Mainly, rational functions have a classic hyperbola shape when graphed.
• They tend to have very specific transformations influenced by their equations, which can help when sketching.

Understanding rational functions is crucial in math because they model situations where one quantity depends inversely on another, frequently appearing in real-world scenarios, like physics and economics.

Asymptotes

Asymptotes are like invisible boundary lines that guide the behavior of a graph.
In the context of rational functions, these are lines that the graph approaches but never really "touches".
There are typically two types we care about:

• Vertical asymptotes: Occur where the denominator of the rational function becomes zero, meaning the function itself is undefined at these points. For \( f(x) = \frac{1}{x} \), the vertical asymptote is at \( x = 0 \), but when transformed to \( g(x) = \frac{1}{x+2} \), the vertical asymptote shifts to \( x = -2 \).
• Horizontal asymptotes: These represent the value that a function approaches as \( x \) becomes very large or very small. In both \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x+2} \), there is a horizontal asymptote at \( y = 0 \).

These asymptotes are crucial for sketching the behavior of rational functions as they expand towards infinity or decrease towards negative infinity.

Horizontal Shifts

A horizontal shift in graph transformations involves moving the graph of a function left or right along the x-axis without affecting its shape.
For an original function \( f(x) \), a transformation written as \( f(x + c) \) would imply a leftward shift by \( c \) units, where \( c \) is a positive number.
In our example, transforming \( f(x) = \frac{1}{x} \) to \( g(x) = \frac{1}{x+2} \) represents a shift to the left by 2 units.
This is because the \( x \) in the denominator is replaced with \( x+2 \), indicating each point on the graph moves leftwards.

• Horizontal shifts are straightforward to identify as they always affect the function indirectly through the input variable, \( x \).
• They help us understand how individual points of the function adjust across the Cartesian plane without changing the function's general structure or orientation.

Grasping these shifts allows us to predict and sketch transformed functions effortlessly.