Problem 40 - Algebra for College Students

Question

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{2}{x^{2}-4 x+4}$$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 2 \), horizontal asymptote at \( y = 0 \), and y-intercept at \( (0, \frac{1}{2}) \).

Step 1: Factor the denominator

Factor the quadratic in the denominator: \( x^2 - 4x + 4 = (x-2)^2 \) So, the function becomes: \( f(x) = \frac{2}{(x-2)^2} \)

Step 2: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero. Set \((x-2)^2 = 0\): \( x - 2 = 0 \) Thus, there is a vertical asymptote at \( x = 2 \).

Step 3: Identify horizontal asymptotes

Since the degree of the polynomial in the denominator \( (x-2)^2 \) is higher than the degree of the numerator (which is 0), the horizontal asymptote is at \( y = 0 \).

Step 4: Find x-intercepts

X-intercepts occur where the numerator is zero. Since the numerator is 2, \( f(x) \) has no x-intercepts.

Step 5: Find y-intercepts

Y-intercepts occur when \( x = 0 \). Thus, \( f(0) = \frac{2}{(0-2)^2} = \frac{2}{4} = \frac{1}{2} \). So, the y-intercept is at \( (0, \frac{1}{2}) \).

Step 6: Sketch the graph

Draw the vertical asymptote at \( x = 2 \), the horizontal asymptote at \( y = 0 \), and plot the y-intercept at \( (0, \frac{1}{2}) \). The graph of the function approaches these asymptotes without crossing them.

Key Concepts

Asymptotes

A rational function can have vertical and horizontal asymptotes, which are lines that the graph approaches but never actually touches. The given function is \( f(x) = \frac{2}{(x-2)^2} \). Here’s a deeper look into asymptotes:

• Vertical Asymptotes: These occur where the denominator of the function is zero and the numerator is non-zero. In our function, \( (x-2)^2 = 0 \) at \( x = 2 \). So, the vertical asymptote is at \( x = 2 \).
• Horizontal Asymptotes: They occur based on the degrees of the numerator and denominator. If the degree of the denominator is higher, the horizontal asymptote is at \( y = 0 \). In our case, the numerator is a constant (degree 0) and the denominator is a quadratic (degree 2). Therefore, \( y = 0 \) is the horizontal asymptote.

Asymptotes play a crucial role in understanding the behavior of the graph away from the origin.

Intercepts

Intercepts are points where the graph crosses the x or y-axis. For the function \( f(x) = \frac{2}{(x-2)^2} \), let’s identify both types of intercepts:

• X-intercepts: These occur where the function equals zero. Since our numerator is a constant 2 and never zero, there are no x-intercepts for this function.
• Y-intercepts: These occur where the function crosses the y-axis, which happens when \( x = 0 \). Substituting \( x = 0 \) into the function, we get \( f(0) = \frac{2}{(0-2)^2} = \frac{2}{4} = \frac{1}{2} \). So, the y-intercept is at \( (0, \frac{1}{2}) \).

Intercepts give us exact points that are easy to plot and provide a starting point for sketching the graph.

Graphing Functions

Graphing rational functions involves several steps to ensure accuracy:

• Sketch Asymptotes: Draw the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 0 \). These lines will guide the overall shape of the graph.
• Plot Intercepts: Mark the y-intercept at \( (0, \frac{1}{2}) \). Since there are no x-intercepts, there is no need to worry about points where the function crosses the x-axis.
• Draw the Graph: The function approaches the asymptotes without crossing them. Starting from the intercept, sketch the curve making sure it gets closer and closer to the asymptotes as it moves away from the intercept but never actually touches these lines.

Graphing gives us a visual representation of the behavior and key characteristics of the function.

Quadratic Factoring

Factoring quadratic expressions is an essential skill in handling rational functions. For our function \( f(x) = \frac{2}{x^2 - 4x + 4} \), the first step involves factoring the denominator:

• Identify Quadratic Form: Recognize that \( x^2 - 4x + 4 \) is a perfect square. This helps to simplify the quadratic.
• Factor Quadratic: Write the quadratic as \( (x-2)^2 \). This form shows the repeating nature of the root and simplifies further steps.
• Simplify the Function: After factoring, the rational function becomes \( f(x) = \frac{2}{(x-2)^2} \). This simplified form makes it easier to identify asymptotes and intercepts.

Factoring simplifies the work with rational functions and reveals the underlying structure crucial for graphing and analysis.