Problem 26 - Algebra 2

Question

Graph each rational function. $$ f(x)=\frac{x-1}{x-3} $$

Step-by-Step Solution

Answer

Vertical asymptote at \( x=3 \), horizontal asymptote at \( y=1 \). Intercepts: \( (1,0) \) and \( \left(0, \frac{1}{3}\right) \).

Step 1: Identify the Function Components

The given function is \( f(x) = \frac{x-1}{x-3} \). The numerator is \( x-1 \), and the denominator is \( x-3 \). This is a rational function of the form \( \frac{a}{b} \).

Step 2: Determine the Vertical Asymptote

Vertical asymptotes occur where the denominator is zero. Set \( x-3 = 0 \). Solving for \( x \), we get \( x = 3 \). So, there is a vertical asymptote at \( x = 3 \).

Step 3: Determine the Horizontal Asymptote

For rational functions \( \frac{a(x)}{b(x)} \), if the degrees of \( a(x) \) and \( b(x) \) are equal, the horizontal asymptote is \( y = \frac{leading \: coefficient \: of \: a(x)}{leading \: coefficient \: of \: b(x)} \). In our function, both numerator and denominator have degree 1 and leading coefficients of 1, giving the asymptote at \( y = \frac{1}{1} = 1 \).

Step 4: Find the X-Intercept

The x-intercept occurs where the function is zero, i.e., \( f(x) = 0 \). Set the numerator \( x-1 = 0 \), giving \( x=1 \). Hence, the x-intercept is at \( (1, 0) \).

Step 5: Find the Y-Intercept

The y-intercept occurs where \( x = 0 \). Substituting into the function: \( f(0) = \frac{0-1}{0-3} = \frac{-1}{-3} = \frac{1}{3} \). Therefore, the y-intercept is at \( \left(0, \frac{1}{3}\right) \).

Step 6: Sketch the Graph

Using the determined features, sketch the graph. Plot the asymptotes at \( x = 3 \) (vertical) and \( y = 1 \) (horizontal). Plot the intercepts at \( (1, 0) \) and \( \left(0, \frac{1}{3}\right) \). The graph will approach the asymptotes and pass through the intercepts.

Key Concepts

Vertical Asymptote

A vertical asymptote in a rational function is a line that the graph of the function approaches but never touches or crosses. It happens due to the denominator of the fraction becoming zero. In the function given:

• The denominator is \(x-3\).
• To find the vertical asymptote, set \(x-3=0\) and solve for \(x\).
• This gives \(x=3\).

This means the graph will approach the line \(x=3\) closely, but it cannot actually touch or cross it. This is because division by zero is undefined in mathematics. To visualize it, picture the graph getting steeper as it approaches \(x=3\), both from the left and right.

Horizontal Asymptote

The horizontal asymptote of a rational function describes the behavior of the graph as \(x\) approaches infinity or negative infinity. It's like a guiding line where the function levels off and helps to predict the end behavior of the graph. When the degrees of the polynomial in the numerator and the denominator are equal, you find the horizontal asymptote by looking at the leading coefficients. In our function:

• Both numerator \(x-1\) and denominator \(x-3\) have degree 1.
• The leading coefficients are both 1.
• The horizontal asymptote is \(y = \frac{1}{1} = 1\).

As \(x\) goes towards positive or negative infinity, the graph gets closer and closer to the line \(y=1\), without actually reaching it. It's like a boundary line that frames the top and bottom of the graph's endless path.

X-Intercept

The x-intercept of a graph is the point where the function crosses the x-axis. At this point, the value of \(f(x)\), or the output of the function, is zero. This happens when the numerator of the rational function equals zero. In the given function:

• The numerator is \(x-1\).
• Set \(x-1=0\) to find the x-intercept.
• This results in \(x=1\).

So the graph crosses the x-axis at the point \((1, 0)\). This means when you plug \(x = 1\) into the function, the result is zero, confirming the location of the x-intercept on the graph.

Y-Intercept

The y-intercept is where the graph crosses the y-axis, meaning \(x\) is zero at this point. To find the y-intercept of a rational function, simply substitute \(x = 0\) into the equation. For the function \(f(x) = \frac{x-1}{x-3}\):

• Substitute \(x=0\) to find the y-value: \(f(0) = \frac{0-1}{0-3} = \frac{-1}{-3} = \frac{1}{3}\).

Thus, the graph intersects the y-axis at \(\left(0, \frac{1}{3}\right)\). This gives a precise point where the graph meets the y-axis, providing a starting reference point when plotting the graph.