Graphing Rational Functions

Recapping the Key Features

As a quick reminder, the features we learned to identify in the last lesson are:

• Domain restrictions: Values of $x$ that make the denominator zero.
• Discontinuities:
  • Vertical Asymptotes: From factors remaining in the denominator after simplification.
  • Holes: From factors that cancel out.
• End Behaviour:
  • Horizontal Asymptotes: When the degree of the numerator is less than or equal to the degree of the denominator.
  • Slant (Oblique) Asymptotes: When the degree of the numerator is exactly one greater than the degree of the denominator.
• Intercepts:
  • y-intercept: The value of $f(0)$.
  • x-intercepts (Roots): The roots of the simplified numerator.

Having these features is essential, but there's one more tool we need to understand how to connect them all into a graph.

The Missing Piece: Sign Analysis

Knowing where the graph crosses the x-axis (at the roots) and where it has vertical asymptotes is great, but what does the graph do between these points? Does it lie above the x-axis (positive) or below it (negative)?

To answer this, we use a sign analysis (or sign diagram). The process is straightforward:

1. Draw a number line.
2. Mark all the function's x-intercepts (roots) and vertical asymptotes. These are the only places where the function can change its sign from positive to negative or vice versa.
3. These marks divide the number line into intervals.
4. Pick a test value within each interval and substitute it into the simplified rational function.
5. All you need to determine is whether the result is positive (+) or negative (-). This sign applies to the entire interval.

This technique is crucial because it tells us which side of the x-axis the graph lives on in each region, and how it behaves around its vertical asymptotes.

A Step-by-Step Guide to Graphing Rational Functions

We can now establish a systematic procedure for sketching any rational function. Following these steps will ensure you don't miss any details.

Here is our version of that process, combining all the concepts we've learned:

Step 1: Analyse the Function
Find all the key features you learned about in the last lesson:

• Factorise the numerator and denominator.
• Find any holes and their coordinates.
• Find the vertical asymptotes.
• Find the horizontal or slant asymptote.
• Find the y-intercept by calculating $f(0)$.
• Find the x-intercepts (roots) by setting the simplified numerator to zero.

Step 2: Set up the Coordinate Plane

• Draw your x and y axes.
• Draw all asymptotes as dashed lines and label them with their equations (e.g., $x=2$, $y=1$).
• Plot the x-intercepts and the y-intercept as solid points.
• Plot any holes as open circles.

Step 3: Perform a Sign Analysis

• Create a number line and mark the x-intercepts and vertical asymptotes.
• Test a value in each interval to determine if the function is positive (+) or negative (-) in that region.

Step 4: Sketch the Graph

• Start from the ends: Use the horizontal or slant asymptote as a guide for the graph's behaviour as $x\to \infty$ and $x\to -\infty$.
• Connect the points: Draw a smooth curve through the plotted intercepts.
• Respect the signs: Ensure your sketch is above the x-axis in intervals marked (+) and below it in intervals marked (-).
• Approach the asymptotes: The sign analysis will tell you whether the graph shoots up to $+\infty$ or down to $-\infty$ as it approaches a vertical asymptote from the left or right. The function should get closer and closer to the vertical asymptotes but never touch them.

Example 1: Function with a Horizontal Asymptote

Let's see this process in action. This video provides an excellent, clear walkthrough of our step-by-step method.

The key insight from the video is how to use test points to determine whether the graph approaches $+\infty$ or $-\infty$ on either side of a vertical asymptote. This is a direct application of our sign analysis.

Example 2: Function with a Hole

Now let's consider a function that has a hole. The process is nearly identical, but we must remember to draw the hole as an open circle on the final graph.

Let's watch the second example from the same video, which deals with a hole.

Example 3: Function with a Slant Asymptote

Our final case is when the degree of the numerator is one greater than the denominator, resulting in a slant asymptote. The only change to our process is that instead of a horizontal line, our "guide rail" for the end behaviour is a diagonal line. You find the equation of this line, $y=mx+c$, using polynomial long division.

Let's watch the final example from the video.

Note that it is possible for a graph to cross its horizontal or slant asymptote, especially in the "middle" of the graph. The asymptote primarily describes the end behaviour as $x$ becomes very large or very small.

Conclusion

You have now learned how to take a rational function from its algebraic form to a complete graphical sketch. The process is systematic and relies on combining the analysis from our last lesson with the new tool of sign analysis.

Key Takeaways:

• The 4-Step Process:
  1. Analyse: Find all intercepts, asymptotes, and holes.
  2. Set up: Draw axes and plot all the features you found.
  3. Sign Analysis: Create a sign chart using roots and vertical asymptotes to determine where the function is positive or negative.
  4. Sketch: Draw the curve, respecting all points, signs, and asymptotic behaviour.
• Sign analysis is crucial: It dictates the graph's behaviour between and around the key points.
• The type of asymptote (horizontal, slant) only changes the "guide rails" for the end behaviour; the rest of the process is the same.

Preview of the Next Lesson:
The sign analysis we used today is a powerful tool in its own right. In the next lesson, we will focus on it directly to solve our final topic in this module: solving polynomial and rational inequalities. You'll find that you've already learned the most important parts of the technique