Grade 11 Math's Unit 2 Part 3 Graph of Rational functions | New Curriculum

Introduction

This tutorial focuses on graphing rational functions, as covered in Grade 11 Math's Unit 2. Understanding how to graph these functions is vital for solving complex mathematical problems and applying these concepts in real-world scenarios. The guide will walk you through key concepts and steps involved in graphing rational functions, focusing on crucial features like intercepts, asymptotes, and behavior at infinity.

Step 1: Understand Rational Functions

Rational functions are expressed as the ratio of two polynomials. The general form is:

f(x) = P(x) / Q(x)

Where:

• P(x) is the numerator polynomial.
• Q(x) is the denominator polynomial.

Practical Tips

• Ensure that Q(x) is not equal to zero; otherwise, the function is undefined at those points.
• Familiarize yourself with polynomial long division if the degree of P(x) is greater than or equal to Q(x).

Step 2: Identify Intercepts

Intercepts are crucial for sketching the graph.

Finding the x-intercepts

• Set the numerator P(x) equal to zero and solve for x.
• Example: If P(x) = x^2 - 4, set x^2 - 4 = 0. Solutions are x = 2 and x = -2.

Finding the y-intercept

• Set x = 0 in the function f(x).
• Example: For f(x) = (x^2 - 4) / (x + 1), calculate f(0) = (-4) / (1) = -4. The y-intercept is at (0, -4).

Step 3: Determine Asymptotes

Asymptotes indicate the behavior of the function at extreme values.

Vertical Asymptotes

• Occur where Q(x) = 0.
• Example: For Q(x) = x + 1, set x + 1 = 0. The vertical asymptote is at x = -1.

Horizontal Asymptotes

• Found by comparing the degrees of P(x) and Q(x).
• If the degree of P(x) < degree of Q(x), the horizontal asymptote is y = 0.
• If the degree of P(x) = degree of Q(x), the horizontal asymptote is y = leading coefficient of P / leading coefficient of Q.

Step 4: Analyze End Behavior

Understanding what happens to the function as x approaches infinity is crucial.

• Use limits to find f(x) as x approaches positive and negative infinity.
• For instance, in the function f(x) = (x^2 - 4) / (x + 1), as x → ∞, f(x) approaches the horizontal asymptote.

Step 5: Sketch the Graph

With all the information gathered, you can now sketch the graph of the rational function.

1. Plot the x-intercepts and y-intercept on the graph.
2. Draw vertical and horizontal asymptotes.
3. Analyze the behavior between and beyond the asymptotes to determine the shape of the graph.
4. Connect the points smoothly, ensuring that the function approaches the asymptotes as necessary.

Conclusion

Graphing rational functions involves understanding their structure, identifying key features like intercepts and asymptotes, and analyzing their behavior. By following these steps, you can successfully graph rational functions and gain a deeper understanding of their properties. For further practice, consider finding additional rational functions and applying these steps to reinforce your learning.