Unlocking Zeros: Mastering Rational Functions

Algebra 2 Grades High School 3:03 Video
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Lesson Description

Learn to find zeros of rational functions, identify removable discontinuities, and connect zeros to x-intercepts. Includes factoring and simplification techniques.

Video Resource

Finding Zeros of Rational Functions

Mario's Math Tutoring

Duration: 3:03
Watch on YouTube

Key Concepts

  • Zeros of rational functions
  • X-intercepts
  • Removable discontinuities (holes)
  • Factoring rational expressions
  • Simplifying rational expressions

Learning Objectives

  • Students will be able to find the zeros of a rational function by setting the numerator equal to zero.
  • Students will be able to identify and account for removable discontinuities (holes) when finding zeros.
  • Students will be able to relate the zeros of a rational function to its x-intercepts.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a zero of a function and its relationship to the x-intercept. Briefly discuss rational functions and their graphical representation.
  • Video Presentation (7 mins)
    Play the video 'Finding Zeros of Rational Functions' by Mario's Math Tutoring. Encourage students to take notes on the methods presented in the video.
  • Example 1: Basic Zeros (5 mins)
    Work through an example similar to the first example in the video, emphasizing setting the numerator equal to zero. Discuss why this method works.
  • Example 2: Removable Discontinuities (10 mins)
    Present an example where factoring is required to identify removable discontinuities (holes). Explain why factors that cancel do not produce zeros but create holes in the graph. Work through locating the hole as shown in the video.
  • Example 3: Complex Factoring (10 mins)
    Tackle a more complex rational function requiring factoring of both the numerator and denominator. Reinforce the concepts of simplifying, identifying removable discontinuities, and finding zeros. Discuss restrictions on the domain.
  • Practice Problems (10 mins)
    Assign practice problems where students find the zeros of rational functions, considering removable discontinuities. Circulate to assist students as needed.
  • Wrap up and Questions (3 mins)
    Summarize the video's key concepts and answer any remaining questions from students about finding zeros of rational functions.

Interactive Exercises

  • Whiteboard Challenge
    Divide students into groups and assign each group a rational function. Have them find the zeros and any removable discontinuities on the whiteboard.
  • Error Analysis
    Present students with worked-out examples of finding zeros, some containing errors. Have them identify and correct the mistakes.

Discussion Questions

  • Why do we set the numerator equal to zero to find the zeros of a rational function?
  • What is the difference between a zero and a removable discontinuity (hole)? How do they affect the graph of a rational function?
  • How does factoring help in finding zeros of rational functions?

Skills Developed

  • Factoring polynomials
  • Solving rational equations
  • Identifying removable discontinuities
  • Analyzing rational functions

Multiple Choice Questions

Question 1:

The zeros of a rational function occur when:

Correct Answer: The numerator is zero

Question 2:

A removable discontinuity (hole) occurs when:

Correct Answer: A factor cancels in the rational function

Question 3:

What is the first step in finding the zeros of a rational function?

Correct Answer: Factor the numerator and denominator

Question 4:

If a rational function has a zero at x = 3, then the graph crosses the x-axis at:

Correct Answer: (3, 0)

Question 5:

Which of the following is true about removable discontinuities?

Correct Answer: They are points where the function is undefined, but can be removed by simplifying

Question 6:

What happens if you try to set a constant value in the numerator equal to 0?

Correct Answer: The function has no zeros or x-intercepts

Question 7:

When a factor in the numerator cancels with a factor in the denominator, what is created in the graph?

Correct Answer: Hole (removable discontinuity)

Question 8:

If a rational function simplifies to (x-2)/(x+3), where is the vertical asymptote?

Correct Answer: x = -3

Question 9:

What should you do after finding a potential zero of a rational function?

Correct Answer: Find the y-intercept

Question 10:

When finding the zeros, you are essentially solving for the function when y equals:

Correct Answer: 0

Fill in the Blank Questions

Question 1:

To find the zeros of a rational function, set the ________ equal to zero.

Correct Answer: numerator

Question 2:

A __________ __________ is a point where the function is undefined but can be removed by simplifying the expression.

Correct Answer: removable discontinuity

Question 3:

Zeros of a rational function correspond to the __________ of the graph.

Correct Answer: x-intercepts

Question 4:

Before finding zeros, it is important to _________ the numerator and denominator of the rational function.

Correct Answer: factor

Question 5:

If a factor cancels out from both the numerator and denominator, it creates a _________ in the graph.

Correct Answer: hole

Question 6:

The x-coordinate of a hole can be found by setting the cancelled factor equal to __________.

Correct Answer: zero

Question 7:

When solving for zeros, we are finding the values of 'x' that make the function equal to _________.

Correct Answer: zero

Question 8:

An undefined point which is not a removable discontinuity creates a _______ ______.

Correct Answer: vertical asymptote

Question 9:

After factoring a rational equation, you need to _________ the equation to find solutions.

Correct Answer: simplify

Question 10:

The y-coordinate of an x-intercept is always _________.

Correct Answer: zero

Educational Standards

A2.A.1.3:Solve one-variable rational equations and check for extraneous solutions. A2.A.2.3:Add, subtract, multiply, divide, and simplify rational expressions. A2.F.1.6:Graph a rational function and identify the domain (including holes), range, x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology (excluding slant or oblique asymptotes).

Teaching Materials

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