How To Find The Vertical Asymptote of a Function

The Organic Chemistry Tutor
9 Sept 201710:53
EducationalLearning
32 Likes 10 Comments

TLDRThis educational video script offers a step-by-step guide on identifying vertical asymptotes of functions. It begins by illustrating the process with a simple function, 1/(x-3), demonstrating that setting the denominator to zero reveals the asymptote at x=3. The script then progresses to more complex examples, including factoring and dealing with imaginary solutions. It explains how to factor expressions like x^2 - 16 and x^2 - 7x + 10, and how to handle cases where the denominator results in imaginary numbers, which do not produce vertical asymptotes. The final example involves factoring a complex rational function, 10x^2 - 5x - 15 / 6x^2 - 17x + 12, to find its vertical asymptote at x=4/3. The script effectively teaches viewers the methodical approach to finding vertical asymptotes in various mathematical scenarios.

Takeaways
  • 🔍 To find the vertical asymptote of a function, set the denominator equal to zero and solve for x.
  • 📉 For the function 1/(x - 3), setting x - 3 to zero gives the vertical asymptote at x = 3.
  • ⚖️ In the function 4/(x^2 - 16), factoring x^2 - 16 as (x + 4)(x - 4) results in vertical asymptotes at x = -4 and x = 4.
  • 🧮 When dealing with 3x - 6 / (x^2 - 7x + 10), factor both numerator and denominator to find vertical asymptotes and holes.
  • 🚫 The vertical asymptote occurs where the denominator cannot be canceled out. In this case, x = 5.
  • ❌ No vertical asymptote exists if the denominator results in an imaginary number, as seen with 8/(x^2 + 4).
  • 🔢 For more complex functions like 10x^2 - 5x - 15 / 6x^2 - 17x + 12, factor both numerator and denominator to identify the asymptotes.
  • 🧩 Factoring the numerator and denominator often involves finding the greatest common factor and using methods like the difference of squares or factoring by grouping.
  • 📏 When terms can be canceled out after factoring, the remaining denominator terms determine the vertical asymptotes.
  • 📈 In the example of 10x^2 - 5x - 15 / 6x^2 - 17x + 12, the final vertical asymptote is found by solving 3x - 4 = 0, resulting in x = 4/3.
Q & A

  • What is a vertical asymptote?

    -A vertical asymptote is a line where a function's value becomes unbounded as it approaches the line from either side. It represents a value of x where the function is undefined and the function's value tends to infinity.

  • How can you find the vertical asymptote of a function?

    -To find the vertical asymptote of a function, set the denominator of the function equal to zero and solve for x. The values of x that make the denominator zero are where the vertical asymptotes are located.

  • What is the vertical asymptote for the function 1/(x-3)?

    -The vertical asymptote for the function 1/(x-3) is at x = 3.

  • Why is x = 3 the vertical asymptote for the function 1/(x-3)?

    -x = 3 is the vertical asymptote because when x is 3, the denominator (x-3) becomes zero, making the function undefined and causing it to approach infinity.

  • How do you find the vertical asymptotes of the function 4/(x^2 - 16)?

    -To find the vertical asymptotes of 4/(x^2 - 16), set the denominator equal to zero: x^2 - 16 = 0. Factor it to (x + 4)(x - 4) = 0, giving the vertical asymptotes at x = -4 and x = 4.

  • What are the steps to factor a quadratic expression to find vertical asymptotes?

    -First, set the quadratic expression equal to zero. Next, factor the quadratic expression using methods such as the difference of squares or factoring by grouping. Finally, solve for the values of x that make the factored expression equal to zero.

  • What is the vertical asymptote of the function 3x-6 / (x^2 - 7x + 10)?

    -To find the vertical asymptote, factor both the numerator and the denominator. The numerator factors to 3(x - 2). The denominator factors to (x - 2)(x - 5). Cancel the common factor (x - 2), leaving 3/(x - 5). The vertical asymptote is at x = 5.

  • Why is x = 2 not a vertical asymptote for the function 3x-6 / (x^2 - 7x + 10)?

    -x = 2 is not a vertical asymptote because it is a removable discontinuity. After factoring and canceling the common factor (x - 2), it does not cause the function to be undefined.

  • What do you do if the denominator of a function cannot be factored using real numbers?

    -If the denominator cannot be factored using real numbers and results in an equation like x^2 = -4, which has imaginary solutions, then the function does not have a vertical asymptote since vertical asymptotes must be real numbers.

  • How do you handle complex trinomials when finding vertical asymptotes?

    -For complex trinomials, factor by grouping. First, multiply the leading coefficient by the constant term. Find two numbers that multiply to this product and add to the middle coefficient. Replace the middle term with these two numbers, factor by grouping, and solve for the values that make the factors equal to zero.

Outlines
00:00
📚 Introduction to Finding Vertical Asymptotes

This paragraph introduces the concept of vertical asymptotes in functions, using the function 1/(x-3) as an example. The method to find a vertical asymptote is explained by setting the denominator equal to zero, which in this case leads to x = 3. The graph of the function is described, indicating that there is a vertical asymptote at x = 3, with the graph lying above the x-axis to the right of the asymptote and below it to the left. The paragraph then proceeds to another example, the function 4/(x^2-16), and explains that factoring the expression may be necessary to find vertical asymptotes, which in this case are at x = -4 and x = 4.

05:00
🔍 Factoring and Simplifying to Find Vertical Asymptotes

The second paragraph delves into a more complex example, the function (3x-6)/(x^2-7x+10), and explains the process of factoring both the numerator and the denominator to find vertical asymptotes. The greatest common factor (GCF) is factored out, and the trinomial in the denominator is factored using the method of finding two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle coefficient. The vertical asymptote is found by setting the factored form of the denominator equal to zero, resulting in x = 5. The paragraph also discusses a case where the denominator results in an imaginary number, such as in the function 8/(x^2+4), and explains that imaginary solutions do not yield vertical asymptotes.

10:02
📘 Advanced Factoring for Vertical Asymptotes

The final paragraph presents an advanced example, the function f(x) = (10x^2-5x-15)/(6x^2-17x+12), and guides through the process of factoring both the numerator and the denominator to identify vertical asymptotes. The GCF is removed from the numerator, and the trinomial is factored by grouping. The denominator is factored by finding two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle coefficient. The vertical asymptote is determined by setting the factored form of the denominator that does not get canceled equal to zero, resulting in the vertical asymptote at x = 4/3.

Mindmap
Keywords
💡Vertical Asymptote
A vertical asymptote is a vertical line that the graph of a function approaches but never crosses as the function's input (x-value) approaches a certain value. In the context of the video, vertical asymptotes are found by setting the denominator of a rational function to zero, as it indicates values where the function becomes undefined. For example, in the function 1/(x-3), setting x-3 to zero gives x=3, which is the vertical asymptote.
💡Denominator
The denominator is the bottom part of a fraction and plays a crucial role in rational functions. In the video, the denominator is set to zero to find vertical asymptotes because a function is undefined when its denominator is zero. This is demonstrated in the function 1/(x-3) where the denominator (x-3) is set to zero to find the asymptote at x=3.
💡Factor
To factor in mathematics means to express a polynomial as the product of its factors. In the video, factoring is used as a method to find vertical asymptotes by setting each factor of the denominator equal to zero. For instance, in the function 4/(x^2-16), the denominator is factored into (x+4)(x-4), leading to two vertical asymptotes at x=-4 and x=4.
💡Difference of Squares
The difference of squares is a factoring technique used for expressions of the form a^2 - b^2, which can be factored into (a+b)(a-b). In the script, this method is applied to the function 4/(x^2-16) where x^2-16 is factored into (x+4)(x-4), illustrating the use of this technique to find vertical asymptotes.
💡Greatest Common Factor (GCF)
The greatest common factor is the largest number that divides two or more numbers without leaving a remainder. In the video, the GCF is used to simplify expressions before factoring. For example, in the function 3x-6/x^2-7x+10, the GCF of 3 is factored out from both the numerator and the denominator, simplifying the expression before further analysis for vertical asymptotes.
💡Tricomial
A trinomial is a polynomial with three terms. In the context of the video, the trinomial x^2-7x+10 is factored to find the vertical asymptote. The script explains how to find two numbers that multiply to the product of the leading coefficient and the constant term (10 in this case) and add up to the middle coefficient (-7), resulting in the factors (x-2)(x-5).
💡Imaginary Numbers
Imaginary numbers are a set of numbers that, when squared, give a negative result. They are represented by the symbol 'i' where i^2 = -1. In the video, when the denominator x^2+4 cannot be factored using real numbers, the square root of -4 is taken, resulting in imaginary numbers, indicating that there is no vertical asymptote for this function.
💡Rational Function
A rational function is a function that is the ratio of two polynomials. The video's main theme revolves around finding vertical asymptotes of rational functions. The function 1/(x-3) is an example of a rational function where the vertical asymptote is found by setting the denominator to zero.
💡Factor by Grouping
Factor by grouping is a technique used to factor polynomials by grouping terms that have a common factor and then factoring out that common factor. In the script, this method is used to factor the numerator 10x^2-5x-15 into 5(x+1)(2x-3), which helps in simplifying the function and finding its vertical asymptotes.
💡Leading Coefficient
The leading coefficient is the coefficient of the term with the highest power in a polynomial. In the video, when factoring polynomials, the leading coefficient is multiplied by the constant term to help find two numbers that will factor the polynomial. For example, in the denominator 6x^2-17x+12, the leading coefficient (6) is multiplied by the constant term (12) to find factors that multiply to 72 and add up to -17.
Highlights

Introduction to finding vertical asymptotes of a function.

Method to find vertical asymptote by setting the denominator equal to zero.

Example of finding vertical asymptote for the function 1/(x-3).

Explanation of the graph's behavior around the vertical asymptote at x=3.

Approach to find vertical asymptotes for the function 4/(x^2-16) by factoring.

Identification of two vertical asymptotes at x=-4 and x=4 for the function 4/(x^2-16).

Guidance on factoring the numerator and denominator in the function 3x-6/(x^2-7x+10).

Cancellation of (x-2) in the function and identification of the vertical asymptote at x=5.

Explanation of why there are no vertical asymptotes for the function 8/(x^2+4) with imaginary solutions.

Process of finding vertical asymptotes for the function 10x^2-5x-15/(6x^2-17x+12).

Removal of the greatest common factor (GCF) in the numerator of the function.

Factoring the trinomial in the numerator using numbers that multiply to -6 and add to -1.

Factoring by grouping in the numerator to get (5x+1)(2x-3).

Factoring the denominator of the function without a GCF.

Finding two numbers that multiply to 72 and add to -17 for the denominator.

Factoring by grouping in the denominator to get (3x-4)(2x-3).

Cancellation of (2x-3) and identification of the vertical asymptote at x=4/3.

Transcripts

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