Mathematics
Grade 10
15 min
Rational functions asymptotes and excluded values
Rational functions asymptotes and excluded values
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Introduction & Learning Objectives
Learning Objectives
Define a rational function and identify its polynomial numerator and denominator.
Determine the excluded values of a rational function by finding the zeros of the denominator.
Identify the equations of vertical asymptotes from the simplified form of a rational function.
Distinguish between a vertical asymptote and a hole (point of discontinuity) by identifying common factors.
Determine the equation of the horizontal asymptote by comparing the degrees of the numerator and denominator.
State asymptotes and excluded values using correct mathematical notation.
Ever wondered why some graphs have invisible 'force fields' they get close to but never cross? 🚧 Let's investigate these mathematical boundaries!
This tutorial explores rational functio...
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Key Concepts & Vocabulary
TermDefinitionExample
Rational FunctionA function that can be written as the ratio of two polynomial functions, P(x) and Q(x), in the form f(x) = P(x) / Q(x), where the denominator Q(x) cannot be zero.f(x) = (x + 5) / (x - 1) is a rational function.
Excluded ValueAny value of 'x' that makes the denominator of a rational function equal to zero. The function is undefined at this x-value.For the function f(x) = 3 / (x - 2), the excluded value is x = 2, because it would lead to division by zero.
Vertical AsymptoteA vertical line, x = c, that the graph of a function approaches but never touches or crosses. It occurs at an excluded value that remains after the function is simplified.The function f(x) = 1 / (x + 4) has a vertical asymptote at x = -4.
Hole (Point of Discontinuity)A sing...
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Core Formulas
Finding Vertical Asymptotes and Holes
For f(x) = \frac{P(x)}{Q(x)}, first factor P(x) and Q(x). Any factor (x-c) that cancels is a hole at x=c. Any factor (x-a) remaining in the denominator creates a vertical asymptote at x=a.
Use this rule to find all vertical lines of discontinuity. Always simplify the function first to distinguish between holes and vertical asymptotes.
Finding Horizontal Asymptotes
Compare the degree of the numerator (n) with the degree of the denominator (m) in f(x) = \frac{ax^n + ...}{bx^m + ...}.
1. If n < m, the horizontal asymptote is y = 0.
2. If n = m, the horizontal asymptote is y = \frac{a}{b} (ratio of leading coefficients).
3. If n > m, there is no horizontal asymptote.
4 more steps in this tutorial
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Challenging
For the function f(x) = \frac{x^2 - 9}{x^2 + kx - 15}, what value of 'k' will create a hole at x = 3?
A.k = 5
B.k = -3
C.k = 2
D.k = -2
Challenging
If a rational function f(x) = P(x)/Q(x) has a horizontal asymptote at y = 0, and P(x) and Q(x) are non-zero polynomials, what must be true about the degrees of P(x) and Q(x)?
A.The degree of Q(x) must be greater than the degree of P(x).
B.The degree of P(x) must be greater than the degree of Q(x).
C.The degrees of P(x) and Q(x) must be equal.
D.The degree of P(x) must be zero.
Challenging
A function's graph has a vertical asymptote at x = -1, a hole at x = 2, and a horizontal asymptote at y = 3. Which of the following functions could represent this graph?
A.f(x) = \frac{3(x-2)}{(x+1)(x-2)}
B.f(x) = \frac{3x^2 - 12}{(x+1)(x-2)}
C.f(x) = \frac{(x-2)(x+1)}{3(x-2)}
D.f(x) = \frac{3x(x-2)}{(x+1)(x-2)}
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