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3.5: Rational Functions

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    239
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    1. Rational Functions (Definition)

    is a quotient of polynomials \(\dfrac{P(x)}{Q(x)}\).

    are all Rational Functions

    \[\{x | x \text{ is not }1 \text{ or } -1\}.\]

    2. Vertical Asymptotes

    of a rational function occurs where the denominator is 0.

    \[f(x) \rightarrow\infty.\]

    Four Types of Vertical Asymptotes

    Below are the four types of vertical asymptotes:

    3. Horizontal Asymptotes

    and we have the the left and right behavior of the graph is a horizontal line \(y = 3\).

    4. Oblique Asymptotes

    Consider the function

    \[f(x) = \dfrac{(x^2 - 3x - 4)}{(x + 3)}\]

    alt

    \(f(x)\) does not have a horizontal asymptote, since

    \[\dfrac{x^2}{x}= x \]

    is not a constant, but we see (on the calculator) that the left and right behavior of the curve is like a line. Our goal is to find the equation of this line.

    We use synthetic division to see that

    \[\dfrac{(x^2 - 3x - 4)}{(x + 3)} = x - 6 + \dfrac{14}{(x+3)}.\]

    For very large \(x\),

    \[\dfrac{14}{x} + 3\]

    is very small, hence \(f(x)\) is approximately equal to

    \[x - 6\]

    on the far left and far right of the graph. We call this line an Oblique Asymptote.

    To graph, we see that there is a vertical asymptote at

    \[x = -3\]

    with behavior:

    left down and right up

    The graph has x-intercepts at 4 and -1, and a y intercept at \(-\frac{4}{3}\).

    \[\dfrac{(x^3 + 8)}{(x^2 - 3x - 4)}\]

    5. Rational Functions With Common Factors

    Consider the graph of

    \[y = \dfrac{x-1}{x-1}\]

    alt

    What is wrong with the picture? When

    \[f(x) = \dfrac{g(x)(x - r)}{h(x)(x - r)}\]

    with neither \(g(r)\) nor \(h(r)\) zero, the graph will have a hole at \(x = r\). We call this hole a removable discontinuity.

    at \((2,2)\).

    We end our discussion with a list of steps for graphing rational functions.

    Steps in graphing rational functions:

    • Step 1 Plug in \(x = 0\) to find the y-intercept
    • Step 2 Factor the numerator and denominator. Cancel any common factors remember to put in the appropriate holes if necessary.
    • Step 3 Set the numerator = 0 to find the x-intercepts
    • Step 4 Set the denominator = 0 to find the vertical asymptotes. Then plug in nearby values to fine the left and right behavior of the vertical asymptotes.
    • Step 5 If the degree of the numerator = degree of the denominator then the graph has a horizontal asymptote. To determine the value of the horizontal asymptote, divide the term highest power of the numerator by the term of highest power of the denominator.
    • If the degree of the numerator = degree of the denominator + 1, then use polynomial or synthetic division to determine the equation of the oblique asymptote.
    • Step 6 Graph it!

    Larry Green (Lake Tahoe Community College)

    • Integrated by Justin Marshall.


    This page titled 3.5: Rational Functions is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

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