How do you graph f ( x ) = x 2 - 2 x - 8 x 2 - 9 using holes, vertical and horizontal asymptotes, x and y intercepts?

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How do you graph             f              (        x        )              =                            x          2                -        2        x        -        8                              x          2                -        9             using holes, vertical and horizontal asymptotes, x and y intercepts?

lufi8c · Accepted Answer

AsymptotesThe denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.solve :             x      2        -    9    =    0    ⇒          x      2        =    9    ⇒    x    =    ±    3        ⇒    x    =    -    3                 and               x    =    3                 are the asymptotes      Horizontal asymptotes occur as            lim              x        →        ±        ∞              ,          f              (        x        )              →    c                 ( a constant)      divide numerator/denominator by the highest power of x, that is             x      2                  f              (        x        )              =                                        x            2                                x            2                          -                              2            x                                x            2                          -                  8                      x            2                                                            x            2                                x            2                          -                  9                      x            2                                =                  1        -                  2          x                -                  8                      x            2                                      1        -                  9                      x            2                              as       x    →    ±    ∞    ,          f              (        x        )              →                  1        -        0        -        0                    1        -        0                  ⇒    y    =    1                 is the asymptote      Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here, hence there are no holes.Intercepts      x    =    0    →    y    =                  -        8                    -        9              =          8      9            ⇒          y-intercept at                     (      0      ,              8        9            )            y    =    0    →          x      2        -    2    x    -    8    =    0    →          (      x      -      4      )              (      x      +      2      )        =    0        ⇒          x-intercepts at                     (      -      2      ,      0      )                     and                     (      4      ,      0      )      graph{(x^2-2x-8)/(x^2-9) [-10, 10, -5, 5]}

How do you graph f(x)=(x^2−2x−8)/(x^2−9) using holes, vertical and horizontal asymptotes, x and y intercepts?

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