How do you graph f ( x ) = x + 3 ( x + 1 ) ( x - 3 ) using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

How do you graph             f              (        x        )              =                  x        +        3                              (          x          +          1          )                          (          x          -          3          )                     using holes, vertical and horizontal asymptotes, x and y intercepts?

Sanaa Holder · 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      +      1      )              (      x      -      3      )        =    0    ⇒    x    =    -    1                 and               x    =    3        ⇒    x    =    -    1                 and               x    =    3                 are the asymptotes      Horizontal asymptotes occur as            lim              x        →        ±        ∞              ,          f              (        x        )              →    c                 ( a constant)      divide terms on numerator/denominator by the highest power of x, that is             x      2                  f              (        x        )              =                            x                      x            2                          +                  3                      x            2                                                            x            2                                x            2                          -                              2            x                                x            2                          -                  3                      x            2                                =                            1          x                +                  3                      x            2                                      1        -                  2          x                -                  3                      x            2                              as       x    →    ±    ∞    ,          f              (        x        )              →                  0        +        0                    1        -        0        -        0                  ⇒    y    =    0                 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    =          3              -        3              =    -    1    ←                           y-intercept                  y    =    0    →    x    +    3    =    0    →    x    =    -    3    ←                           x-intercept            graph{(x+3)/(x^2-2x-3) [-10, 10, -5, 5]}

How do you graph f(x)=(x+3)/(x+1)(x−3) using holes, vertical and horizontal asymptotes, x and y intercepts?

Answered question

Answer & Explanation

New Questions in Algebra II