Quartic

  1. degree: even
    coefficient: negative

  2. 4 x intercepts

  3. 3 direction changes
    +1=4

  4. The minimum degree is 4

  5. x-intercepts are: -3, -1, 2, 5
    Written as factors: (x+3)(x+1)(x-2)(x-5)

  6. f(x)=a(x+3)(x+1)(x-2)(x-5)

  7. Plug in the point (0,-15) and solve for a
    -15=a(x+3)(x+1)(x-2)(x-5)
    -15=a(0+3)(0+1)(0-2)(0-5)
    -15=a(3)(1)(-2)(-5)
    -15=30a
    a=\frac{-1}{2}

  8. The function for this graph is
    f(x)=\frac{-1}{2}(x+3)(x+1)(x-2)(x-5)
    or
    f(x)=\frac{-1}{2}(x^4-3x^3-15x^2+19x+30)