Exercises


  1. For each of the following functions, determine whether it is
    1. an even or odd degree polynomial function.
    2. an even function, odd function, or neither.
    1. Cubic, positive leading coefficient, zeros at x=-2, 0, 2
    2. Cubic, positive leading coeffiicient, zero at x = -2.5, double zero at x = 0
    3. Cubic, negative leading coefficient, triple zero at x = 0
    4. Quartic, positive leading coefficient, turning pts (-1, -3), (0, -2), (1, -3)
    5. \( f(x) = 3x^2 + 4 \)
    6. \( f(x) = -2x + 5 \)
    7. \( f(x) = 2x^2 + 3x \)
    8. \( f(x) = -3x^3 + x \)
    9. \( f(x) = -x^4 + 6x^2 - 1 \)
    10. \( f(x) = x^3 + x^2 - x \)
  2. Determine algebraically if the function is an even function, odd function, or neither.
    1. \( f(x) = 3 - x^2 + 5x^4 \)
    2. \( f(x) = 3x^5 + x^3 - 1 \)
    3. \( g(x) = 4x - 3x^3 + x^5 \)
  3. Are the following statements true or false?
    1. An even polynomial function must be an even degree polynomial function.
    2. An odd degree polynomial function must be an odd function.
    3. An even polynomial function must have an even number of \( x \)-intercepts.
    4. An odd polynomial function must have an odd number of \( x \)-intercepts.
    5. An odd polynomial function must have an odd number of turning points.
    6. An odd polynomial function must pass through the origin.
    1. Sketch the graph of \( y = -\frac{1}{2}x(x - 4)^2(x + 4)^2 \).
    2. From your sketch, is it possible to determine if the function is even, odd, or neither? Explain.
    3. Determine, algebraically, if \( y = -2x(x - 4)^2(x + 4)^2 \) is even, odd, or neither.
    1. Sketch the graph of a polynomial function with the following characteristics:
      1. the function is an even function
      2. as \( x \rightarrow \infty, y \rightarrow -\infty \)
      3. the function has exactly \( 3 \) \( x \)-intercepts
    2. Sketch the graph of two polynomial functions of different degree with the following characteristics:
      1. the function is an odd function
      2. as \( x \rightarrow \infty, y \rightarrow \infty \)
      3. the function has exactly \( 3 \) \( x \)-intercepts
    1. Determine an equation, in factored form, for an even polynomial function with a turning point at \( (2, 0) \). Show algebraically that the function is even.
    2. Determine an equation, in factored form, for an odd polynomial function with one of its zeros at \( x = 2 \). Show algebraically that the function is odd.
    1. If \( y = f(x) \) is an odd polynomial function, which of the following are odd polynomial functions?
      1. \( y = f(x + 2) \)
      2. \( y = -f(x) \)
      3. \( y = f(2x) \)
      4. \( y = 2f(x) \)
      5. \( y = f(x) + 2 \)
      6. \( y = f(-x) \)
    2. If \( y = f(x) \) is an even polynomial function, which of the functions in part a) would be even polynomial functions?