2. Polynomial Functions
  3. Finite Differences of Polynomial Functions

Exercises


  1. For the following tables of values,
    1. Determine if the function is linear, quadratic, cubic, or quartic.
    2. Determine the leading coefficient for each polynomial function in part (i).
    1. \( x \) \( y \)
      \( -2 \) \( 0 \)
      \( -1 \) \( 7 \)
      \( 0 \) \( 8 \)
      \( 1 \) \( 3 \)
      \( 2 \) \( -8 \)
      \( 3 \) \( -25 \)
    2. \( x \) \( y \)
      \( 1 \) \( -70 \)
      \( 2 \) \( -26 \)
      \( 3 \) \( -10 \)
      \( 4 \) \( -4 \)
      \( 5 \) \( 10 \)
      \( 6 \) \( 50 \)
    3. \( x \) \( y \)
      \( -3 \) \( 139 \)
      \( -2 \) \( 24 \)
      \( -1 \) \( 3 \)
      \( 0 \) \( 4 \)
      \( 1 \) \( 3 \)
      \( 2 \) \( 24 \)
  2. Given the function \( y = -4x^5 + 6x^2 + 1 \):
    1. Which finite difference is constant?
    2. What is the value of the constant difference when \( \Delta{x} = 1 \)?
  3. Sketch a possible graph for a polynomial whose \( 4^{th} \) finite difference is a constant value of \( -36 \) when \( \Delta{x} = 1 \) and the function has exactly \( 3 \) distinct zeros at \( -4, -1, \) and \( 3 \).
  4. A polynomial function has a constant \( 3^{rd} \) difference of \( -20 \) when \( \Delta{x} = 1 \), zeros at \( x = 0 \) and \( -5 \), and a turning point at \( x = 0 \). Find the equation of the function.
  5. A polynomial function has a constant \( 4^{th} \) difference of \( 12 \) when \( \Delta{x} = 1 \), a zero at \( x = \frac{1}{2} \), and a point of inflection at \( (-3, 0) \). Find the equation of the function.
  6. Show that the data in the given tables can be modeled by a cubic polynomial function. Determine the equation of the function in standard form, \( f(x) = ax^3 + bx^2 + cx + d \).
    1. \( x \) \( y \)
      \( -2 \) \( -56 \)
      \( -1 \) \( -21 \)
      \( 0 \) \( -4 \)
      \( 1 \) \( 7 \)
      \( 2 \) \( 24 \)
      \( 3 \) \( 59 \)
    2. \( x \) \( y \)
      \( -3 \) \( 4 \)
      \( -2 \) \( -29 \)
      \( -1 \) \( -26 \)
      \( 0 \) \( -5 \)
      \( 1 \) \( 16 \)
      \( 2 \) \( 19 \)
    3. \( x \) \( y \)
      \( -4 \) \( 114 \)
      \( -2 \) \( 22 \)
      \( 0 \) \( 2 \)
      \( 2 \) \( 6 \)
      \( 4 \) \( -14 \)
      \( 6 \) \( -106 \)
    1. The population of a small town from \( 1992 \) to \( 1997 \) is given in the table. The data can be modeled by a polynomial function. Determine an equation that models the population, \( P \), in terms of time, \( t \), where \( t = 0 \) in \( 1992 \).
      \( \text{Year} \) \( \text{Population} \)
      \( 1992 \) \( 8538 \)
      \( 1993 \) \( 8515 \)
      \( 1994 \) \( 8444 \)
      \( 1995 \) \( 8331 \)
      \( 1996 \) \( 8182 \)
      \( 1997 \) \( 8003 \)
    2. According to the model, what was the population in \( 1988 \)?
    3. Assuming the function will continue to describe the population accurately, what will the population be in the year 2030?
    4. Is it realistic to expect that this function will always accurately describe the population? Explain.
  8. The first three square pyramidal numbers are \(1, 5, \) and \( 14 \), as shown in the diagram. find the next three pyramidal numbers and determine the equation of a polynomial function that gives the \( x^{th} \) square pyramidal number.
    First three square pyramidal numbers, 1, 5, 14
  9. Determine an equation of a polynomial function \( y = f(n) \), where \( f(n) \) is the maximum number of regions that \( n \) lines divide the plane into.