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Due to length restrictions, we kindly invite to check the explanation of this question to understand the derivation of the polynomic expressions.

How to determine a family of cubic functions

Cubic functions are polynomials of grade 3. In this case, we have pairs of cubic functions of the following form:

y = (x - h)³ + k       (1)

y = - (x - h)³ + k     (2)

a) Where (h, k) are the coordinates of the vertex of each cubic function. There is a translation of (x, y) = (3, 1) between each two consecutive pairs of cubic functions. Hence, we have the following fourteen cubic functions:

  1. y = (x + 9)³ - 3
  2. y = - (x + 9)³ - 3
  3. y = (x + 6)³ - 2
  4. y = - (x + 6)³ - 2
  5. y = (x + 3)³ - 1
  6. y = - (x + 3)³ - 1
  7. y = x³
  8. y = - x³
  9. y = (x - 3)³ + 1
  10. y = - (x - 3)³ + 1
  11. y = (x - 6)³ + 2
  12. y = - (x - 6)³ + 2
  13. y = (x - 9)³ + 3
  14. y = - (x - 9)³ + 3

b) Another family of functions with a similar pattern is shown below:

  1. y = (x + 9)² - 3
  2. y = - (x + 9)² - 3
  3. y = (x + 6)² - 2
  4. y = - (x + 6)² - 2
  5. y = (x + 3)² - 1
  6. y = - (x + 3)² - 1
  7. y = x²
  8. y = - x²
  9. y = (x - 3)² + 1
  10. y = - (x - 3)² + 1
  11. y = (x - 6)² + 2
  12. y = - (x - 6)² + 2
  13. y = (x - 9)² + 3
  14. y = - (x - 9)² + 3

To learn more on cubic functions: https://brainly.com/question/25732149

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