The 3 parabolas graphed in the standard (x,y) coordinate plane

The 3 parabolas graphed in the standard (x,y) coordinate plane below are from a family of parabolas. A general equation that defines this family of parabolas contains the variable n in addition to x and y. For one of the parabolas shown, n = 1; for another, n = 2; and for the third, n = 3. Which of the following could be a general equation that defines this family of parabolas for all n ≥ 1 ?

  1. y = nx2 + 1
  2. y = 1/n x2 + 1
  3. y = x2 + n
  4. y = -nx2 + 1
  5. y = -1/n x2 + 1

Answer

The correct answer is A.

All the parabolas open upward. This rules out answer choices D and E. All the parabolas have the same y-intercept, (0, 1). This rules out answer choice C, which has y-intercept equal to n, which varies.

The parabolas in the family go up more quickly as the value of n increases. This means the coefficient of x must get larger as n gets larger. That happens in A but not in B.