Question

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or −1. There may be more than one correct answer.

The y-intercept is (0, 9). The x-intercepts are (−3, 0), (3, 0). Degree is 2. End behavior: as x → −∞, f(x) → −∞, as x → ∞, f(x) → −∞.

Answer 1

Consider a polynomial function of degree 2.

The x-intercepts are (-3, 0), (3, 0).

The y-intercept is (0, 9)

 

End behavior of the polynomial function is,

x → ∞, f(x) → -∞

x → -∞, f(x) → -∞

 

The end behavior shows that the graph is downwards. This means that the leading coefficient is -1.

Consider the following general form of a polynomial function of degree 2.

f(x) = ax2 + bx + c

 

Take a = -1,

f(x) = -x2 + bx + c

 

Put the point of y-intercept (0, 9) in the equation,

f(x) = -x2 + bx + c

   9 = 0 + 0 + c

   c = 9

 

Therefore,

f(x) = -x2 + bx + 9

 

Put the point of x-intercept (-3, 0) in the function,

f(x) = -x2 + bx + 9

   0 = -(-3)2 -3b + 9

   0 = -9 -3b + 9

   b = 0

 

Put the value in the function,

f(x) = -x2 + bx + 9

      = -x2 + 0 + 9

      = -x2 + 9

Hence, the polynomial function is f(x) = -x2 + 9.

Hence, the polynomial function is f(x) = -x2 + 9.