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, −4). The x-intercepts are (−2, 0), (2, 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 (-2, 0), (2, 0).

The y-intercept is (0, -4).

 

End behavior of the polynomial function is,

x → ∞, f(x) → ∞

x → -∞, f(x) → ∞

 

The end behavior shows that the graph is upwards. 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, -4) in the equation,

f(x) = x2 + bx + c

  -4 = 0 + 0 + c

   c = -4

 

Therefore,

f(x) = x2 + bx - 4

 

Put the point of x-intercept (-2, 0) in the function,

f(-2) = (-2)2 + b(-2) – 4

     0 = 4 -2b -4

   2b = 0

     b = 0

 

Put the value in the function,

f(x) = x2 + bx – 4

f(x) = x2 - 4

Hence, the polynomial function is f(x) = x2 - 4.

Hence, the polynomial function is f(x) = x2 - 4.