Use the image to answer the question.

Question

Four points are plotted on a coordinate plane, forming a parabolic curve.

The quadratic function f(x)=x2+2x−15
is shown in the graph. Determine the domain and range of this function.

(1 point)
Responses

domain R
; range y≥−16
domain the real numbers ; range y is greater than or equal to negative 16

domain R
; range y≤−16
domain negative infinity less-than-or-equal-to x less-than-or-equal-to infinity ; range y is less than or equal to negative 16

domain R
; range R
domain the real numbers ; range the real numbers

domain R
; range y≥16
domain the real numbers ; range y is greater than or equal to 16
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Answer 1

To determine the domain and range of the quadratic function \( f(x) = x^2 + 2x - 15 \), we can analyze the function.

1. Domain:

   • The domain of any quadratic function is all real numbers. Thus, the domain is \( \mathbb{R} \) or "the real numbers".

2. Range:

   • To find the range, we need to identify the vertex of the parabola, which gives the minimum (or maximum) value of the function depending on whether the parabola opens upwards or downwards.
   • The formula for the vertex \( x \) coordinate of the quadratic \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 2 \).
   • Plugging in these values: \[ x = -\frac{2}{2(1)} = -1 \]
   • Now, we find \( f(-1) \): \[ f(-1) = (-1)^2 + 2(-1) - 15 = 1 - 2 - 15 = -16 \]
   • Since the coefficient of \( x^2 \) is positive, the parabola opens upwards. Therefore, the minimum value of the function is \( -16 \), and it can take on all values greater than or equal to this point.
   • Thus, the range is \( y \ge -16 \).

Putting this all together, the correct response is:

Domain: the real numbers; Range: \( y \ge -16 \).