Question

use your graphing calculator to sketch the graph of the following function. then find the domain and range.
$f(x)=x^2 + 2x - 15$

find the domain of the function. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the domain is
( type an inequality or a compound inequality. use integers or fractions for any numbers in the expression. )
b. the domain is all real numbers.
find the range of the function. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the range is
( type an inequality or a compound inequality. use integers or fractions for any numbers in the expression. )
b. the range is all real numbers.

Explanation:

Step1: Analyze the function type

The function \( f(x) = x^2 + 2x - 15 \) is a quadratic function in the form \( f(x)=ax^2 + bx + c \) where \( a = 1 \), \( b = 2 \), and \( c=-15 \). Since \( a>0 \), the parabola opens upwards.

Step2: Find the domain of the quadratic function

For any quadratic function (a polynomial function of degree 2), there are no restrictions on the values of \( x \) (no division by zero, no square roots of negative numbers, etc.). So the domain of a quadratic function is all real numbers. Thus, the domain is all real numbers (option B).

Step3: Find the vertex of the parabola to determine the range

The x - coordinate of the vertex of a parabola \( f(x)=ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = 2 \), we get \( x=-\frac{2}{2\times1}=- 1 \).
To find the y - coordinate of the vertex, substitute \( x=-1 \) into the function:

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Since the parabola opens upwards (because \( a = 1>0 \)), the minimum value of the function is \( - 16 \), and the function can take on all values greater than or equal to \( -16 \). So the range of the function is \( y\geq - 16 \). So for the range, we choose option A with the inequality \( y\geq - 16 \).

Answer:

For the domain: B. The domain is all real numbers.
For the range: A. The range is \( y \geq -16 \) (or in boxed form, if needed, \(\boxed{y \geq -16}\))