Question

the quadratic function f (x) = x^2 + 2x - 15 is shown in the graph. determine the domain and range of this function.

Answer 1

The function \( f(x) = x^2 + 2x - 15 \) is a quadratic function, which is represented by a parabola.

### Domain:
The domain of any quadratic function is all real numbers, since you can input any real number for \( x \). Therefore, the domain of \( f(x) \) is:
\[
\text{Domain: } (-\infty, \infty)
\]

### Range:
To determine the range, we need to find the vertex of the parabola, as it opens upwards (the coefficient of \( x^2 \) is positive). The vertex form can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \).

In the function \( f(x) = x^2 + 2x - 15 \):
- \( a = 1 \)
- \( b = 2 \)

Now, we can calculate the x-coordinate of the vertex:
\[
x = -\frac{2}{2(1)} = -1
\]

Next, we find the y-coordinate of the vertex by plugging \( x = -1 \) back into the function:
\[
f(-1) = (-1)^2 + 2(-1) - 15 = 1 - 2 - 15 = -16
\]

Thus, the vertex is at the point \((-1, -16)\). Since this is the minimum point of the upwards-opening parabola, the range starts from this y-value and goes to positive infinity.

Thus, the range of the function is:
\[
\text{Range: } [-16, \infty)
\]

### Summary:
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [-16, \infty) \)