- Identify that the function \(f(x) = -x^2 - 2x + 15\) is a quadratic polynomial, so its domain is all real numbers.
- Calculate the vertex \((h, k)\) using \(h = -b/(2a) = -1\) and \(k = f(-1) = 16\).
- Determine that the parabola opens downward, so the range is all values \(y \leq 16\).
- Select option C as it matches the domain and range found.
- The correct option is \(\boxed{C}\).
Explanation
- Identify the domain of the function
The given function is \(f(x) = -x^2 - 2x + 15\). This is a polynomial of degree 2, which is a quadratic function. A polynomial function is defined for all real numbers, meaning its domain is all real numbers, often denoted as \((-\infty, \infty)\).
- Find the vertex and maximum value
Since the leading coefficient (the coefficient of \(x^2\)) is \(a = -1\), which is less than 0, the graph of this function is a parabola that opens downward. A downward-opening parabola has a maximum value at its vertex. The x-coordinate of the vertex \((h)\) is calculated using the formula \(h = -\frac{b}{2a}\). With \(a = -1\) and \(b = -2\), we have:
\[h = -\frac{-2}{2 \times (-1)} = \frac{2}{-2} = -1.\]
To find the y-coordinate of the vertex \((k)\), we evaluate the function at \(h = -1\):
\[k = f(-1) = -(-1)^2 - 2(-1) + 15 = -1 + 2 + 15 = 16.\]
Therefore, the maximum value of the function is 16.
- Determine the range and select the correct option
Since the parabola opens downward and reaches a maximum value of 16 at the vertex, the function can take any value less than or equal to 16. Thus, the range of the function is all real numbers \(y\) such that \(y \leq 16\), written in interval notation as \((-\infty, 16]\). Matching our results with the provided options, we find that the domain is all real numbers and the range is \(\{y \mid y \leq 16\}\), which corresponds to option C.
Examples
Quadratic functions are widely used in physics and engineering to model projectile motion. For example, if you throw a ball into the air, its height as a function of time can be modeled by a downward-opening parabola. By determining the domain (time the ball is in the air) and the range (the minimum and maximum heights the ball reaches), you can predict when and where the ball will land. Mastering these concepts is essential for understanding motion, optimization in business, and various engineering design tasks.