Problem 59 - College Algebra

Question

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. Vertex \((-100,100),\) opens up.

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \([100, \infty)\).

Step 1: Understanding the Vertex Form

The vertex form of a quadratic function is given by: \[ f(x) = a(x-h)^2 + k \] where \((h, k)\) is the vertex of the parabola. In this problem, the vertex is given as \((-100, 100)\). Therefore, we have \(h = -100\) and \(k = 100\). The function opens up, indicating \(a > 0\).

Step 2: Determining the Domain

The domain of a quadratic function is the set of all possible input values \(x\). For any quadratic function of the form \(f(x) = ax^2 + bx + c\), the domain is all real numbers because a parabola extends infinitely left and right on a graph. Thus, the domain is:\[ (-\infty, \infty) \]

Step 3: Determining the Range

The range of a quadratic function depends on the vertex and the direction it opens. Since the parabola opens upward, the minimum value of the function is at the vertex \(k = 100\). Thus, the range includes all values from \(100\) upwards. Therefore, the range is:\[ [100, \infty) \]

Key Concepts

Vertex Form

The vertex form of a quadratic function is one of the most insightful ways to understand a parabola. Mathematically, it is expressed as \( f(x) = a(x-h)^2 + k \). Here, \((h, k)\) represents the vertex of the parabola. The value "\(a\)" determines how wide or narrow the parabola is, as well as its direction of opening. In our exercise, the vertex provided is \((-100, 100)\), which points directly to \(h = -100\) and \(k = 100\).
This equation form is particularly useful because it readily shows the maximum or minimum point of the parabola, which is the vertex. This fact helps in quickly finding the range of the function, especially if we know which direction the parabola opens.

Domain and Range

For any quadratic function, the domain is the set of all possible values that \(x\) can take. Quadratics are defined for all real numbers; therefore, their domain is always \((-\infty, \infty)\).

However, the range is different for every function as it depends on both the vertex and the direction of the parabola. In our problem, since the vertex is \(100\) and the parabola opens upwards, the function's minimum value is \(100\). Thus, the range of this quadratic function starts from \(100\) and goes to infinity, written as \([100, \infty)\). This parameter is crucial because it tells us the span of possible output values the quadratic function can have.

Parabola Orientation

The direction a parabola opens is determined by the coefficient \(a\) in the quadratic function. If \(a > 0\), the parabola opens upwards; conversely, if \(a < 0\), it opens downwards. This positional feature affects the range of the function significantly.

For the example function in our exercise, the parabola is said to open upwards, which implies that \(a > 0\). This orientation indicates that the parabola has a minimum point at its vertex. Thus, the range starts at this minimum point and extends to positive infinity. Recognizing a parabola's orientation helps in visually interpreting and sketching the graph, making analysis much easier and intuitive for students.