Problem 58 - 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 \((-5,11),\) opens down.

Step-by-Step Solution

Answer

Domain is all real numbers; range is \((-\infty, 11]\).

Step 1: Understand the Vertex Form of a Quadratic Function

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 and \(a\) determines whether the parabola opens upward (\(a > 0\)) or downward (\(a < 0\)). In this case, the vertex is \((-5, 11)\) and the parabola opens downward.

Step 2: Determine the Domain of the Function

The domain of any quadratic function is always all real numbers because there are no restrictions on the values that \(x\) can take. Therefore, the domain is \( (-\infty, \infty) \).

Step 3: Determine the Range of the Function

Since the parabola opens downward, the maximum value of the function is at the vertex. Therefore, the range of the function begins from minus infinity up to and including the \(y\)-coordinate of the vertex, which is 11. Thus, the range is \(( -\infty, 11 ]\).

Key Concepts

Vertex Form

The vertex form of a quadratic function allows us to understand the key elements of a parabola with ease. It is expressed as \( f(x) = a(x - h)^2 + k \), where the pair \((h, k)\) represents the vertex of the parabola. The vertex is a crucial point, as it is either the highest or lowest point on the graph, depending on the direction the parabola opens. In this specific exercise, the vertex is given as \((-5, 11)\). This tells us exactly where the top (or the bottom) of the parabola lies on the graph. Knowing the vertex helps us to determine other important properties of the quadratic function, such as its range. The term \(a\) in the equation plays a crucial role in determining the parabola's direction of opening, which we will explore further below. In summary, mastering the vertex form is essential for understanding and manipulating quadratic functions effectively.

Domain and Range

In mathematics, the domain of a function refers to all the possible input values (or \(x\)-values) that the function can take. For quadratic functions like the ones described by their vertex form, the domain is all real numbers, represented as \((-\infty, \infty)\). This means \(x\) can be any value from negative infinity to positive infinity without any restriction.

On the other hand, the range of a function describes all the possible output values (or \(y\)-values). In the case of a parabola that opens downward, the maximum \(y\)-value is found at the vertex. From the exercise, we know the vertex is at \(( -5, 11)\), which means the maximum \(y\)-value is 11. Since the parabola opens downward, it stretches indefinitely towards negative infinity, leading us to express the range as \(( -\infty, 11 ]\). Understanding both domain and range gives us a complete picture of the function’s behavior and the limits within which it operates.

Parabola Opens Direction

The direction in which a parabola opens is determined by the sign of the coefficient \(a\) in the quadratic function expressed in vertex form. When \(a > 0\), the parabola opens upwards like a smile. Conversely, when \(a < 0\), the parabola opens downward like a frown. This is a key element of understanding quadratic functions as it affects both the range and the maximum or minimum point of the function.

In the given exercise, it is specified that the parabola opens downward. This immediately tells us that \(a\) is less than zero, although the exact value of \(a\) is not necessary to know for specifying the direction alone. The significance of the direction is primarily found in the range, as a downward-opening parabola reaches its maximum point at the vertex and then decreases infinitely. Therefore, knowing the direction aids in finding the maximum values for the function, as well as visualizing how the graph extends within the coordinate system.