Problem 57 - College Algebra with Corequisite Support

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 (-1,2) opens down.

Step-by-Step Solution

Answer

Domain: \\((-\infty, \infty)\\); Range: \\(( -\infty, 2])\\).

Step 1: Understanding the Vertex

The problem states that the vertex of the quadratic function is at point (-1,2). This means the maximum or minimum point of the function is located at these coordinates.

Step 2: Identifying the Direction

The graph opens downward, indicating that the quadratic function is in the form of a downward-facing parabola. This implies the highest point of the graph is at the vertex, making it a maximum point.

Step 3: Determining the Domain

Quadratic functions have a domain of all real numbers because they continue indefinitely from left to right on the x-axis. Thus, the domain is \((-\infty, \infty)\).

Step 4: Determining the Range

Because the parabola opens downward with its vertex at the highest point, all other points are below or at y = 2. Therefore, the range is \(( -\infty, 2])\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a specific way to write the equation of a parabola. It is given by: \[ f(x) = a(x-h)^2 + k \]where

• \( h \) and \( k \) are the coordinates of the vertex \((h, k)\).
• \( a \) determines the width and the direction (upward or downward) of the parabola.

Understanding this form is helpful as it directly provides the vertex of the parabola, allowing you to easily identify the highest or lowest point. In this exercise, the vertex is given as \((-1, 2)\). This makes interpreting the function and drawing the graph more straightforward, since the vertex form aligns with the physical position of the vertex on the graph. For our downward opening parabola, knowing the vertex lets us unravel the function properties swiftly.

Domain and Range

The domain and range are fundamental characteristics of functions that outline where the function 'lives' in the coordinate plane. For quadratic functions, the domain is always all real numbers. This is because, no matter how the parabola opens, it extends infinitely in both horizontal directions.
For the given function with vertex \((-1, 2)\) opening downwards, the range tells us the vertical span of the graph. Since the parabola opens downward from \( y = 2 \), and goes infinitely downward, our range is all the real numbers less than or equal to 2. Mathematically, this is expressed as \[ (-\infty, 2] \].