Problem 57 - 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 \((-1,2)\) opens down.

Step-by-Step Solution

Answer

Domain is all real numbers; Range is \(y \leq 2\).

Step 1: Understand the Problem

The function has a vertex at \((-1, 2)\) and opens downward, which means it is a quadratic function with a maximum point at its vertex.

Step 2: Determine the Domain

For any quadratic function, the domain is the set of all possible x-values. Since there are no restrictions on the x-values for a quadratic function, the domain is all real numbers.

Step 3: Determine the Range

Since the function opens downward, the vertex provides the highest y-value. The range consists of all y-values less than or equal to the y-coordinate of the vertex, which is 2. Therefore, the range is \(y \leq 2\).

Key Concepts

Domain and Range

When dealing with quadratic functions, it's essential to understand the concepts of domain and range as they outline the extents of the function.

The **domain** of a quadratic function is straightforward. It encompasses all real numbers, meaning there are no restrictions on the input values (x-values). Whether the graph of the quadratic function opens upwards or downwards, the x-values can take any real number. Therefore, it can be represented as:

• The domain is \( x \in \mathbb{R} \), meaning all real numbers.

The **range** of a quadratic function, on the other hand, differs based on whether the parabola opens upwards or downwards. This characteristic changes the set of possible y-values.

For a parabola that opens downward (like our example), the vertex represents the highest point on the graph. Thus, all other points have a y-value less than or equal to the y-value of the vertex. In this case, if the vertex is at \( (-1, 2) \) and the parabola opens downward, the range is:

• All y-values such that \( y \leq 2 \).

Understanding the domain and range is crucial because it allows us to visualize the full scope of the function.

Vertex of a Quadratic Function

The vertex of a quadratic function provides us with valuable information about the graph of the function. It's the most critical point in the function because it indicates either the highest or lowest point depending on the parabola's direction.

In a standard quadratic equation of the form \( y = ax^2 + bx + c \), the vertex can be found using the vertex formula \( x = -\frac{b}{2a} \). The vertex coordinates are \( (h, k) \), where \( h \) comes from the formula, and \( k \) is the y-value when you plug \( h \) back into the equation.

Once you have the vertex, you can determine important characteristics of the quadratic function:

• If the parabola opens upwards, the vertex is the lowest point, called the minimum.
• If it opens downwards, like in our example, it is the highest point, called the maximum.

Understanding the vertex helps in sketching the graph and predicting the behavior of the quadratic function in various scenarios.

Graph of Quadratic Functions

Graphing a quadratic function involves plotting its characteristic U-shape called a parabola. The shape of the parabola depends on the direction it opens, which influences the function's range and behavior.

There are several steps to visualize the graph of a quadratic function:

• **Identify the Vertex:** This is your starting point since it's the maximum or minimum point depending on the opening direction.
• **Determine the Axis of Symmetry:** A vertical line through the vertex divides the parabola into two mirror images. For our example, the axis of symmetry is \( x = -1 \).
• **Identify if the Parabola Opens Upwards or Downwards:** Check the sign of \( a \) in the equation \( y = ax^2 + bx + c \). If \( a > 0 \), it opens upwards, and if \( a < 0 \), it opens downwards.
• **Plot Additional Points:** Select x-values on either side of the vertex, calculate corresponding y-values, and plot to understand the curve's spread.

Drawing out these details will help you accurately represent and interpret quadratic functions graphically. The visual representation is powerful for predicting trends and solving quadratic-related problems more effectively.