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

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((- fty, 11]\).

Step 1: Understanding the Vertex

The vertex of the quadratic function is given as \((-5, 11)\). This means that the graph of the quadratic reaches its highest or lowest point at this coordinate. Since the graph opens down, the vertex represents the maximum point of the graph.

Step 2: Identifying the Direction of Opening

The problem states that the graph "opens down." This implies that the quadratic function has a parabola shape with the vertex at the top. This affects the range because the vertex is at the maximum value, and all other points on the graph have lesser y-values than the vertex.

Step 3: Determining the Domain

For any quadratic function, the domain is all real numbers. This is because you can input any real number into the function and obtain a corresponding y-value.

Step 4: Determining the Range

Given that the vertex is at the maximum point \((11)\) and the graph opens down, the range of the function includes all y-values less than or equal to 11. Therefore, the range is \((-fty, 11]\).

Key Concepts

Vertex of Parabola

In a quadratic function, one important feature is the vertex of the parabola. The vertex is essentially the tip or turning point of the parabola. It can be either the highest point or the lowest point on the curve, depending on the direction in which the parabola opens.

For instance, in the quadratic function presented in the exercise, the vertex is given as \((-5, 11)\). This tells us that at \(x = -5\), the function reaches a vital turning point. Since the parabola opens downwards, this vertex represents a maximum point.

To find this vertex when only the quadratic equation is given, you can use the formula \(x = -\frac{b}{2a}\) to find the x-coordinate. Then, substitute this x-value back into the equation to find the corresponding y-coordinate. Understanding the vertex's role is crucial because it helps you determine the range and sketch the basic shape of the graph.

Domain of Quadratic Functions

The domain of a quadratic function is all about understanding the set of possible input values, or x-values, that you can use in the function. For any quadratic function, the domain is always all real numbers.

This is because there are no restrictions on what values x can take. No matter what x-value you plug into the quadratic equation, you will always get a valid y-value as an output. In mathematical terms, we write this as \((-\infty, \infty)\).

• This unbounded nature makes quadratic functions incredibly versatile.
• It allows them to represent a wide variety of real-world scenarios.

For example, you could substitute any real number into the equation without worrying about undefined regions, unlike functions with fractions or square roots, which can have restrictions.

Range of Quadratic Functions

The range of a quadratic function concerns the possible output values or y-values. Finding the range is crucial, as it tells us the limits within which the function's graph lies, vertically.

In the given exercise, the vertex \((11)\) signifies the highest y-value since the parabola opens down. Therefore, every point on the parabola has a y-value less than or equal to 11.

• This creates a range which is written as \((-\infty, 11]\).
• Thus, it includes all real numbers less than or equal to 11.

A parabola that opens upwards would have its range starting from the vertex going up to infinity. Thus, understanding the direction of on which the parabola opens is key to determining the correct range.