Determining The Domain And Range Of Quadratic Functions With Vertex At (-5, -7)

#Domain and range are fundamental concepts in mathematics, especially when dealing with functions. Today, we're diving deep into understanding the domain and range of a quadratic function, specifically one where the vertex and direction of opening are known. This article will focus on how to determine the domain and range when given that the vertex is (-5, -7) and the parabola opens upwards. Let's explore this in detail.

Understanding Quadratic Functions

Before we jump into the specifics, let's establish a clear understanding of quadratic functions. A quadratic function is a polynomial function of degree two, generally represented in the form: f(x) = ax^2 + bx + c, where a, b, and c are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that can open either upwards or downwards.

The direction in which the parabola opens is determined by the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. The vertex of the parabola is a crucial point, representing either the minimum value (if the parabola opens upwards) or the maximum value (if the parabola opens downwards) of the function. The vertex form of a quadratic function, f(x) = a(x - h)^2 + k, is particularly useful because it directly reveals the vertex (h, k).

Key Characteristics of Quadratic Functions

Understanding the key characteristics helps greatly in determining both the domain and the range. These characteristics include:

• Vertex: This is the point where the parabola changes direction. It's the minimum point for parabolas opening upwards and the maximum point for those opening downwards.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = h, where h is the x-coordinate of the vertex.
• Direction of Opening: Determined by the sign of the coefficient 'a'. A positive 'a' means the parabola opens upwards, and a negative 'a' means it opens downwards.

By understanding these characteristics, we can effectively analyze and determine the domain and range of any quadratic function. Now, let's apply this knowledge to our specific problem.

Domain of a Quadratic Function

In mathematical terms, the domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For quadratic functions, determining the domain is straightforward because there are no restrictions on the x-values that can be inputted. Unlike rational functions (which have denominators that cannot be zero) or square root functions (which require non-negative values under the square root), quadratic functions are defined for all real numbers.

Why Quadratic Functions Have a Domain of All Real Numbers

The reason quadratic functions have a domain of all real numbers lies in their structure. The general form of a quadratic function, f(x) = ax^2 + bx + c, involves only addition, subtraction, and multiplication operations on the variable x. These operations are defined for every real number. There are no divisions or radicals that could potentially lead to undefined values. For example, squaring any real number results in a real number, and multiplying or adding real numbers will always yield real numbers.

Expressing the Domain in Interval Notation

To express the domain in interval notation, we represent all possible x-values. Since quadratic functions are defined for all real numbers, the domain spans from negative infinity to positive infinity. In interval notation, this is written as (-∞, ∞). This notation indicates that the domain includes all real numbers without any breaks or gaps.

Applying to Our Case

Given our quadratic function with a vertex at (-5, -7) and opening upwards, the domain remains all real numbers. The specific characteristics of the parabola, such as the vertex or direction of opening, do not affect the domain. Regardless of these factors, the function is always defined for any x-value we choose. Therefore, for our specific quadratic function, the domain is (-∞, ∞).

Thus, the domain of the given quadratic function is (-∞, ∞), a fundamental property of all quadratic functions.

Range of a Quadratic Function

The range of a function is the set of all possible output values (y-values) that the function can produce. Unlike the domain, the range of a quadratic function is restricted because the parabola has either a minimum or a maximum value, depending on its direction of opening. In our case, the parabola opens upwards, meaning it has a minimum value. This minimum value is the y-coordinate of the vertex.

How the Vertex Affects the Range

The vertex is the turning point of the parabola. For a parabola that opens upwards, the vertex represents the lowest point on the graph. Therefore, the y-coordinate of the vertex is the minimum y-value the function can achieve. All other y-values will be greater than this minimum.

In our scenario, the vertex is given as (-5, -7). This means the minimum y-value of our function is -7. Since the parabola opens upwards, the function's y-values will extend from -7 to positive infinity. Thus, the range is all real numbers greater than or equal to -7.

Expressing the Range in Interval Notation

To express the range in interval notation, we use the minimum y-value as the lower bound and positive infinity as the upper bound. Since the function includes the minimum value (-7), we use a square bracket to indicate that -7 is part of the range. Infinity is always represented with a parenthesis because it is not a specific number and cannot be included.

Therefore, the range of our quadratic function is [-7, ∞). This interval notation signifies that the function's output values start at -7 and extend indefinitely upwards.

Understanding Different Scenarios

It's important to note that if the parabola opened downwards, the vertex would represent the maximum point, and the range would be from negative infinity up to the y-coordinate of the vertex. For example, if a parabola opened downwards with a vertex at (2, 5), the range would be (-∞, 5].

Conclusion on Range

Given the vertex of our quadratic function is (-5, -7) and it opens upwards, the range is [-7, ∞). This means the function's output values are all real numbers greater than or equal to -7. The vertex and the direction of opening are crucial in determining the range of a quadratic function.

Combining Domain and Range for the Given Quadratic Function

Now that we have determined both the domain and range of our quadratic function, let’s combine our findings for a comprehensive understanding. The quadratic function in question has a vertex at (-5, -7) and opens upwards. This information is sufficient to define both the input values (domain) and the output values (range) of the function.

Summarizing the Domain

As we previously discussed, the domain of a quadratic function encompasses all possible x-values for which the function is defined. Quadratic functions do not have any inherent restrictions on the input values, meaning any real number can be used as an input. Therefore, the domain is all real numbers.

In interval notation, the domain is represented as (-∞, ∞). This notation signifies that the function is defined for every real number, extending from negative infinity to positive infinity without any breaks or interruptions.

For our specific quadratic function, the domain remains (-∞, ∞). The position of the vertex or the direction in which the parabola opens does not impact the domain. Regardless of these characteristics, the function will always accept any real number as an input.

Summarizing the Range

The range, on the other hand, is affected by the vertex and the direction of opening. Since our parabola opens upwards, it has a minimum value, which is the y-coordinate of the vertex. The vertex is given as (-5, -7), so the minimum y-value is -7. This means the function’s output values will be -7 or greater.

The range includes all y-values from -7 upwards to positive infinity. In interval notation, this is represented as [-7, ∞). The square bracket indicates that -7 is included in the range, while the parenthesis indicates that infinity is not a specific number and is not included.

Putting It All Together

To summarize, for the quadratic function with a vertex at (-5, -7) that opens upwards:

• Domain: (-∞, ∞)
• Range: [-7, ∞)

This comprehensive view allows us to understand the full scope of the function's behavior. The domain tells us what inputs are valid, and the range tells us what outputs to expect. This information is crucial for graphing the function, solving equations involving the function, and understanding its real-world applications.

Implications and Applications

Understanding the domain and range is not just a theoretical exercise; it has practical implications. For instance, in modeling real-world scenarios with quadratic functions (such as projectile motion or optimization problems), knowing the domain and range helps ensure that the solutions are meaningful within the context of the problem.

For example, if a quadratic function models the height of a projectile over time, the domain might be restricted to non-negative time values, and the range would provide information about the maximum height the projectile reaches.

Conclusion

In conclusion, determining the domain and range of a quadratic function involves understanding its fundamental characteristics, particularly the vertex and the direction of opening. The domain of a quadratic function is always all real numbers, expressed as (-∞, ∞). The range, however, depends on the vertex and the direction of opening. For a parabola that opens upwards, the range is from the y-coordinate of the vertex to positive infinity, represented as [k, ∞), where k is the y-coordinate of the vertex.

In our specific case, with a vertex at (-5, -7) and the parabola opening upwards, the domain is (-∞, ∞), and the range is [-7, ∞). This detailed analysis provides a complete picture of the function’s input and output values, essential for further mathematical analysis and real-world applications.

Understanding these concepts is critical for students and professionals alike in various fields, from mathematics and physics to engineering and economics. By mastering the domain and range of quadratic functions, one can effectively analyze and solve a wide array of problems.