Domain And Range Of The Parabola F(x) = -3x² + 12x - 7

When exploring quadratic functions, understanding the domain and range is crucial for grasping their behavior and characteristics. In this article, we will delve into how to determine the domain and range of a given parabola, specifically the function f(x) = -3x² + 12x - 7. This function represents a parabola that opens downwards, and we will dissect its properties to accurately define its domain and range. The domain of a function encompasses all possible input values (x-values) for which the function is defined, while the range includes all possible output values (y-values or f(x) values) that the function can produce. Determining these aspects provides a comprehensive understanding of the parabola's extent and behavior on the coordinate plane.

1. Understanding the Domain of a Parabola

When discussing the domain of a parabola, it's essential to recognize the nature of quadratic functions. Quadratic functions, represented by the general form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, are defined for all real numbers. This characteristic stems from the fact that squaring any real number and performing addition or subtraction operations will always result in a real number. Therefore, there are no restrictions on the values of 'x' that can be inputted into the function. In the context of our specific function, f(x) = -3x² + 12x - 7, we can substitute any real number for 'x' and obtain a valid output.

This inherent property of quadratic functions means that the domain of any parabola, including the one we're examining, is the set of all real numbers. Mathematically, this is expressed as (-∞, ∞), indicating that 'x' can take on any value from negative infinity to positive infinity. Understanding this fundamental aspect of parabolas simplifies the process of analyzing their behavior and characteristics. It allows us to focus on other properties, such as the range, vertex, and axis of symmetry, without worrying about input restrictions. Knowing that the domain is all real numbers provides a solid foundation for further exploration and analysis of the quadratic function.

2. Finding the Range of the Parabola

The range of a parabola, which represents the set of all possible output values (y-values), is intrinsically linked to the parabola's vertex and its direction of opening. The vertex, being the highest or lowest point on the parabola, plays a pivotal role in determining the range. When a parabola opens upwards (a > 0), the vertex represents the minimum point, and the range includes all y-values greater than or equal to the y-coordinate of the vertex. Conversely, when a parabola opens downwards (a < 0), the vertex is the maximum point, and the range consists of all y-values less than or equal to the y-coordinate of the vertex. For our given function, f(x) = -3x² + 12x - 7, the coefficient of the x² term is -3, indicating that the parabola opens downwards. This means that the vertex will be the highest point on the parabola, and the range will include all y-values less than or equal to the y-coordinate of the vertex.

To determine the range, we first need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b / 2a, where 'a' and 'b' are the coefficients of the quadratic equation. In our case, a = -3 and b = 12, so the x-coordinate of the vertex is x = -12 / (2 * -3) = 2. Next, we substitute this x-value back into the function to find the y-coordinate of the vertex: f(2) = -3(2)² + 12(2) - 7 = -12 + 24 - 7 = 5. Therefore, the vertex of the parabola is (2, 5). Since the parabola opens downwards, the y-coordinate of the vertex represents the maximum value of the function. This means that the range of the function includes all y-values less than or equal to 5. Mathematically, this is expressed as f(x) ≤ 5. Understanding this process allows us to accurately determine the range of any parabola, given its equation.

3. Step-by-Step Calculation of the Range

To accurately determine the range of the parabola defined by f(x) = -3x² + 12x - 7, we need to follow a step-by-step calculation process that focuses on finding the vertex. The vertex is the crucial point that dictates the range, as it represents either the maximum or minimum value of the function. Since the coefficient of the x² term is negative (-3), we know that the parabola opens downwards, indicating that the vertex will be the maximum point. This means the range will include all y-values less than or equal to the y-coordinate of the vertex.

3.1. Finding the x-coordinate of the vertex

The first step is to find the x-coordinate of the vertex. This can be calculated using the formula x = -b / 2a, where 'a' and 'b' are the coefficients from the quadratic equation in the form f(x) = ax² + bx + c. In our case, a = -3 and b = 12. Substituting these values into the formula, we get: x = -12 / (2 * -3) = -12 / -6 = 2. This tells us that the x-coordinate of the vertex is 2.

3.2. Calculating the y-coordinate of the vertex

Next, we need to find the y-coordinate of the vertex. This is done by substituting the x-coordinate we just found (x = 2) back into the original function: f(2) = -3(2)² + 12(2) - 7. Performing the calculations, we get: f(2) = -3(4) + 24 - 7 = -12 + 24 - 7 = 5. So, the y-coordinate of the vertex is 5. This means the vertex of the parabola is at the point (2, 5).

3.3. Determining the range based on the vertex

Now that we have the vertex (2, 5) and know that the parabola opens downwards, we can determine the range. Since the vertex is the maximum point, the range will include all y-values less than or equal to the y-coordinate of the vertex. Therefore, the range of the function f(x) = -3x² + 12x - 7 is f(x) ≤ 5. This means that the function will never output a value greater than 5. Understanding these steps allows us to confidently find the range of any quadratic function by focusing on the properties and calculations related to the vertex.

4. Visualizing the Parabola and its Range

Visualizing the parabola helps in reinforcing our understanding of its domain and range. When we graph the function f(x) = -3x² + 12x - 7, we observe a curve that opens downwards, confirming our earlier analysis based on the negative coefficient of the x² term. The vertex, which we calculated to be at the point (2, 5), is clearly the highest point on the graph. This visual representation vividly demonstrates that the function's output values (y-values) will never exceed 5. The parabola extends downwards indefinitely, indicating that there is no lower bound to the y-values, but it is capped at the vertex.

If you were to draw a horizontal line at y = 5, you would see that this line touches the parabola at its highest point (the vertex) and that the entire parabola lies below this line. This visual confirmation aligns perfectly with our calculated range of f(x) ≤ 5. Furthermore, the graph extends infinitely to the left and right, reaffirming that the domain is all real numbers. No matter what x-value we choose, there will always be a corresponding point on the parabola. This visualization is a powerful tool for solidifying the concepts of domain and range, making it easier to apply these principles to other quadratic functions. By understanding how the graph relates to the algebraic representation, we can develop a more intuitive grasp of the behavior of parabolas and their properties.

5. Conclusion: Domain and Range of f(x) = -3x² + 12x - 7

In summary, after a comprehensive analysis of the quadratic function f(x) = -3x² + 12x - 7, we have determined both its domain and range. The domain, which represents all possible input values (x-values), is the set of all real numbers. This is because quadratic functions are defined for any real number input, and there are no restrictions on the values that 'x' can take. This means that we can substitute any real number into the function, and it will produce a valid output. This fundamental property is consistent across all parabolas, making the domain a straightforward aspect to identify.

The range, representing all possible output values (y-values), is determined by the vertex of the parabola and its direction of opening. Since the coefficient of the x² term is negative (-3), the parabola opens downwards, indicating that the vertex is the maximum point. We calculated the vertex to be at the point (2, 5), meaning the maximum y-value the function can achieve is 5. Therefore, the range includes all y-values less than or equal to 5, which is expressed as f(x) ≤ 5. This means the function will never output a value greater than 5, and all output values will be 5 or less. Understanding how to find the vertex and relate it to the direction of opening is crucial for accurately determining the range of any quadratic function.

Therefore, the correct answer is: Domain is all real numbers, and the range is f(x) ≤ 5. This conclusion is supported by our step-by-step calculations, the properties of quadratic functions, and the visual representation of the parabola. This detailed exploration provides a solid understanding of how to determine the domain and range of quadratic functions, a fundamental skill in algebra and calculus.