Calculate Values Of Quadratic Function F(x) = 6x² - 6x + 6
This article explores the quadratic function f(x) = 6x² - 6x + 6 and demonstrates the calculation of its values for specific inputs. We will delve into the process of substituting different values for 'x' into the function and simplifying the expression to find the corresponding output values. This exercise provides a practical understanding of how quadratic functions behave and how their outputs change based on the input values. By calculating f(-2), f(-1), f(0), f(1), and f(2), we gain insights into the function's behavior over a range of inputs, which is crucial for various applications in mathematics, physics, and engineering.
Understanding Quadratic Functions
Before we dive into the calculations, let's briefly discuss quadratic functions. A quadratic function is a polynomial function of the form f(x) = ax² + 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. The coefficients 'a', 'b', and 'c' determine the shape and position of the parabola. In our case, f(x) = 6x² - 6x + 6, we have a = 6, b = -6, and c = 6. Understanding these coefficients is crucial for analyzing the function's behavior and predicting its outputs for different inputs. For example, the positive value of 'a' indicates that the parabola opens upwards, meaning it has a minimum point. The axis of symmetry, a vertical line that divides the parabola into two symmetrical halves, can be found using the formula x = -b / 2a. This value will give us the x-coordinate of the vertex, the minimum point of the parabola. By substituting different values of 'x', we can map out the parabola and understand the relationship between the input and output of the function. This foundation is essential for accurately calculating and interpreting the values we will find in the following sections.
Calculating f(-2)
To calculate f(-2), we substitute x = -2 into the function f(x) = 6x² - 6x + 6. This means replacing every instance of 'x' in the function's equation with '-2'. The calculation proceeds as follows:
f(-2) = 6(-2)² - 6(-2) + 6
First, we evaluate the exponent: (-2)² = 4.
f(-2) = 6(4) - 6(-2) + 6
Next, we perform the multiplications:
f(-2) = 24 + 12 + 6
Finally, we add the terms together:
f(-2) = 42
Therefore, the value of the function at x = -2 is 42. This single calculation illustrates the core process of evaluating a function at a specific point. We replace the variable with the given value, follow the order of operations (PEMDAS/BODMAS), and simplify to obtain the result. This process is fundamental in mathematics and is used extensively in various fields that rely on mathematical modeling. Understanding how to perform this substitution and simplification is crucial for working with functions and interpreting their behavior. The value of 42 tells us that the point (-2, 42) lies on the graph of the quadratic function. This process will be repeated for other values of 'x' to build a comprehensive understanding of the function's behavior.
Calculating f(-1)
Now, let's calculate f(-1). We substitute x = -1 into the function f(x) = 6x² - 6x + 6:
f(-1) = 6(-1)² - 6(-1) + 6
Again, we start by evaluating the exponent: (-1)² = 1.
f(-1) = 6(1) - 6(-1) + 6
Next, perform the multiplications:
f(-1) = 6 + 6 + 6
Finally, add the terms:
f(-1) = 18
Thus, f(-1) = 18. This result, along with the previous calculation of f(-2), begins to give us a sense of the function's increasing nature as we move from negative values towards the vertex of the parabola. Each calculation provides a specific point on the graph of the function, and these points help to visualize the overall shape and trend of the curve. Understanding this relationship between input and output is critical for applications such as optimization problems, where the goal is to find the minimum or maximum value of a function within a given range. The process of substituting and simplifying, as demonstrated here, is a fundamental skill in mathematics and is essential for analyzing and manipulating functions.
Calculating f(0)
To find f(0), we substitute x = 0 into the function f(x) = 6x² - 6x + 6:
f(0) = 6(0)² - 6(0) + 6
Evaluating the terms involving zero:
f(0) = 6(0) - 6(0) + 6
f(0) = 0 - 0 + 6
Therefore, f(0) = 6. This calculation is particularly interesting because it gives us the y-intercept of the parabola. The y-intercept is the point where the graph of the function intersects the y-axis, and it occurs when x = 0. In this case, the y-intercept is (0, 6). The y-intercept is a key feature of any function, as it provides a direct indication of the function's value when the input is zero. It is also a crucial point for sketching the graph of the function and understanding its behavior near the y-axis. In the context of real-world applications, the y-intercept can represent an initial condition or a starting value of a system being modeled by the function.
Calculating f(1)
Let's calculate f(1) by substituting x = 1 into the function f(x) = 6x² - 6x + 6:
f(1) = 6(1)² - 6(1) + 6
First, we evaluate the exponent: (1)² = 1.
f(1) = 6(1) - 6(1) + 6
Next, perform the multiplications:
f(1) = 6 - 6 + 6
Finally, add the terms:
f(1) = 6
Thus, f(1) = 6. This result is significant because it reveals a symmetry in the function's values. We found that f(0) = 6 and f(1) = 6, indicating that these points have the same y-value. This suggests that the axis of symmetry of the parabola likely lies between x = 0 and x = 1. The symmetry of a parabola is a fundamental property that simplifies its analysis and allows for efficient graphing. By knowing the axis of symmetry, we can predict the behavior of the function on one side of the parabola based on its behavior on the other side. This symmetry also plays a crucial role in optimization problems, as the vertex, which represents the minimum or maximum value of the function, lies on the axis of symmetry.
Calculating f(2)
Finally, let's calculate f(2). We substitute x = 2 into the function f(x) = 6x² - 6x + 6:
f(2) = 6(2)² - 6(2) + 6
Evaluate the exponent: (2)² = 4.
f(2) = 6(4) - 6(2) + 6
Perform the multiplications:
f(2) = 24 - 12 + 6
Add the terms:
f(2) = 18
Therefore, f(2) = 18. Comparing this result with our previous calculations, we observe that f(-1) = 18 and f(2) = 18. This further reinforces the symmetry of the parabola. The equal values at x = -1 and x = 2 indicate that these points are equidistant from the axis of symmetry. By knowing these points, we can more accurately sketch the parabola and understand its overall shape and position. The increasing value of the function as we move away from the axis of symmetry is characteristic of a parabola that opens upwards. This final calculation provides a complete set of values that allow us to visualize the function's behavior over the given range of inputs. These values can be plotted on a graph to create a visual representation of the quadratic function and its properties.
Summary of Calculated Values
Here’s a summary of the calculated values for the function f(x) = 6x² - 6x + 6:
- f(-2) = 42
- f(-1) = 18
- f(0) = 6
- f(1) = 6
- f(2) = 18
These values provide a comprehensive understanding of the function's behavior over the range x = -2 to x = 2. The symmetry observed in the values, with f(-1) = f(2) = 18 and f(0) = f(1) = 6, highlights the parabolic nature of the function. The minimum value of the function occurs between x = 0 and x = 1, as these points have the lowest calculated value of 6. This indicates that the vertex of the parabola lies within this interval. The calculated values can be used to sketch the graph of the function, providing a visual representation of its behavior. By plotting these points and connecting them with a smooth curve, we can see the U-shaped form of the parabola and understand its key features, such as the vertex and the axis of symmetry. This exercise demonstrates the importance of evaluating functions at specific points to understand their behavior and characteristics. The calculated values serve as a foundation for further analysis and applications of the quadratic function.
Conclusion
In conclusion, we have successfully calculated the values of the quadratic function f(x) = 6x² - 6x + 6 for x = -2, -1, 0, 1, and 2. These calculations provide valuable insights into the function's behavior and characteristics. The process involved substituting each value of 'x' into the function's equation and simplifying the expression to obtain the corresponding output value. The results demonstrated the parabolic nature of the function and revealed its symmetry. The calculated values can be used to plot the graph of the function, which visually represents the relationship between the input and output values. This exercise is a fundamental step in understanding quadratic functions and their applications in various fields. The ability to evaluate functions at specific points is a crucial skill in mathematics and is essential for solving a wide range of problems. By understanding the behavior of quadratic functions, we can model and analyze real-world phenomena that exhibit parabolic relationships, such as the trajectory of a projectile or the shape of a suspension bridge. The knowledge gained from this exercise provides a solid foundation for further exploration of mathematical concepts and their applications.
