Key Features Of The Quadratic Function F(x)=-0.3(x-5)^2+5

In the realm of mathematics, understanding the characteristics of functions is paramount. Among these functions, quadratic functions hold a significant position due to their unique parabolic nature and wide-ranging applications. In this article, we will delve into the intricacies of the quadratic function f(x)=-0.3(x-5)^2+5, dissecting its key features to gain a comprehensive understanding of its behavior and graphical representation. By examining its domain, axis of symmetry, and other essential aspects, we will unravel the properties that make this function so intriguing.

Before we embark on the journey of exploring the specifics of f(x)=-0.3(x-5)^2+5, let's establish a firm grasp on the fundamentals of quadratic functions. A quadratic function is a polynomial function of degree two, generally expressed in the form f(x)=ax^2+bx+c, where a, b, and c are constants and a≠0. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The parabola's orientation (whether it opens upwards or downwards) and its overall shape are determined by the coefficient a. When a>0, the parabola opens upwards, resembling a smile, while when a<0, it opens downwards, resembling a frown. The vertex of the parabola, which is the point where the curve changes direction, plays a crucial role in defining the function's behavior. The x-coordinate of the vertex is given by -b/2a, and the y-coordinate is the value of the function at that x-coordinate. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. It is represented by the equation x=-b/2a. The domain of a quadratic function is the set of all real numbers, meaning that the function is defined for any input value. However, the range, which is the set of all possible output values, is restricted by the vertex. If the parabola opens upwards, the range consists of all values greater than or equal to the y-coordinate of the vertex, while if it opens downwards, the range consists of all values less than or equal to the y-coordinate of the vertex. Understanding these fundamental concepts is essential for analyzing and interpreting the behavior of any quadratic function, including the one we are about to explore.

Now, let's turn our attention to the specific quadratic function f(x)=-0.3(x-5)^2+5 and dissect its key features. This function is presented in vertex form, which provides valuable insights into its characteristics. The vertex form of a quadratic function is given by f(x)=a(x-h)^2+k, where (h,k) represents the vertex of the parabola. In our case, a=-0.3, h=5, and k=5. From this form, we can immediately identify several key features. First, the coefficient a=-0.3 is negative, indicating that the parabola opens downwards. This means that the function has a maximum value at its vertex. Second, the vertex is located at the point (5,5). This point represents the highest point on the parabola, and its coordinates provide crucial information about the function's behavior. Third, the axis of symmetry is the vertical line that passes through the vertex, which is given by the equation x=5. This line divides the parabola into two symmetrical halves, reflecting the function's symmetry. Fourth, the domain of the function is the set of all real numbers, as it is a quadratic function and is defined for any input value. Fifth, the range of the function is the set of all values less than or equal to the y-coordinate of the vertex, which is 5. This is because the parabola opens downwards, and the vertex represents the maximum value of the function. In summary, the key features of f(x)=-0.3(x-5)^2+5 include a downward-opening parabola, a vertex at (5,5), an axis of symmetry at x=5, a domain of all real numbers, and a range of all values less than or equal to 5. These features provide a comprehensive understanding of the function's behavior and graphical representation.

The domain of a function is the set of all possible input values (often represented by x) for which the function is defined. In simpler terms, it's the range of x-values that you can plug into the function without encountering any mathematical impossibilities, such as division by zero or taking the square root of a negative number. For the quadratic function f(x) = -0.3(x - 5)^2 + 5, the domain is particularly straightforward. Quadratic functions, by their very nature, are defined for all real numbers. This means there are no restrictions on the values you can substitute for x. You can input any real number, whether it's positive, negative, zero, an integer, a fraction, or an irrational number, and the function will produce a valid output. To express this mathematically, we say that the domain of f(x) is the set of all real numbers, which can be written as {x | x is a real number} or, in interval notation, as (-∞, ∞). This vast domain is a characteristic feature of quadratic functions and stems from the fact that the squaring operation and the other arithmetic operations involved (multiplication, subtraction, and addition) are all defined for all real numbers. Therefore, when analyzing the function f(x) = -0.3(x - 5)^2 + 5, you can confidently consider any real number as a potential input, knowing that the function will yield a corresponding output value. This broad domain is one of the fundamental aspects that distinguishes quadratic functions and contributes to their wide applicability in various mathematical and real-world contexts.

The axis of symmetry is a crucial characteristic of a parabola, the U-shaped curve that represents the graph of a quadratic function. It is an imaginary vertical line that cuts the parabola into two perfectly symmetrical halves. Imagine folding the parabola along this line; the two halves would match up exactly. The axis of symmetry not only provides a visual sense of the parabola's symmetry but also plays a key role in determining its vertex, which is the point where the parabola changes direction. For a quadratic function in the standard form f(x) = ax^2 + bx + c, the axis of symmetry is given by the vertical line x = -b / 2a. However, our function, f(x) = -0.3(x - 5)^2 + 5, is presented in vertex form, which makes identifying the axis of symmetry even easier. The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. In this form, the axis of symmetry is simply the vertical line x = h. Comparing our function to the vertex form, we can see that h = 5. Therefore, the axis of symmetry for f(x) = -0.3(x - 5)^2 + 5 is the vertical line x = 5. This means that the parabola is symmetrical about the line that passes through the point x = 5 on the x-axis. The axis of symmetry is a valuable tool for graphing the parabola, as it helps to determine the parabola's position and orientation on the coordinate plane. It also provides insight into the function's behavior, as the parabola's values increase or decrease symmetrically on either side of the axis of symmetry.

In conclusion, by meticulously examining the quadratic function f(x)=-0.3(x-5)^2+5, we have unveiled its key features, including its domain, axis of symmetry, and other significant characteristics. We established that the domain of the function encompasses all real numbers, signifying its applicability across a wide range of input values. Furthermore, we identified the axis of symmetry as the vertical line x=5, which divides the parabola into two symmetrical halves, providing valuable insights into its graphical representation. These features, along with the function's vertex form, offer a comprehensive understanding of its behavior and properties. By mastering the analysis of quadratic functions like this one, we equip ourselves with essential tools for tackling mathematical challenges and real-world applications where these functions play a crucial role. The exploration of quadratic functions not only enhances our mathematical proficiency but also fosters a deeper appreciation for the elegance and utility of these fundamental mathematical constructs.

Key Features of the Quadratic Function f(x)=-0.3(x-5)^2+5