Parabola Analysis F(x) = 100 - 9x² Finding Intercepts, Vertex, Symmetry, Domain And Range
This article delves into the characteristics of the parabola defined by the equation f(x) = 100 - 9x². We will explore key features such as the y-intercept, x-intercepts, vertex, axis of symmetry, domain, and range. By understanding these elements, we can gain a comprehensive understanding of this parabolic function and its graphical representation. This guide will provide a step-by-step approach to identifying these features, ensuring clarity and ease of comprehension for readers of all backgrounds. Whether you're a student learning about parabolas for the first time or someone seeking a refresher, this article will serve as a valuable resource.
A) Determining the Y-Intercept
The y-intercept is a crucial point on the parabola, representing the location where the curve intersects the y-axis. To find the y-intercept, we need to determine the value of the function, f(x), when x is equal to 0. This is because any point on the y-axis has an x-coordinate of 0. Substituting x = 0 into our equation, f(x) = 100 - 9x², we get:
f(0) = 100 - 9(0)² = 100 - 0 = 100
Therefore, the y-intercept is the point where the parabola crosses the y-axis, and its coordinates are (0, 100). This means that when x is 0, the function's value is 100. In graphical terms, this point is located 100 units above the origin on the y-axis. The y-intercept is a key feature of the parabola, providing a starting point for sketching the graph and understanding its vertical position. It also gives us a sense of the parabola's overall shape and direction, as it is the point where the curve begins to turn away from the y-axis. Understanding the y-intercept is fundamental to analyzing and interpreting parabolic functions.
B) Finding the X-Intercepts
To find the x-intercepts of the parabola, we need to determine the points where the curve intersects the x-axis. These points are also known as the roots or zeros of the function. At the x-intercepts, the value of the function, f(x), is equal to 0. Therefore, we need to solve the equation 100 - 9x² = 0 for x. This will give us the x-coordinates of the points where the parabola crosses the x-axis.
Let's solve the equation:
100 - 9x² = 0
First, we can rearrange the equation:
9x² = 100
Next, divide both sides by 9:
x² = 100/9
Now, take the square root of both sides:
x = ±√(100/9)
x = ±(10/3)
Thus, we have two x-intercepts: x = 10/3 and x = -10/3. These correspond to the points (10/3, 0) and (-10/3, 0) on the Cartesian plane. The x-intercepts are critical points as they define where the parabola crosses the horizontal axis. They, together with the y-intercept, provide a frame of reference that is essential in accurately graphing the parabola and analyzing its behavior. By finding the x-intercepts, we gain a deeper understanding of the function's zeros and its relationship to the x-axis. Knowing the x-intercepts also allows us to determine the intervals where the function is positive or negative, which is valuable in various applications of parabolic functions.
C) Locating the Vertex
The vertex of a parabola is the point where the curve changes direction. It is either the minimum or maximum point of the parabola, depending on whether the parabola opens upwards or downwards. For a parabola in the form f(x) = ax² + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a. In our case, f(x) = 100 - 9x², which can be rewritten as f(x) = -9x² + 0x + 100. Therefore, a = -9, b = 0, and c = 100.
Using the formula for the x-coordinate of the vertex:
x = -b / 2a = -0 / (2 * -9) = 0
To find the y-coordinate of the vertex, we substitute x = 0 back into the original equation:
f(0) = 100 - 9(0)² = 100
Therefore, the vertex of the parabola is the point (0, 100). Since the coefficient of the x² term (a) is negative (-9), the parabola opens downwards, meaning the vertex represents the maximum point of the function. The vertex plays a central role in understanding the parabola's shape and behavior. It is the point of symmetry, and its coordinates provide critical information about the parabola's position in the coordinate plane. The vertex is also useful in solving optimization problems where we need to find the maximum or minimum value of a quadratic function. Recognizing and accurately locating the vertex is a fundamental skill in the study of parabolas and their applications.
D) Defining the Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. This line acts as a mirror, reflecting one side of the parabola onto the other. The equation of the axis of symmetry is always of the form x = k, where k is the x-coordinate of the vertex. In our case, the vertex of the parabola f(x) = 100 - 9x² is (0, 100), as we calculated in the previous section.
Therefore, the axis of symmetry is the vertical line x = 0. This means that the parabola is symmetrical about the y-axis. The axis of symmetry is an essential characteristic of a parabola as it helps to visualize the symmetry of the curve and simplifies the process of graphing the parabola. It also aids in understanding the relationship between points on either side of the vertex. For example, if we know a point on one side of the axis of symmetry, we can easily find its corresponding point on the other side. Recognizing the axis of symmetry not only helps in graphing but also in solving problems involving parabolas, particularly in optimization and reflection problems.
E) Determining the Domain and Range
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a parabola defined by a quadratic equation, the domain is typically all real numbers, unless there are specific restrictions imposed by the context of the problem. In our case, the function f(x) = 100 - 9x² is defined for all real numbers, as there are no restrictions on the values that x can take.
Therefore, the domain of the parabola is all real numbers, which can be expressed in interval notation as (-∞, ∞). This means that we can plug in any real number for x, and the function will produce a valid output. The domain is a fundamental aspect of a function, providing the scope within which the function operates. For parabolas, the domain is typically unrestricted, reflecting the continuous nature of the quadratic equation.
The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range of the parabola f(x) = 100 - 9x², we need to consider whether the parabola opens upwards or downwards and the y-coordinate of the vertex. As we established earlier, the parabola opens downwards because the coefficient of the x² term is negative (-9). The vertex, (0, 100), represents the maximum point of the parabola.
Therefore, the range of the function is all y-values less than or equal to 100. In interval notation, this can be expressed as (-∞, 100]. This means that the function's output will never be greater than 100, and it can take on any value below that. The range is a crucial characteristic of a function as it defines the limits of its output. For parabolas that open downwards, the range is bounded above by the y-coordinate of the vertex, while for parabolas that open upwards, the range is bounded below by the y-coordinate of the vertex. Understanding the range helps in analyzing the function's behavior and predicting its output values.
Conclusion
In conclusion, by analyzing the equation f(x) = 100 - 9x², we have successfully identified key features of the parabola, including the y-intercept, x-intercepts, vertex, axis of symmetry, domain, and range. The y-intercept is (0, 100), the x-intercepts are (10/3, 0) and (-10/3, 0), the vertex is (0, 100), the axis of symmetry is x = 0, the domain is all real numbers (-∞, ∞), and the range is (-∞, 100]. These elements collectively paint a comprehensive picture of the parabola's shape, position, and behavior in the coordinate plane. This detailed analysis serves as a valuable foundation for further exploration of quadratic functions and their applications in various fields.
