Analyzing The Quadratic Function D(x) = 3x² - 12x + 11 Vertex Intercepts And Graph
This article delves into the comprehensive analysis of the quadratic function d(x) = 3x² - 12x + 11. We will explore various aspects of this function, including converting it to vertex form, identifying its vertex, determining its x and y-intercepts, sketching its graph, and discussing its mathematical properties. Understanding quadratic functions is crucial in various fields like physics, engineering, and economics, where they model parabolic trajectories, optimization problems, and more. Let's embark on this journey to dissect and understand the intricacies of this particular quadratic function.
(a) Transforming to Vertex Form
To begin our analysis, let's transform the given quadratic function, d(x) = 3x² - 12x + 11, into its vertex form. The vertex form of a quadratic function is expressed as d(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. Converting to vertex form allows us to easily identify the vertex and understand the function's transformations. The process involves completing the square, a technique used to rewrite a quadratic expression as a squared term plus a constant.
First, we factor out the coefficient of the x² term (which is 3) from the first two terms of the function:
d(x) = 3(x² - 4x) + 11
Next, we complete the square inside the parentheses. To do this, we take half of the coefficient of the x term (-4), square it ((-2)² = 4), and add and subtract it inside the parentheses:
d(x) = 3(x² - 4x + 4 - 4) + 11
Now, we rewrite the expression inside the parentheses as a squared term:
d(x) = 3((x - 2)² - 4) + 11
Distribute the 3 to both terms inside the parentheses:
d(x) = 3(x - 2)² - 12 + 11
Finally, simplify the expression:
d(x) = 3(x - 2)² - 1
Therefore, the vertex form of the function d(x) = 3x² - 12x + 11 is d(x) = 3(x - 2)² - 1. This form immediately reveals the vertex of the parabola, which we will discuss in the next section.
(b) Identifying the Vertex
Now that we have the function in vertex form, d(x) = 3(x - 2)² - 1, identifying the vertex becomes straightforward. As mentioned earlier, the vertex form is given by d(x) = a(x - h)² + k, where the vertex is the point (h, k). By comparing our vertex form with the general form, we can directly read off the coordinates of the vertex.
In our case, we have h = 2 and k = -1. Therefore, the vertex of the parabola represented by the function d(x) = 3x² - 12x + 11 is (2, -1). The vertex is a crucial point on the parabola, as it represents either the minimum or maximum value of the function. In this case, since the coefficient of the (x - 2)² term is positive (3), the parabola opens upwards, and the vertex represents the minimum point of the function. This means that the function d(x) has a minimum value of -1, which occurs when x = 2.
Understanding the vertex helps us visualize the graph of the quadratic function and its behavior. We know that the parabola is symmetric about the vertical line passing through the vertex (the axis of symmetry), which in this case is the line x = 2. This symmetry is a fundamental property of parabolas and is useful in sketching the graph of the function.
(c) Determining the x-intercept(s)
The x-intercepts of a function are the points where the graph of the function intersects the x-axis. At these points, the value of the function is zero, i.e., d(x) = 0. To find the x-intercepts of d(x) = 3x² - 12x + 11, we need to solve the quadratic equation 3x² - 12x + 11 = 0. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is not readily apparent, so we will use the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. For our equation 3x² - 12x + 11 = 0, we have a = 3, b = -12, and c = 11. Plugging these values into the quadratic formula, we get:
x = (12 ± √((-12)² - 4 * 3 * 11)) / (2 * 3)
x = (12 ± √(144 - 132)) / 6
x = (12 ± √12) / 6
x = (12 ± 2√3) / 6
Simplifying further, we get two distinct x-intercepts:
x₁ = (6 + √3) / 3
x₂ = (6 - √3) / 3
Therefore, the x-intercepts of the function d(x) = 3x² - 12x + 11 are approximately x₁ ≈ 2.577 and x₂ ≈ 1.423. These points represent where the parabola crosses the x-axis.
(d) Determining the y-intercept(s)
The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when x = 0. To find the y-intercept of d(x) = 3x² - 12x + 11, we simply substitute x = 0 into the function:
d(0) = 3(0)² - 12(0) + 11
d(0) = 11
Therefore, the y-intercept of the function d(x) = 3x² - 12x + 11 is 11. This means the parabola intersects the y-axis at the point (0, 11). The y-intercept provides another key point for sketching the graph of the function.
(e) Sketching the Function
Now that we have determined the vertex, x-intercepts, and y-intercept, we can sketch the graph of the function d(x) = 3x² - 12x + 11. We know the following information:
- Vertex: (2, -1)
- x-intercepts: x₁ ≈ 2.577 and x₂ ≈ 1.423
- y-intercept: 11
Since the coefficient of the x² term is positive (3), the parabola opens upwards. The vertex (2, -1) represents the minimum point of the function. The parabola is symmetric about the vertical line x = 2. We can plot these points on a coordinate plane and draw a smooth curve connecting them to represent the parabola. The graph will pass through the x-intercepts at approximately x = 1.423 and x = 2.577, cross the y-axis at y = 11, and have its lowest point at the vertex (2, -1).
The sketch provides a visual representation of the function's behavior, showing its shape, direction, and key points. This graphical representation is often helpful in understanding the function's properties and its relationship to real-world applications.
(f) Discussion: Category - Mathematics
The function d(x) = 3x² - 12x + 11 falls under the mathematics category, specifically within the subfield of algebra, dealing with quadratic functions. Quadratic functions are polynomial functions of degree 2, meaning the highest power of the variable x is 2. They are characterized by their parabolic shape when graphed and are described by the general form ax² + bx + c, where a, b, and c are constants and a ≠ 0. The properties and characteristics of quadratic functions, such as their vertex, intercepts, and axis of symmetry, are fundamental concepts in mathematics and have wide-ranging applications.
This specific function, d(x) = 3x² - 12x + 11, exemplifies the key attributes of quadratic functions. Its vertex form, d(x) = 3(x - 2)² - 1, highlights the transformations applied to the basic parabola y = x². The vertex (2, -1) represents a shift of the basic parabola 2 units to the right and 1 unit downwards. The coefficient 3 stretches the parabola vertically, making it narrower than the basic parabola. The x-intercepts and y-intercept provide further information about the function's behavior and its relationship to the coordinate axes. Analyzing this function provides a solid understanding of quadratic functions and their properties, which is essential for various mathematical and scientific applications.
In conclusion, we have thoroughly analyzed the quadratic function d(x) = 3x² - 12x + 11, converting it to vertex form, identifying its vertex, determining its intercepts, sketching its graph, and discussing its mathematical categorization. This comprehensive analysis demonstrates the importance of understanding the different forms of a quadratic function and how to extract meaningful information from them. The principles and techniques explored in this analysis are applicable to a wide range of quadratic functions and are valuable tools in mathematical problem-solving and modeling.
