Finding The Vertex Of A Quadratic Function F(x)=-3x^2+12x+1

In mathematics, quadratic functions play a crucial role in various applications, from modeling projectile motion to optimizing business processes. Understanding the properties of quadratic functions is essential for solving real-world problems. One of the most important features of a quadratic function is its vertex, which represents the maximum or minimum point of the parabola. In this comprehensive guide, we will explore the concept of the vertex and delve into different methods for finding it. We will focus on the quadratic function $f(x) = -3x^2 + 12x + 1$ as a primary example, illustrating each method with detailed steps and explanations. Whether you are a student learning about quadratic functions for the first time or a professional seeking a refresher, this guide will provide you with the knowledge and tools to confidently find the vertex of any quadratic function.

Understanding Quadratic Functions and Their Vertex

To effectively find the vertex, it's crucial to understand the basic form and properties of quadratic functions. A quadratic function is a polynomial function of degree two, generally expressed in the standard form as:

$$f(x) = ax^2 + bx + c$$

where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient a. If a > 0, the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, if a < 0, the parabola opens downwards, and the vertex represents the maximum point. In our example function, $f(x) = -3x^2 + 12x + 1$, a = -3, b = 12, and c = 1. Since a is negative, the parabola opens downwards, and the vertex will be the maximum point.

The vertex of a parabola is the point where the parabola changes direction. It is the highest or lowest point on the graph. The vertex is defined by its coordinates (h, k), where h represents the x-coordinate and k represents the y-coordinate. The x-coordinate h can be found using the formula:

$$h = -\frac{b}{2a}$$

Once we have the value of h, we can find the y-coordinate k by substituting h into the original function:

$$k = f(h)$$

The vertex form of a quadratic function is given by:

$$f(x) = a(x - h)^2 + k$$

This form directly reveals the vertex (h, k) and the direction the parabola opens (a). Understanding these fundamentals is key to effectively finding and interpreting the vertex of any quadratic function. In the following sections, we will explore various methods for finding the vertex, applying these concepts to our example function and beyond.

Method 1: Using the Vertex Formula

The most straightforward method for finding the vertex of a quadratic function is by using the vertex formula. This method is efficient and directly provides the coordinates of the vertex without requiring additional algebraic manipulations. The vertex formula is derived from the process of completing the square and is a fundamental tool in quadratic function analysis. As we established earlier, for a quadratic function in the standard form $f(x) = ax^2 + bx + c$, the x-coordinate (h) of the vertex is given by:

$$h = -\frac{b}{2a}$$

And the y-coordinate (k) of the vertex is found by substituting h into the function:

$$k = f(h)$$

Let's apply this method to our example function, $f(x) = -3x^2 + 12x + 1$. Here, a = -3 and b = 12. We begin by calculating the x-coordinate h:

$$h = -\frac{12}{2(-3)} = -\frac{12}{-6} = 2$$

Now that we have h = 2, we can find the y-coordinate k by substituting h into the function:

$$k = f(2) = -3(2)^2 + 12(2) + 1 = -3(4) + 24 + 1 = -12 + 24 + 1 = 13$$

Therefore, the vertex of the quadratic function $f(x) = -3x^2 + 12x + 1$ is (2, 13). This means the maximum value of the function occurs at x = 2, and the maximum value is 13. This method offers a quick and reliable way to identify the vertex, making it a valuable tool for solving optimization problems and understanding the behavior of quadratic functions. In the next sections, we will explore alternative methods for finding the vertex, providing a broader perspective on this essential concept.

Method 2: Completing the Square

Completing the square is another powerful technique for finding the vertex of a quadratic function. This method involves transforming the standard form of the quadratic function, $f(x) = ax^2 + bx + c$, into the vertex form, $f(x) = a(x - h)^2 + k$, where (h, k) represents the vertex. Completing the square not only allows us to find the vertex but also provides valuable insights into the function's properties, such as its symmetry and transformations. The process involves manipulating the quadratic expression to create a perfect square trinomial. Let's illustrate this method using our example function, $f(x) = -3x^2 + 12x + 1$.

First, we factor out the coefficient of the $x^2$ term, which is -3, from the first two terms:

$$f(x) = -3(x^2 - 4x) + 1$$

Next, we complete the square inside the parentheses. To do this, we take half of the coefficient of the x term (-4), square it, and add it inside the parentheses. Half of -4 is -2, and (-2)^2 is 4. However, since we are adding 4 inside the parentheses, which is being multiplied by -3, we must subtract -3(4) = -12 outside the parentheses to maintain the equality:

$$f(x) = -3(x^2 - 4x + 4) + 1 - (-12)$$

Now, we can rewrite the expression inside the parentheses as a perfect square:

$$f(x) = -3(x - 2)^2 + 1 + 12$$

Simplifying the expression, we get:

$$f(x) = -3(x - 2)^2 + 13$$

This is now in the vertex form, $f(x) = a(x - h)^2 + k$, where a = -3, h = 2, and k = 13. Therefore, the vertex of the quadratic function is (2, 13), which confirms the result we obtained using the vertex formula. Completing the square provides a deeper understanding of the function's structure and its relationship to the vertex. It is a valuable technique for solving quadratic equations and analyzing quadratic functions in various contexts. In the following sections, we will explore another method for finding the vertex, further expanding our toolkit for quadratic function analysis.

Method 3: Using Symmetry of the Parabola

The symmetry of a parabola offers another elegant method for finding the vertex of a quadratic function. A parabola is symmetric about its axis of symmetry, which is a vertical line that passes through the vertex. This symmetry means that the x-coordinate of the vertex lies exactly in the middle of any two points on the parabola with the same y-coordinate. This property is particularly useful when we know the x-intercepts (roots) of the quadratic function, as they are symmetric about the axis of symmetry. Let's explore how we can use this symmetry to find the vertex of our example function, $f(x) = -3x^2 + 12x + 1$.

First, while we could find the x-intercepts by setting $f(x) = 0$ and solving for x, it often involves more complex calculations (using the quadratic formula). Instead, let's consider a simpler approach using any two points with the same y-coordinate. To illustrate, we can find two convenient points by choosing x values and calculating the corresponding f(x) values. Let's choose x = 0 and x = 4:

For x = 0:

$$f(0) = -3(0)^2 + 12(0) + 1 = 1$$

For x = 4:

$$f(4) = -3(4)^2 + 12(4) + 1 = -3(16) + 48 + 1 = -48 + 48 + 1 = 1$$

We found two points (0, 1) and (4, 1) on the parabola that have the same y-coordinate. The x-coordinate of the vertex, h, lies exactly in the middle of the x-coordinates of these two points. We can find h by taking the average of the x-coordinates:

$$h = \frac{0 + 4}{2} = 2$$

Now that we have h = 2, we can find the y-coordinate of the vertex, k, by substituting h into the function:

$$k = f(2) = -3(2)^2 + 12(2) + 1 = -3(4) + 24 + 1 = -12 + 24 + 1 = 13$$

Thus, the vertex of the quadratic function is (2, 13), which again confirms our previous results. This method highlights the inherent symmetry of parabolas and provides an alternative approach to finding the vertex, especially when dealing with functions where finding x-intercepts might be cumbersome. In summary, leveraging the symmetry of the parabola is a valuable tool in understanding and analyzing quadratic functions, offering a geometric perspective on finding the vertex.

In this comprehensive guide, we explored the essential concept of the vertex of a quadratic function and delved into three distinct methods for finding it. The vertex, representing the maximum or minimum point of a parabola, is crucial for understanding the behavior and properties of quadratic functions. We examined the quadratic function $f(x) = -3x^2 + 12x + 1$ as our primary example, demonstrating each method with detailed steps and explanations. First, we employed the vertex formula, a direct and efficient method that utilizes the coefficients of the quadratic function to calculate the coordinates of the vertex. This method provides a quick and reliable way to identify the vertex, making it a valuable tool for various applications.

Next, we explored the technique of completing the square, which involves transforming the standard form of the quadratic function into the vertex form. This method not only allows us to find the vertex but also provides deeper insights into the function's structure and its transformations. Completing the square is a powerful algebraic technique with broad applications beyond finding the vertex. Finally, we leveraged the symmetry of the parabola to determine the vertex. This method utilizes the fact that the vertex lies on the axis of symmetry, which is the midpoint between any two points on the parabola with the same y-coordinate. By finding two such points, we can easily calculate the x-coordinate of the vertex and subsequently find the y-coordinate.

Each of these methods offers a unique perspective on finding the vertex, and the choice of method may depend on the specific context and the information available. Understanding these methods equips you with a robust toolkit for analyzing quadratic functions and solving related problems. Whether you are dealing with optimization problems, projectile motion, or any other application involving quadratic functions, the ability to find the vertex is an invaluable skill. By mastering these techniques, you can confidently tackle a wide range of mathematical challenges and gain a deeper appreciation for the elegance and power of quadratic functions. Remember to practice these methods with various examples to solidify your understanding and build your problem-solving skills. The journey to mastering quadratic functions is a rewarding one, and finding the vertex is a key milestone in that journey.