Find The Vertex The Y-value Of F(x)=-(x-3)(x+11) Explained

Navigating the realm of quadratic functions often leads us to a pivotal point known as the vertex. This point holds significant importance as it represents either the maximum or minimum value of the function, depending on the parabola's orientation. In this comprehensive exploration, we delve into the intricacies of determining the $y$-value of the vertex for the function $f(x) = -(x-3)(x+11)$. Our journey will encompass a detailed examination of the function's properties, the vertex formula, and step-by-step calculations to arrive at the solution. This detailed explanation aims to provide a clear and concise understanding for students, educators, and anyone with an interest in quadratic functions.

Understanding Quadratic Functions

Before we embark on the quest to find the $y$-value of the vertex, it's crucial to establish a firm grasp of 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 opens upwards if $a > 0$ and downwards if $a < 0$.

In the given function, $f(x) = -(x-3)(x+11)$, we observe a slightly different form. This is the factored form of a quadratic function, which can be expanded to the standard form. Factored form provides valuable insights into the roots or zeros of the function, which are the $x$-values where the function intersects the $x$-axis. These roots are found by setting the function equal to zero and solving for $x$. In this case, the roots are $x = 3$ and $x = -11$.

The vertex of the parabola is the point where the function reaches its maximum or minimum value. For a parabola opening upwards ($a > 0$), the vertex represents the minimum point, while for a parabola opening downwards ($a < 0$), the vertex represents the maximum point. The vertex is a crucial feature of the parabola, and its coordinates $(h, k)$ are essential for understanding the function's behavior. The $x$-coordinate, $h$, represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The $y$-coordinate, $k$, represents the maximum or minimum value of the function.

Understanding these fundamental concepts of quadratic functions is paramount to effectively determine the $y$-value of the vertex. By recognizing the relationship between the function's form, its graph, and its key features, we can approach the problem with clarity and precision.

Deconstructing the Function $f(x) = -(x-3)(x+11)$

Now, let's turn our attention to the specific function at hand: $f(x) = -(x-3)(x+11)$. This function is presented in factored form, which offers us a direct pathway to understanding its roots and, consequently, the $x$-coordinate of the vertex. To extract the most information, we will take the following steps:

First, we need to expand the factored form to obtain the standard quadratic form, $f(x) = ax^2 + bx + c$. This expansion involves multiplying the binomials and distributing the negative sign:

$f(x) = -(x-3)(x+11) $$f(x) = -(x^2 + 11x - 3x - 33) $$f(x) = -(x^2 + 8x - 33) $$f(x) = -x^2 - 8x + 33$

Now we have the quadratic function in standard form, where $a = -1$, $b = -8$, and $c = 33$. The coefficient $a = -1$ immediately tells us that the parabola opens downwards, indicating that the vertex will represent the maximum point of the function. This is a critical piece of information as it sets the stage for interpreting the $y$-value of the vertex as the maximum value of the function.

Next, we identify the roots of the function. The factored form directly reveals the roots, which are the $x$-values where the function equals zero. These are the points where the parabola intersects the $x$-axis. From the factored form $f(x) = -(x-3)(x+11)$, we can see that the roots are $x = 3$ and $x = -11$. Setting each factor equal to zero gives us:

$x - 3 = 0 => x = 3$ $x + 11 = 0 => x = -11$

These roots are essential for finding the axis of symmetry, which passes through the vertex. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It's $x$-coordinate is exactly halfway between the two roots. This crucial insight allows us to calculate the $x$-coordinate of the vertex with ease.

By carefully deconstructing the function $f(x) = -(x-3)(x+11)$, we have gathered vital information about its shape, direction, and roots. This knowledge sets the foundation for our next step: determining the vertex of the parabola.

Pinpointing the Vertex: A Step-by-Step Guide

The vertex of a parabola is the cornerstone for understanding its behavior, representing the function's maximum or minimum value. To pinpoint the vertex of our function, $f(x) = -(x-3)(x+11)$, we'll embark on a step-by-step journey, leveraging the insights we've gained thus far.

Step 1: Determine the $x$-coordinate of the vertex ($h$)

The $x$-coordinate of the vertex, often denoted as $h$, lies precisely on the axis of symmetry. As we previously discussed, the axis of symmetry is situated midway between the roots of the quadratic function. We know the roots are $x = 3$ and $x = -11$. Thus, we can find $h$ by calculating the average of the roots:

$h = (root_1 + root_2) / 2$ $h = (3 + (-11)) / 2$ $h = -8 / 2$ $h = -4$

Therefore, the $x$-coordinate of the vertex is $-4$. This value signifies the horizontal position of the vertex on the coordinate plane and is a crucial component in defining the vertex's location.

Step 2: Calculate the $y$-coordinate of the vertex ($k$)

The $y$-coordinate of the vertex, denoted as $k$, represents the function's maximum or minimum value. To find $k$, we simply substitute the $x$-coordinate of the vertex ($h = -4$) back into the original function, $f(x) = -(x-3)(x+11)$:

$k = f(h)$ $k = f(-4)$ $k = -(-4 - 3)(-4 + 11)$ $k = -(-7)(7)$ $k = -(-49)$ $k = 49$

Thus, the $y$-coordinate of the vertex is $49$. This value represents the maximum value of the function, as we know the parabola opens downwards. It signifies the vertical position of the vertex and completes our understanding of the vertex's location.

Step 3: Express the vertex as a coordinate pair

Having determined both the $x$ and $y$ coordinates, we can now express the vertex as a coordinate pair $(h, k)$. In our case, the vertex is located at $(-4, 49)$. This coordinate pair provides a concise representation of the vertex's position on the coordinate plane and encapsulates the essential information about the function's maximum value and axis of symmetry.

By meticulously following these steps, we have successfully pinpointed the vertex of the function $f(x) = -(x-3)(x+11)$. The vertex, located at $(-4, 49)$, reveals that the function reaches its maximum value of $49$ at $x = -4$. This understanding is invaluable for analyzing the function's behavior and its graphical representation.

The $y$-value of the Vertex: The Grand Finale

After our thorough investigation, we have arrived at the heart of our quest: the $y$-value of the vertex. As we meticulously calculated in the previous section, the vertex of the function $f(x) = -(x-3)(x+11)$ is located at the point $(-4, 49)$. The $y$-coordinate of this point is $49$.

Therefore, the $y$-value of the vertex of the function $f(x) = -(x-3)(x+11)$ is 49. This value holds significant meaning. Since the coefficient of the $x^2$ term in the expanded form of the function ($f(x) = -x^2 - 8x + 33$) is negative, we know that the parabola opens downwards. This implies that the vertex represents the maximum point of the function. Consequently, the $y$-value of the vertex, $49$, is the maximum value that the function can attain.

This finding is crucial for understanding the function's range, which is the set of all possible output values. Since the maximum value is $49$ and the parabola opens downwards, the range of the function is all real numbers less than or equal to $49$. In interval notation, this is expressed as $(-\infty, 49]$.

The $y$-value of the vertex also provides insights into the function's behavior and its graphical representation. When sketching the graph of the function, knowing the vertex helps us to accurately plot the turning point of the parabola. Furthermore, it allows us to visualize the symmetry of the parabola around the axis of symmetry, which passes through the vertex.

In summary, the $y$-value of the vertex, $49$, is not merely a numerical answer. It is a key piece of information that unlocks a deeper understanding of the function's properties, behavior, and graphical representation. It signifies the maximum value of the function, defines the upper bound of its range, and aids in visualizing its parabolic shape.

Conclusion: The Significance of the Vertex

In this comprehensive exploration, we have successfully determined the $y$-value of the vertex for the function $f(x) = -(x-3)(x+11)$. Through a step-by-step analysis, we expanded the function, identified its roots, calculated the $x$-coordinate of the vertex, and ultimately found the $y$-coordinate to be $49$. This value, $49$, represents the maximum value of the function, a critical insight into its behavior and characteristics.

The vertex, as we've seen, is a pivotal point in understanding quadratic functions. It marks the turning point of the parabola, revealing whether the function has a maximum or minimum value. The coordinates of the vertex provide a wealth of information, including the axis of symmetry, the range of the function, and the overall shape of the parabola. For the function $f(x) = -(x-3)(x+11)$, the vertex at $(-4, 49)$ tells us that the function reaches its peak at $x = -4$, with a maximum value of $49$.

Understanding the vertex is not just an academic exercise; it has practical applications in various fields. In physics, for example, the vertex can represent the maximum height of a projectile's trajectory. In engineering, it can help determine the optimal design parameters for structures. In economics, it can represent the maximum profit or minimum cost in a business model.

The ability to find the vertex of a quadratic function is a fundamental skill in mathematics. It requires a solid understanding of quadratic equations, their properties, and their graphical representations. By mastering this skill, we gain a powerful tool for analyzing and interpreting real-world phenomena that can be modeled by quadratic functions.

In conclusion, the journey to find the $y$-value of the vertex has been more than just a calculation. It has been an exploration of the rich landscape of quadratic functions, highlighting the importance of the vertex as a key feature and a gateway to deeper understanding. The $y$-value of the vertex, $49$, stands as a testament to the power of mathematical analysis and its ability to reveal the hidden patterns and relationships within the world around us.