Find The Vertex Of Parabola G(x)=3x^2-30x-93 Using Vertex Formula

Finding the vertex of a parabola is a fundamental concept in algebra, particularly when dealing with quadratic functions. The vertex represents the highest or lowest point on the parabola, making it a crucial feature for understanding the function's behavior. In this article, we will explore how to determine the vertex of a parabola by applying the vertex formula. We will use the example function $g(x) = 3x^2 - 30x - 93$ to illustrate the process step by step. Understanding how to find the vertex is essential for various applications, such as optimization problems, graphing quadratic functions, and analyzing real-world scenarios modeled by parabolas.

Understanding the Vertex Formula

The vertex formula is a powerful tool for finding the vertex of a parabola defined by a quadratic equation in the form $f(x) = ax^2 + bx + c$. The vertex is a point $(h, k)$, where $h$ represents the x-coordinate and $k$ represents the y-coordinate. The vertex formula provides a straightforward method for calculating these coordinates directly from the coefficients $a$, $b$, and $c$ of the quadratic equation.

The x-coordinate, $h$, of the vertex is given by the formula:

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

This formula is derived from completing the square or using calculus to find the critical point of the quadratic function. The y-coordinate, $k$, of the vertex can then be found by substituting the value of $h$ back into the original quadratic equation:

$k = f(h) = a(h)^2 + b(h) + c$

By calculating $h$ and $k$, we can precisely locate the vertex of the parabola. Understanding the vertex formula is crucial for analyzing quadratic functions and their graphical representations. The vertex provides key information about the parabola's symmetry, direction, and extreme values. For a parabola that opens upwards (where $a > 0$), the vertex represents the minimum point, while for a parabola that opens downwards (where $a < 0$), the vertex represents the maximum point. Therefore, the vertex formula is not just a mathematical tool but also a practical method for solving optimization problems in various fields, including physics, engineering, and economics.

Step-by-Step Calculation for $g(x) = 3x^2 - 30x - 93$

To find the vertex of the parabola represented by the quadratic function $g(x) = 3x^2 - 30x - 93$, we will apply the vertex formula step by step. This process involves identifying the coefficients $a$, $b$, and $c$, calculating the x-coordinate ($h$) of the vertex, and then finding the y-coordinate ($k$) by substituting $h$ back into the function. Breaking down the calculation into manageable steps helps ensure accuracy and clarity in the solution.

Step 1: Identify the Coefficients

First, we need to identify the coefficients $a$, $b$, and $c$ from the given quadratic function $g(x) = 3x^2 - 30x - 93$. Comparing this function to the standard form $f(x) = ax^2 + bx + c$, we can see that:

• $a = 3$
• $b = -30$
• $c = -93$

These coefficients are essential for applying the vertex formula. The coefficient $a$ determines the parabola's direction (whether it opens upwards or downwards), while $b$ and $a$ together determine the x-coordinate of the vertex. The coefficient $c$ represents the y-intercept of the parabola.

Step 2: Calculate the x-coordinate ($h$)

Next, we calculate the x-coordinate ($h$) of the vertex using the formula:

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

Substituting the values we identified in Step 1, we get:

$h = -\frac{-30}{2(3)} = -\frac{-30}{6} = 5$

So, the x-coordinate of the vertex is $h = 5$. This value represents the axis of symmetry for the parabola, meaning the parabola is symmetric about the vertical line $x = 5$. Knowing the x-coordinate of the vertex is a critical step in determining the complete location of the vertex and understanding the parabola's symmetrical properties.

Step 3: Calculate the y-coordinate ($k$)

Now that we have the x-coordinate $h = 5$, we can find the y-coordinate ($k$) by substituting $h$ back into the original function $g(x)$:

$k = g(h) = g(5) = 3(5)^2 - 30(5) - 93$

Calculating this, we get:

$k = 3(25) - 150 - 93 = 75 - 150 - 93 = -168$

Thus, the y-coordinate of the vertex is $k = -168$. This value represents the minimum value of the function $g(x)$, since the parabola opens upwards (because $a = 3 > 0$).

Step 4: Write the Vertex

Finally, we can write the vertex as a coordinate pair $(h, k)$. Using the values we calculated, the vertex of the parabola $g(x) = 3x^2 - 30x - 93$ is:

Vertex = $(5, -168)$

This point represents the lowest point on the parabola. Knowing the vertex allows us to sketch the graph of the function accurately and understand its behavior. The vertex, along with the axis of symmetry, provides a clear picture of the parabola's shape and position on the coordinate plane.

Practical Applications of Finding the Vertex

Finding the vertex of a parabola has numerous practical applications across various fields. The vertex represents the extreme point (either maximum or minimum) of the quadratic function, making it invaluable for optimization problems. Understanding these applications highlights the importance of mastering the vertex formula and its implications.

Optimization Problems

One of the most common applications of finding the vertex is in solving optimization problems. These problems often involve finding the maximum or minimum value of a quantity, such as profit, cost, or area, that can be modeled by a quadratic function. The vertex directly provides this optimal value. For instance, if a business wants to maximize its profit, which is modeled by a quadratic function, finding the vertex will reveal the production level that yields the highest profit. Similarly, in engineering, the vertex can help determine the minimum amount of material needed to construct a parabolic structure or the maximum height a projectile can reach.

Graphing Quadratic Functions

The vertex is a crucial point for graphing quadratic functions accurately. Along with the axis of symmetry and the y-intercept, the vertex helps define the shape and position of the parabola on the coordinate plane. By plotting the vertex and a few additional points, one can quickly sketch the graph of the function. This graphical representation provides a visual understanding of the function's behavior, including its roots (x-intercepts) and its range of values. In fields like physics and engineering, where parabolic trajectories and shapes are common, accurately graphing quadratic functions is essential for analysis and design.

Real-World Scenarios

Many real-world scenarios can be modeled using quadratic functions, and finding the vertex provides valuable insights into these situations. For example, the trajectory of a projectile, such as a ball thrown into the air, follows a parabolic path. The vertex of this parabola represents the maximum height the projectile reaches. Similarly, the shape of suspension bridge cables, satellite dishes, and architectural arches can be modeled using parabolas, and the vertex plays a significant role in their design and analysis. In economics, quadratic functions can model cost and revenue curves, where the vertex helps determine break-even points and optimal production levels.

Physics

In physics, understanding projectile motion is a classic application of quadratic functions. The height of a projectile over time can be modeled by a parabolic equation, where the vertex represents the highest point the projectile reaches. By finding the vertex, physicists can determine the maximum height, the time it takes to reach that height, and the total flight time. This information is crucial in fields like ballistics, sports science, and aerospace engineering.

Engineering

Engineers use parabolic shapes in various designs, including bridges, antennas, and reflectors. The vertex of the parabola is a critical point for these designs. For example, in a parabolic antenna, the vertex is the focal point where signals are concentrated. Understanding the vertex's position and properties allows engineers to optimize the antenna's performance. Similarly, in bridge design, the parabolic shape of suspension cables helps distribute weight evenly, and the vertex plays a role in calculating the cable's tension and stability.

Conclusion

In conclusion, finding the vertex of a parabola using the vertex formula is a fundamental skill in mathematics with wide-ranging applications. By understanding the steps involved in applying the formula and recognizing the practical implications of the vertex, one can effectively analyze quadratic functions and solve optimization problems in various fields. The example of $g(x) = 3x^2 - 30x - 93$ illustrates the process clearly, and the discussion of applications highlights the importance of this concept in real-world scenarios. Mastering the vertex formula is essential for anyone working with quadratic functions, whether in academics, engineering, physics, or any other field where parabolic relationships are significant. The vertex is more than just a point on a graph; it's a key to understanding the behavior and optimizing the outcomes in many practical situations.