Calculating The Vertex Coordinates Of Y=12x+20+(3/4)x^2

Finding the vertex of a quadratic equation is a fundamental concept in algebra with numerous applications in various fields, from physics to economics. In this comprehensive guide, we will delve into the process of calculating the coordinates of the vertex for the equation $y = 12x + 20 + \frac{3}{4}x^2$. We will break down the steps involved, provide clear explanations, and illustrate the concepts with examples to ensure a thorough understanding. This knowledge is crucial for anyone studying quadratic functions and their properties.

Understanding Quadratic Equations and Their Properties

Before we dive into the calculation, it's essential to understand the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is given by:

$y = ax^2 + bx + c$

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic equation is a parabola, a U-shaped curve. The parabola can open upwards (if 'a' is positive) or downwards (if 'a' is negative). The vertex of the parabola is the point where the curve changes direction. It is the minimum point if the parabola opens upwards and the maximum point if the parabola opens downwards. Understanding the nature of the parabola is critical for interpreting the solutions of quadratic equations and their applications.

The vertex of a parabola holds significant information about the quadratic equation. Its x-coordinate represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The y-coordinate of the vertex represents the minimum or maximum value of the quadratic function. Finding the vertex is therefore crucial in optimization problems, where we aim to find the maximum or minimum value of a function.

The importance of the vertex extends beyond pure mathematics. In physics, for instance, the vertex can represent the highest point reached by a projectile. In economics, it can represent the point of maximum profit or minimum cost. Therefore, mastering the technique of finding the vertex is a valuable skill in various disciplines.

Rewriting the Equation in Standard Form

The given equation is $y = 12x + 20 + \frac{3}{4}x^2$. To make it easier to work with, we need to rewrite it in the standard quadratic form, which is $y = ax^2 + bx + c$. This involves rearranging the terms so that the term with $x^2$ comes first, followed by the term with x, and then the constant term. This rearrangement allows us to easily identify the coefficients 'a', 'b', and 'c', which are essential for calculating the vertex.

By rearranging the terms in the given equation, we get:

$y = \frac{3}{4}x^2 + 12x + 20$

Now, we can clearly identify the coefficients:

• a = $\frac{3}{4}$
• b = 12
• c = 20

Having the equation in standard form is a crucial step as it simplifies the subsequent calculations and allows us to apply the vertex formula correctly. It also helps in visualizing the parabola's orientation (whether it opens upwards or downwards) based on the sign of the coefficient 'a'. In this case, since 'a' is positive, we know that the parabola opens upwards and the vertex represents the minimum point.

The process of rewriting the equation highlights the importance of algebraic manipulation in solving mathematical problems. By rearranging terms and expressing the equation in a more convenient form, we can unlock the underlying structure and make it easier to apply relevant formulas and techniques.

Calculating the x-coordinate of the Vertex

The x-coordinate of the vertex, often denoted as $h$, can be calculated using the following formula:

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

This formula is derived from the process of completing the square, a technique used to rewrite the quadratic equation in vertex form. The vertex form of a quadratic equation is given by:

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

where (h, k) are the coordinates of the vertex. The formula $h = -\frac{b}{2a}$ directly provides the x-coordinate of the vertex without the need to complete the square every time. This formula is a cornerstone in finding the vertex of a parabola.

Now, let's apply this formula to our equation. We have a = $\frac{3}{4}$ and b = 12. Substituting these values into the formula, we get:

$h = -\frac{12}{2(\frac{3}{4})}$

Simplifying the expression:

$h = -\frac{12}{\frac{3}{2}}$

$h = -12 \cdot \frac{2}{3}$

$h = -8$

Therefore, the x-coordinate of the vertex is -8. This value represents the axis of symmetry of the parabola, meaning the parabola is symmetrical about the vertical line x = -8. Accurately calculating this value is essential for determining the location of the vertex and understanding the parabola's symmetry.

Calculating the y-coordinate of the Vertex

To find the y-coordinate of the vertex, often denoted as $k$, we substitute the x-coordinate ($h$) that we just calculated back into the original equation. This is because the vertex lies on the parabola, so its coordinates must satisfy the equation. This substitution is a direct application of the definition of a function and its graph.

Substituting $x = -8$ into the equation $y = \frac{3}{4}x^2 + 12x + 20$, we get:

$k = \frac{3}{4}(-8)^2 + 12(-8) + 20$

Now, we simplify the expression:

$k = \frac{3}{4}(64) - 96 + 20$

$k = 3 \cdot 16 - 96 + 20$

$k = 48 - 96 + 20$

$k = -28$

Therefore, the y-coordinate of the vertex is -28. This value represents the minimum value of the quadratic function, as the parabola opens upwards. Understanding the significance of this value is crucial for solving optimization problems.

The Vertex Coordinates and Their Significance

We have now calculated both the x-coordinate ($h = -8$) and the y-coordinate ($k = -28$) of the vertex. Therefore, the coordinates of the vertex are (-8, -28). This point is the minimum point of the parabola represented by the equation $y = \frac{3}{4}x^2 + 12x + 20$.

The vertex coordinates provide valuable information about the quadratic function and its graph. The x-coordinate, -8, represents the axis of symmetry. The y-coordinate, -28, represents the minimum value of the function. These coordinates are key indicators of the parabola's behavior.

Furthermore, the vertex form of the equation, $y = a(x - h)^2 + k$, can be directly obtained using the vertex coordinates. In this case, the vertex form is:

$y = \frac{3}{4}(x + 8)^2 - 28$

The vertex form provides a clear representation of the vertex and the parabola's stretch or compression factor ('a'). This form is particularly useful for graphing the parabola and understanding its transformations. The relationship between the standard form and the vertex form is a fundamental concept in understanding quadratic functions.

Conclusion

In this comprehensive guide, we have explored the process of calculating the vertex coordinates of the quadratic equation $y = 12x + 20 + \frac{3}{4}x^2$. We started by understanding the basics of quadratic equations and their properties, including the concept of the vertex and its significance. We then rewrote the equation in standard form, identified the coefficients, and applied the vertex formula to calculate the x-coordinate. We substituted this value back into the equation to find the y-coordinate. Finally, we interpreted the vertex coordinates and their significance in the context of the parabola and the quadratic function.

Mastering this process is essential for anyone studying algebra and its applications. The ability to find the vertex of a quadratic equation opens doors to solving a wide range of problems in mathematics, physics, economics, and other fields. By understanding the concepts and practicing the techniques, you can confidently tackle quadratic equations and their applications. Remember, the vertex is not just a point on a graph; it is a key to understanding the behavior of a quadratic function.

This comprehensive guide provides a strong foundation for further exploration of quadratic functions and their applications. We encourage you to practice with different equations and explore various scenarios to solidify your understanding. The more you practice, the more confident you will become in your ability to solve quadratic equations and interpret their solutions.