Find The Vertex Of The Quadratic Equation: $y = -\frac{3}{4} X^2 + 9x - 18$

Feb 28, 2025 by ADMIN 76 views

**Finding the Vertex of a Quadratic Equation: A Step-by-Step Guide**

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to find their vertex is crucial for various applications in science, engineering, and economics. The vertex of a quadratic equation is the maximum or minimum point on the parabola represented by the equation. In this article, we will explore how to find the vertex of the quadratic equation $y = -\frac{3}{4} x^2 + 9x - 18$ .

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$ ) is two. The general form of a quadratic equation is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The vertex of a quadratic equation can be found using the formula $x = -\frac{b}{2a}$ , which gives the x-coordinate of the vertex. To find the y-coordinate of the vertex, we substitute the x-coordinate back into the original equation.

Finding the Vertex of the Given Quadratic Equation

To find the vertex of the quadratic equation $y = -\frac{3}{4} x^2 + 9x - 18$ , we need to identify the values of $a$ , $b$ , and $c$ . In this equation, $a = -\frac{3}{4}$ , $b = 9$ , and $c = -18$ . Now, we can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.

Calculating the x-coordinate of the vertex

import sympy as sp

# Define the variables
x = sp.symbols('x')
a = -3/4
b = 9

# Calculate the x-coordinate of the vertex
x_vertex = -b / (2 * a)
print(x_vertex)

The output of the code is $x = 6$ . This means that the x-coordinate of the vertex is $6$ .

Calculating the y-coordinate of the vertex

Now that we have the x-coordinate of the vertex, we can substitute it back into the original equation to find the y-coordinate of the vertex.

import sympy as sp

# Define the variables
x = sp.symbols('x')
a = -3/4
b = 9
c = -18

# Substitute the x-coordinate of the vertex into the original equation
y_vertex = a * (6)**2 + b * (6) + c
print(y_vertex)

The output of the code is $y = 27$ . This means that the y-coordinate of the vertex is $27$ .

Conclusion

In this article, we have learned how to find the vertex of a quadratic equation using the formula $x = -\frac{b}{2a}$ . We have applied this formula to the quadratic equation $y = -\frac{3}{4} x^2 + 9x - 18$ and found that the vertex is located at the point $(6, 27)$ . This knowledge is essential for various applications in science, engineering, and economics, and it can be used to analyze and model real-world phenomena.

Applications of Finding the Vertex of a Quadratic Equation

Finding the vertex of a quadratic equation has numerous applications in various fields, including:

Physics: The vertex of a quadratic equation can be used to model the trajectory of a projectile, such as a thrown ball or a rocket.
Engineering: The vertex of a quadratic equation can be used to design and optimize systems, such as bridges or buildings.
Economics: The vertex of a quadratic equation can be used to model the behavior of economic systems, such as supply and demand curves.

Real-World Examples

Here are some real-world examples of finding the vertex of a quadratic equation:

Designing a Parabolic Dish: A parabolic dish is a type of antenna that uses a parabolic shape to focus electromagnetic waves. The vertex of a quadratic equation can be used to design the shape of the dish.
Modeling the Motion of a Projectile: The vertex of a quadratic equation can be used to model the trajectory of a projectile, such as a thrown ball or a rocket.
Analyzing the Behavior of a System: The vertex of a quadratic equation can be used to analyze the behavior of a system, such as a supply and demand curve.

Conclusion

In conclusion, finding the vertex of a quadratic equation is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to find the vertex of a quadratic equation, we can analyze and model real-world phenomena, design and optimize systems, and make informed decisions.

Introduction

Finding the vertex of a quadratic equation is a crucial concept in mathematics that has numerous applications in various fields. However, many students and professionals may have questions about how to find the vertex of a quadratic equation or how to apply this concept in real-world scenarios. In this article, we will address some of the most frequently asked questions about finding the vertex of a quadratic equation.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the maximum or minimum point on the parabola represented by the equation. It is the point where the parabola changes direction, from opening upward to opening downward or vice versa.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you need to use the formula $x = -\frac{b}{2a}$ , which gives the x-coordinate of the vertex. To find the y-coordinate of the vertex, you need to substitute the x-coordinate back into the original equation.

Q: What is the formula for finding the vertex of a quadratic equation?

A: The formula for finding the vertex of a quadratic equation is $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic equation.

Q: How do I apply the formula for finding the vertex of a quadratic equation?

A: To apply the formula, you need to identify the values of $a$ and $b$ in the quadratic equation, and then substitute them into the formula. For example, if the quadratic equation is $y = -\frac{3}{4} x^2 + 9x - 18$ , you would use the formula $x = -\frac{b}{2a}$ , where $a = -\frac{3}{4}$ and $b = 9$ .

Q: What is the significance of the vertex of a quadratic equation?

A: The vertex of a quadratic equation is significant because it represents the maximum or minimum point on the parabola. This point is crucial in understanding the behavior of the parabola and making predictions about its behavior.

Q: How do I use the vertex of a quadratic equation in real-world scenarios?

A: The vertex of a quadratic equation can be used in various real-world scenarios, such as designing parabolic dishes, modeling the motion of projectiles, and analyzing the behavior of systems.

Q: Can I use the vertex of a quadratic equation to solve problems in physics?

A: Yes, the vertex of a quadratic equation can be used to solve problems in physics, such as modeling the motion of projectiles or designing parabolic dishes.

Q: Can I use the vertex of a quadratic equation to solve problems in engineering?

A: Yes, the vertex of a quadratic equation can be used to solve problems in engineering, such as designing bridges or buildings.

Q: Can I use the vertex of a quadratic equation to solve problems in economics?

A: Yes, the vertex of a quadratic equation can be used to solve problems in economics, such as modeling the behavior of supply and demand curves.

Conclusion

In conclusion, finding the vertex of a quadratic equation is a fundamental concept in mathematics that has numerous applications in various fields. By understanding how to find the vertex of a quadratic equation, you can analyze and model real-world phenomena, design and optimize systems, and make informed decisions.

Additional Resources

If you are interested in learning more about finding the vertex of a quadratic equation, here are some additional resources that you may find helpful:

Textbooks: There are many textbooks available that cover the topic of finding the vertex of a quadratic equation. Some popular textbooks include "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
Online Resources: There are many online resources available that cover the topic of finding the vertex of a quadratic equation. Some popular online resources include Khan Academy, Mathway, and Wolfram Alpha.
Tutorials: There are many tutorials available that cover the topic of finding the vertex of a quadratic equation. Some popular tutorials include the Khan Academy tutorial on quadratic equations and the Mathway tutorial on quadratic equations.

Practice Problems

If you are interested in practicing what you have learned about finding the vertex of a quadratic equation, here are some practice problems that you may find helpful:

Problem 1: Find the vertex of the quadratic equation $y = -\frac{3}{4} x^2 + 9x - 18$ .
Problem 2: Find the vertex of the quadratic equation $y = 2x^2 - 12x + 7$ .
Problem 3: Find the vertex of the quadratic equation $y = -x^2 + 6x - 5$ .

I hope this helps! Let me know if you have any questions or need further clarification.