Coefficients of the Quadratic Equation

Oct 13, 2025 by miodeya.com 101 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding their characteristics is crucial for solving various problems in algebra and beyond. In this article, we will explore the characteristics of a given quadratic equation and determine which table correctly represents these characteristics.

The Quadratic Equation

The given quadratic equation is $y = 8x^2 - 8$ . To determine the characteristics of this equation, we need to identify the values of $a$ , $b$ , and $c$ .

Coefficients of the Quadratic Equation

$a$ is the coefficient of the squared term, which is $8$ in this case.
$b$ is the coefficient of the linear term, which is $0$ in this case.
$c$ is the constant term, which is $-8$ in this case.

Determining the Characteristics of the Quadratic Equation

Now that we have identified the values of $a$ , $b$ , and $c$ , we can determine the characteristics of the quadratic equation.

Up or Down: Since the coefficient of the squared term ( $a$ ) is positive, the parabola opens upward.
y-intercept: To find the $y$ -intercept, we need to substitute $x = 0$ into the equation. This gives us $y = 8(0)^2 - 8 = -8$ . Therefore, the $y$ -intercept is $-8$ .

Table A

$a$	$b$	$c$	Up or Down	$y$ -intercept
8	0	-8	Up	-8

Table B

$a$	$b$	$c$	Up or Down	$y$ -intercept
8	0	8	Up	8

Table C

$a$	$b$	$c$	Up or Down	$y$ -intercept
8	0	-8	Down	-8

Conclusion

Based on the characteristics of the quadratic equation $y = 8x^2 - 8$ , we can conclude that the correct table is Table A. This is because the coefficient of the squared term ( $a$ ) is positive, indicating that the parabola opens upward, and the $y$ -intercept is $-8$ , which is consistent with the equation.

Discussion

The characteristics of quadratic equations are essential for solving various problems in algebra and beyond. Understanding the up or down direction of the parabola and the $y$ -intercept is crucial for graphing and solving quadratic equations. In this article, we have explored the characteristics of a given quadratic equation and determined which table correctly represents these characteristics.

Key Takeaways

The coefficient of the squared term ( $a$ ) determines the up or down direction of the parabola.
The $y$ -intercept can be found by substituting $x = 0$ into the equation.
Understanding the characteristics of quadratic equations is essential for solving various problems in algebra and beyond.

References

[1] Algebra and Trigonometry, Michael Sullivan, 11th edition.
[2] Calculus, James Stewart, 8th edition.

Additional Resources

Khan Academy: Quadratic Equations
Mathway: Quadratic Equations
Wolfram Alpha: Quadratic Equations
Quadratic Equations: A Comprehensive Q&A Guide =====================================================

Introduction

Q1: What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually $x$ ) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q2: How Do I Determine the Characteristics of a Quadratic Equation?

To determine the characteristics of a quadratic equation, you need to identify the values of $a$ , $b$ , and $c$ . The coefficient of the squared term ( $a$ ) determines the up or down direction of the parabola. If $a$ is positive, the parabola opens upward, and if $a$ is negative, the parabola opens downward. The $y$ -intercept can be found by substituting $x = 0$ into the equation.

Q3: What is the Difference Between a Quadratic Equation and a Linear Equation?

A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is $ax + b = 0$ , where $a$ and $b$ are constants.

Q4: How Do I Solve a Quadratic Equation?

There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are constants.

Q5: What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are constants.

Q6: How Do I Use the Quadratic Formula?

To use the quadratic formula, you need to substitute the values of $a$ , $b$ , and $c$ into the formula and simplify. The formula will give you two solutions, which are the values of $x$ that satisfy the quadratic equation.

Q7: What is the Significance of the Discriminant?

The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$ . The discriminant determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.

Q8: How Do I Graph a Quadratic Equation?

To graph a quadratic equation, you need to identify the vertex of the parabola, which is the point where the parabola changes direction. The vertex can be found using the formula $x = -\frac{b}{2a}$ . Once you have found the vertex, you can use it to graph the parabola.

Q9: What is the Relationship Between Quadratic Equations and Real-World Applications?

Quadratic equations have numerous real-world applications, including physics, engineering, and economics. They are used to model the motion of objects, the growth of populations, and the behavior of financial markets.

Q10: How Do I Use Quadratic Equations in Real-World Situations?

Quadratic equations can be used to solve a wide range of real-world problems, including:

Modeling the motion of objects under the influence of gravity
Determining the maximum or minimum value of a function
Finding the area or perimeter of a shape
Solving optimization problems

Conclusion

Quadratic equations are a fundamental concept in mathematics, and understanding their characteristics is crucial for solving various problems in algebra and beyond. In this article, we have provided a comprehensive Q&A guide to help you better understand quadratic equations. Whether you are a student or a professional, this guide will help you to develop a deeper understanding of quadratic equations and their applications.

Additional Resources

Khan Academy: Quadratic Equations
Mathway: Quadratic Equations
Wolfram Alpha: Quadratic Equations
Algebra and Trigonometry, Michael Sullivan, 11th edition.
Calculus, James Stewart, 8th edition.