Characteristics of Quadratic Equations

Oct 13, 2025 by recoveryshake.com 128 views

**Selecting the Equation with a Graph Whose Vertex Has Both a Positive x- and a Positive y-Coordinate**

Understanding the Problem

When dealing with quadratic equations, it's essential to understand the characteristics of their graphs. One crucial aspect is the location of the vertex, which is the maximum or minimum point of the parabola. In this article, we'll explore how to select the equation with a graph whose vertex has both a positive x- and a positive y-coordinate.

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, and a cannot be zero.

Characteristics of Quadratic Equations

Quadratic equations can be classified into three main categories based on their discriminant (b^2 - 4ac):

Positive discriminant: The equation has two real and distinct roots.
Zero discriminant: The equation has one real root (repeated root).
Negative discriminant: The equation has no real roots.

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is:

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

where (h, k) is the vertex of the parabola.

Selecting the Equation with a Positive x- and y-Coordinate Vertex

To select the equation with a graph whose vertex has both a positive x- and a positive y-coordinate, we need to analyze each option:

Option A: y = -(x - 4)^2 + 3

The vertex of this parabola is (4, 3), which has both a positive x- and a positive y-coordinate.
The coefficient of the squared term is negative, indicating a downward-opening parabola.

Option B: y = 0.5(x + 2)^2 + 6

The vertex of this parabola is (-2, 6), which has a negative x-coordinate.
The coefficient of the squared term is positive, indicating an upward-opening parabola.

Option C: y = -2(x + 3)^2 - 1

The vertex of this parabola is (-3, -1), which has a negative x-coordinate.
The coefficient of the squared term is negative, indicating a downward-opening parabola.

Option D: y = x^2 - 8

The vertex of this parabola is (0, -8), which has a negative y-coordinate.
The coefficient of the squared term is positive, indicating an upward-opening parabola.

Conclusion

Based on the analysis, the equation with a graph whose vertex has both a positive x- and a positive y-coordinate is:

y = -(x - 4)^2 + 3

This equation represents a downward-opening parabola with a vertex at (4, 3), which meets the given criteria.

Frequently Asked Questions

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

A: The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Q: How do I determine the direction of the parabola?

A: The direction of the parabola is determined by the coefficient of the squared term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: What is the significance of the discriminant in a quadratic equation?

A: The discriminant (b^2 - 4ac) determines the nature of the roots of the quadratic equation. A positive discriminant indicates two real and distinct roots, a zero discriminant indicates one real root, and a negative discriminant indicates no real roots.

Q: How do I select the equation with a graph whose vertex has both a positive x- and a positive y-coordinate?

A: To select the equation, analyze each option by identifying the vertex and the direction of the parabola. The equation with a positive x- and y-coordinate vertex is the one that meets the given criteria.

Additional Resources

For more information on quadratic equations and their graphs, refer to the following resources:

Khan Academy: Quadratic Equations
Mathway: Quadratic Equations
Wolfram MathWorld: Quadratic Equation

By understanding the characteristics of quadratic equations and their graphs, you can select the equation with a graph whose vertex has both a positive x- and a positive y-coordinate.