Select The Correct Answer.Select The Graph Of The Equation Below.$ Y = -\frac{1}{4} X^2 + 1 $A. B.

Feb 26, 2025 by ADMIN 102 views

**Selecting the Correct Graph: A Guide to Understanding Quadratic Equations**

Introduction

In mathematics, quadratic equations are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In this article, we will focus on understanding the graph of a quadratic equation and selecting the correct graph from a given set of options.

Understanding Quadratic Equations

A quadratic equation can be written in the form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic equation is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of a. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards.

The Graph of y = -\frac{1}{4} x^2 + 1

The given equation is y = -\frac{1}{4} x^2 + 1. To understand the graph of this equation, we need to analyze the coefficients of the equation. The coefficient of x^2 is -\frac{1}{4}, which is negative. This means that the parabola will open downwards. The coefficient of x is zero, which means that the parabola will not have any linear component. The constant term is 1, which means that the parabola will be shifted upwards by 1 unit.

Key Features of the Graph

The graph of y = -\frac{1}{4} x^2 + 1 has the following key features:

Vertex: The vertex of the parabola is the lowest point on the graph. Since the parabola opens downwards, the vertex will be the highest point on the graph. The x-coordinate of the vertex can be found using the formula -b/2a. In this case, b = 0, so the x-coordinate of the vertex is 0. The y-coordinate of the vertex can be found by substituting x = 0 into the equation. This gives y = 1.
Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is at (0, 1), the axis of symmetry is the y-axis.
Intercepts: The x-intercepts of the parabola are the points where the graph intersects the x-axis. To find the x-intercepts, we need to set y = 0 and solve for x. This gives the equation -\frac{1}{4} x^2 + 1 = 0. Solving for x, we get x = ±2√2.

Selecting the Correct Graph

Now that we have analyzed the graph of y = -\frac{1}{4} x^2 + 1, we can select the correct graph from the given options. The graph should have the following features:

Downward Opening: The graph should open downwards, since the coefficient of x^2 is negative.
No Linear Component: The graph should not have any linear component, since the coefficient of x is zero.
Shifted Upwards: The graph should be shifted upwards by 1 unit, since the constant term is 1.

Conclusion

In conclusion, the graph of y = -\frac{1}{4} x^2 + 1 is a downward-opening parabola with no linear component and shifted upwards by 1 unit. The graph has a vertex at (0, 1) and x-intercepts at ±2√2. By analyzing the key features of the graph, we can select the correct graph from the given options.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Graphing Quadratic Equations" by Purplemath

Additional Resources

[1] Khan Academy: Quadratic Equations
[2] Mathway: Quadratic Equation Solver

Final Answer

The final answer is:

Introduction

Quadratic equations are a fundamental concept in mathematics that plays a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and their applications.

Q: What is a quadratic equation?

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

Q: What is the graph of a quadratic equation?

A: The graph of a quadratic equation is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of a. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards.

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

A: To find the vertex of a quadratic equation, you can use the formula -b/2a. This will give you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you can substitute the x-coordinate into the equation.

Q: What is the axis of symmetry of a quadratic equation?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is at (0, 1), the axis of symmetry is the y-axis.

Q: How do I find the x-intercepts of a quadratic equation?

A: To find the x-intercepts of a quadratic equation, you need to set y = 0 and solve for x. This will give you the equation ax^2 + bx + c = 0. You can then solve for x using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the solutions to a quadratic equation. The formula is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. This will give you the solutions to the quadratic equation.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including:

Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

Write the equation: Write the quadratic equation in the form ax^2 + bx + c = 0.
Find the vertex: Find the vertex of the parabola using the formula -b/2a.
Find the x-intercepts: Find the x-intercepts of the parabola by setting y = 0 and solving for x.
Use the quadratic formula: Use the quadratic formula to find the solutions to the quadratic equation.