Select The Correct Answer.A Parabola Has A Maximum Value Of 4 At $x = -1$, A $y$-intercept Of 3, And An $x$-intercept Of 1. Which Graph Matches The Description?A. B. C.

Introduction

A parabola is a fundamental concept in mathematics, representing a quadratic function in the form of $y = ax^2 + bx + c$. It has a unique shape, with a single turning point, known as the vertex. In this article, we will explore the characteristics of a parabola and how to identify the correct graph based on its description.

Characteristics of a Parabola

A parabola can be described by its vertex, axis of symmetry, and intercepts. The vertex is the point where the parabola changes direction, and it is represented by the coordinates $(h, k)$. The axis of symmetry is a vertical line that passes through the vertex, and it is represented by the equation $x = h$. The $y$-intercept is the point where the parabola intersects the $y$-axis, and it is represented by the coordinates $(0, k)$. The $x$-intercept is the point where the parabola intersects the $x$-axis, and it is represented by the coordinates $(h, 0)$.

Given Information

The given information describes a parabola with a maximum value of 4 at $x = -1$, a $y$-intercept of 3, and an $x$-intercept of 1. This information can be used to identify the vertex, axis of symmetry, and intercepts of the parabola.

Vertex

The vertex of the parabola is the point where the parabola changes direction. Since the parabola has a maximum value of 4 at $x = -1$, the vertex is located at the point $(-1, 4)$.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. Since the vertex is located at the point $(-1, 4)$, the axis of symmetry is represented by the equation $x = -1$.

Intercepts

The $y$-intercept is the point where the parabola intersects the $y$-axis. Since the $y$-intercept is 3, the parabola intersects the $y$-axis at the point $(0, 3)$.

The $x$-intercept is the point where the parabola intersects the $x$-axis. Since the $x$-intercept is 1, the parabola intersects the $x$-axis at the point $(1, 0)$.

Graph Identification

Based on the given information, we can identify the correct graph by analyzing the characteristics of the parabola.

• The vertex is located at the point $(-1, 4)$.
• The axis of symmetry is represented by the equation $x = -1$.
• The $y$-intercept is 3, and the parabola intersects the $y$-axis at the point $(0, 3)$.
• The $x$-intercept is 1, and the parabola intersects the $x$-axis at the point $(1, 0)$.

Conclusion

In conclusion, the correct graph that matches the description is the one that has a vertex at the point $(-1, 4)$, an axis of symmetry represented by the equation $x = -1$, a $y$-intercept of 3, and an $x$-intercept of 1.

Graph Options

A. Graph A

B. Graph B

C. Graph C

Answer

The correct answer is Graph A.

Explanation

Graph A has a vertex at the point $(-1, 4)$, an axis of symmetry represented by the equation $x = -1$, a $y$-intercept of 3, and an $x$-intercept of 1. This matches the description provided in the problem.

Final Thoughts

Introduction

In our previous article, we explored the characteristics of a parabola and how to identify the correct graph based on its description. In this article, we will answer some frequently asked questions about parabolas to help you better understand this fundamental concept in mathematics.

Q: What is a parabola?

A: A parabola is a quadratic function in the form of $y = ax^2 + bx + c$. It has a unique shape, with a single turning point, known as the vertex.

Q: What are the characteristics of a parabola?

A: A parabola can be described by its vertex, axis of symmetry, and intercepts. The vertex is the point where the parabola changes direction, and it is represented by the coordinates $(h, k)$. The axis of symmetry is a vertical line that passes through the vertex, and it is represented by the equation $x = h$. The $y$-intercept is the point where the parabola intersects the $y$-axis, and it is represented by the coordinates $(0, k)$. The $x$-intercept is the point where the parabola intersects the $x$-axis, and it is represented by the coordinates $(h, 0)$.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to find the values of $h$ and $k$ in the equation $y = ax^2 + bx + c$. The vertex is located at the point $(h, k)$.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex. It is represented by the equation $x = h$.

Q: How do I find the $y$-intercept of a parabola?

A: To find the $y$-intercept of a parabola, you need to find the value of $k$ in the equation $y = ax^2 + bx + c$. The $y$-intercept is the point where the parabola intersects the $y$-axis, and it is represented by the coordinates $(0, k)$.

Q: How do I find the $x$-intercept of a parabola?

A: To find the $x$-intercept of a parabola, you need to find the value of $h$ in the equation $y = ax^2 + bx + c$. The $x$-intercept is the point where the parabola intersects the $x$-axis, and it is represented by the coordinates $(h, 0)$.

Q: What is the difference between a parabola and a quadratic function?

A: A parabola is a graphical representation of a quadratic function, while a quadratic function is a mathematical expression in the form of $y = ax^2 + bx + c$. A parabola can be represented by a quadratic function, but not all quadratic functions represent a parabola.

Q: Can a parabola have more than one turning point?

A: No, a parabola can only have one turning point, known as the vertex.

Q: Can a parabola be represented by a linear equation?

A: No, a parabola cannot be represented by a linear equation. A linear equation is in the form of $y = mx + b$, while a parabola is in the form of $y = ax^2 + bx + c$.

Conclusion

In conclusion, we have answered some frequently asked questions about parabolas to help you better understand this fundamental concept in mathematics. We hope this article has provided you with a deeper understanding of parabolas and their characteristics.

Additional Resources

If you want to learn more about parabolas, we recommend checking out the following resources:

• Khan Academy: Parabolas
• Mathway: Parabolas
• Wolfram Alpha: Parabolas

Final Thoughts

In this article, we have explored some frequently asked questions about parabolas. We hope this article has provided you with a deeper understanding of parabolas and their characteristics. If you have any further questions, please don't hesitate to ask.