Select The Correct Answer.A Parabola Has A Minimum Value Of 0, A $y$-intercept Of 4, And An Axis Of Symmetry At $x = -2$.Which Graph Matches The Description?
Introduction to Parabolas
A parabola is a fundamental concept in mathematics, particularly in algebra and geometry. It is a U-shaped curve that can be described by a quadratic equation in the form of . In this article, we will explore the characteristics of a parabola and use them to select the correct graph that matches a given description.
Characteristics of a Parabola
A parabola has several key characteristics that can be used to identify it. These include:
- Minimum Value: The minimum value of a parabola is the lowest point on the curve. It can be either a minimum or a maximum value, depending on the direction of the parabola.
- Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is a line of symmetry, meaning that the parabola is reflected on either side of this line.
- Vertex: The vertex is the point on the parabola where the axis of symmetry intersects the curve. It is the highest or lowest point on the curve, depending on the direction of the parabola.
- Y-Intercept: The y-intercept is the point where the parabola intersects the y-axis. It is the value of y when x is equal to 0.
Given Description
The given description states that the parabola has a minimum value of 0, a y-intercept of 4, and an axis of symmetry at x = -2. We will use these characteristics to select the correct graph that matches this description.
Graph Analysis
To select the correct graph, we need to analyze each graph and determine which one matches the given description. We will look for the following characteristics:
- Minimum Value: The minimum value of the parabola should be 0.
- Axis of Symmetry: The axis of symmetry should be at x = -2.
- Y-Intercept: The y-intercept should be 4.
Graph 1
Graph 1 has a minimum value of 0, but the axis of symmetry is at x = 0, not x = -2. The y-intercept is also not 4. Therefore, Graph 1 does not match the given description.
Graph 2
Graph 2 has a minimum value of 0 and an axis of symmetry at x = -2. However, the y-intercept is not 4. Therefore, Graph 2 does not match the given description.
Graph 3
Graph 3 has a minimum value of 0, an axis of symmetry at x = -2, and a y-intercept of 4. Therefore, Graph 3 matches the given description.
Conclusion
In conclusion, the correct graph that matches the given description is Graph 3. This graph has a minimum value of 0, an axis of symmetry at x = -2, and a y-intercept of 4. These characteristics are consistent with the description provided.
Understanding Parabolas
Parabolas are an essential concept in mathematics, particularly in algebra and geometry. They are used to model real-world situations, such as the trajectory of a projectile or the shape of a satellite dish. Understanding the characteristics of a parabola is crucial in selecting the correct graph that matches a given description.
Key Takeaways
- A parabola has a minimum value, axis of symmetry, vertex, and y-intercept.
- The minimum value of a parabola can be either a minimum or a maximum value, depending on the direction of the parabola.
- The axis of symmetry is a vertical line that passes through the vertex of the parabola.
- The vertex is the point on the parabola where the axis of symmetry intersects the curve.
- The y-intercept is the point where the parabola intersects the y-axis.
Real-World Applications
Parabolas have numerous real-world applications, including:
- Projectile Motion: Parabolas are used to model the trajectory of a projectile, such as a thrown ball or a launched rocket.
- Satellite Dish Design: Parabolas are used to design satellite dishes, which are used to receive and transmit signals from satellites.
- Optics: Parabolas are used in optics to design lenses and mirrors, which are used to focus and redirect light.
Conclusion
In conclusion, parabolas are a fundamental concept in mathematics, particularly in algebra and geometry. Understanding the characteristics of a parabola is crucial in selecting the correct graph that matches a given description. Parabolas have numerous real-world applications, including projectile motion, satellite dish design, and optics.
Introduction
Parabolas are a fundamental concept in mathematics, particularly in algebra and geometry. They are used to model real-world situations, such as the trajectory of a projectile or the shape of a satellite dish. In this article, we will answer some frequently asked questions about parabolas and provide a deeper understanding of this essential concept.
Q: What is a parabola?
A: A parabola is a U-shaped curve that can be described by a quadratic equation in the form of . It has a minimum or maximum value, an axis of symmetry, and a vertex.
Q: What are the characteristics of a parabola?
A: The characteristics of a parabola include:
- Minimum Value: The minimum value of a parabola is the lowest point on the curve. It can be either a minimum or a maximum value, depending on the direction of the parabola.
- Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is a line of symmetry, meaning that the parabola is reflected on either side of this line.
- Vertex: The vertex is the point on the parabola where the axis of symmetry intersects the curve. It is the highest or lowest point on the curve, depending on the direction of the parabola.
- Y-Intercept: The y-intercept is the point where the parabola intersects the y-axis. It is the value of y when x is equal to 0.
Q: How do I determine the axis of symmetry of a parabola?
A: To determine the axis of symmetry of a parabola, you need to find the vertex of the parabola. The vertex is the point on the parabola where the axis of symmetry intersects the curve. The axis of symmetry is a vertical line that passes through the vertex.
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point on the parabola where the axis of symmetry intersects the curve. It is the highest or lowest point on the curve, depending on the direction of the parabola.
Q: How do I determine the y-intercept of a parabola?
A: To determine the y-intercept of a parabola, you need to find the point where the parabola intersects the y-axis. The y-intercept is the value of y when x is equal to 0.
Q: What is the minimum value of a parabola?
A: The minimum value of a parabola is the lowest point on the curve. It can be either a minimum or a maximum value, depending on the direction of the parabola.
Q: How do I determine the minimum value of a parabola?
A: To determine the minimum value of a parabola, you need to find the vertex of the parabola. The vertex is the point on the parabola where the axis of symmetry intersects the curve. The minimum value is the value of y at the vertex.
Q: What are some real-world applications of parabolas?
A: Parabolas have numerous real-world applications, including:
- Projectile Motion: Parabolas are used to model the trajectory of a projectile, such as a thrown ball or a launched rocket.
- Satellite Dish Design: Parabolas are used to design satellite dishes, which are used to receive and transmit signals from satellites.
- Optics: Parabolas are used in optics to design lenses and mirrors, which are used to focus and redirect light.
Conclusion
In conclusion, parabolas are a fundamental concept in mathematics, particularly in algebra and geometry. Understanding the characteristics of a parabola is crucial in selecting the correct graph that matches a given description. Parabolas have numerous real-world applications, including projectile motion, satellite dish design, and optics.