Select The Correct Answer.The Graph Of A Function Is A Parabola That Has A Minimum At The Point { (-3, 9)$}$. Which Equation Could Represent The Function { F$}$?A. { G(x) = 3(x - 3)^2 + 9$} B . \[ B. \[ B . \[ G(x) = -\frac{1}{2}(x

In mathematics, a parabola is a type of quadratic function that can be represented in various forms. One of the key characteristics of a parabola is its minimum or maximum point, which can be used to determine the equation of the function. In this article, we will explore the concept of a parabola with a minimum point and how to select the correct equation that represents the function.

What is a Parabola with a Minimum Point?

A parabola with a minimum point is a quadratic function that has a vertex, which is the lowest or highest point on the graph. The vertex is represented by the point ${(-3, 9)}$ in the given problem. This means that the parabola opens upwards and has a minimum value at the point ${(-3, 9)}$.

Key Characteristics of a Parabola with a Minimum Point

There are several key characteristics of a parabola with a minimum point that can be used to determine the equation of the function. These characteristics include:

• Vertex: The vertex is the point on the graph where the parabola has a minimum or maximum value. In this case, the vertex is ${(-3, 9)}$.
• Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the parabola. In this case, the axis of symmetry is ${x = -3}$.
• Direction of Opening: The parabola opens upwards, which means that it has a minimum value at the vertex.

Equation of a Parabola with a Minimum Point

The equation of a parabola with a minimum point can be represented in the form ${f(x) = a(x - h)^2 + k}$, where ${a}$ is the coefficient of the squared term, ${h}$ is the x-coordinate of the vertex, and ${k}$ is the y-coordinate of the vertex.

Selecting the Correct Equation

Given the information provided in the problem, we need to select the correct equation that represents the function ${f}$. The two options provided are:

A. ${g(x) = 3(x - 3)^2 + 9}$

B. ${g(x) = -\frac{1}{2}(x + 3)^2 + 9}$

To select the correct equation, we need to analyze the characteristics of the parabola and compare them with the options provided.

Analyzing Option A

Option A is ${g(x) = 3(x - 3)^2 + 9}$. Let's analyze the characteristics of this equation:

• Vertex: The vertex of this equation is ${(3, 9)}$, which is not the same as the given vertex ${(-3, 9)}$.
• Axis of Symmetry: The axis of symmetry of this equation is ${x = 3}$, which is not the same as the given axis of symmetry ${x = -3}$.
• Direction of Opening: The parabola opens upwards, which is consistent with the given information.

However, the vertex and axis of symmetry do not match the given information, so this option is not correct.

Analyzing Option B

Option B is ${g(x) = -\frac{1}{2}(x + 3)^2 + 9}$. Let's analyze the characteristics of this equation:

• Vertex: The vertex of this equation is ${(-3, 9)}$, which matches the given vertex.
• Axis of Symmetry: The axis of symmetry of this equation is ${x = -3}$, which matches the given axis of symmetry.
• Direction of Opening: The parabola opens upwards, which is consistent with the given information.

All the characteristics of this equation match the given information, so this option is correct.

Conclusion

In conclusion, the correct equation that represents the function ${f}$ is ${g(x) = -\frac{1}{2}(x + 3)^2 + 9}$. This equation matches the characteristics of the parabola, including the vertex, axis of symmetry, and direction of opening.

Final Answer

The final answer is:

In our previous article, we explored the concept of a parabola with a minimum point and how to select the correct equation that represents the function. In this article, we will answer some frequently asked questions related to the graph of a function - a parabola with a minimum point.

Q: What is the vertex of a parabola with a minimum point?

A: The vertex of a parabola with a minimum point is the lowest point on the graph. It is represented by the point ${(-3, 9)}$ in the given problem.

Q: What is the axis of symmetry of a parabola with a minimum point?

A: The axis of symmetry of a parabola with a minimum point is a vertical line that passes through the vertex and is perpendicular to the parabola. In this case, the axis of symmetry is ${x = -3}$.

Q: How do I determine the equation of a parabola with a minimum point?

A: To determine the equation of a parabola with a minimum point, you need to use the vertex form of a quadratic function, which is ${f(x) = a(x - h)^2 + k}$, where ${a}$ is the coefficient of the squared term, ${h}$ is the x-coordinate of the vertex, and ${k}$ is the y-coordinate of the vertex.

Q: What is the direction of opening of a parabola with a minimum point?

A: The direction of opening of a parabola with a minimum point is upwards, which means that it has a minimum value at the vertex.

Q: How do I select the correct equation that represents the function?

A: To select the correct equation that represents the function, you need to analyze the characteristics of the parabola, including the vertex, axis of symmetry, and direction of opening, and compare them with the options provided.

Q: What is the final answer to the problem?

A: The final answer to the problem is ${g(x) = -\frac{1}{2}(x + 3)^2 + 9}$.

Q: Can you provide more examples of parabolas with a minimum point?

A: Yes, here are a few more examples of parabolas with a minimum point:

• ${f(x) = 2(x - 2)^2 + 1}$
• ${g(x) = -\frac{1}{3}(x + 1)^2 + 4}$
• ${h(x) = 4(x - 1)^2 - 2}$

Q: How do I graph a parabola with a minimum point?

A: To graph a parabola with a minimum point, you need to use a graphing calculator or a computer program to plot the points on the graph. You can also use a table of values to help you graph the parabola.

Q: What are some real-world applications of parabolas with a minimum point?

A: Parabolas with a minimum point have many real-world applications, including:

• Optimization problems: Parabolas with a minimum point can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
• Physics: Parabolas with a minimum point can be used to model the motion of objects under the influence of gravity.
• Engineering: Parabolas with a minimum point can be used to design curves and surfaces for buildings and other structures.

Conclusion

In conclusion, the graph of a function - a parabola with a minimum point - is a fundamental concept in mathematics that has many real-world applications. By understanding the characteristics of a parabola with a minimum point, you can select the correct equation that represents the function and solve optimization problems, model the motion of objects, and design curves and surfaces for buildings and other structures.