The Vertex Of This Parabola Is At ( 1 , 3 (1,3 ( 1 , 3 ]. Which Of The Following Could Be Its Equation?A. X = 3 ( Y − 3 ) 2 − 1 X = 3(y - 3)^2 - 1 X = 3 ( Y − 3 ) 2 − 1 B. Y = 3 ( X + 1 ) 2 − 3 Y = 3(x + 1)^2 - 3 Y = 3 ( X + 1 ) 2 − 3 C. Y = 3 ( X + 1 ) 2 + 3 Y = 3(x + 1)^2 + 3 Y = 3 ( X + 1 ) 2 + 3 D. X = 3 ( Y − 3 ) 2 + 1 X = 3(y - 3)^2 + 1 X = 3 ( Y − 3 ) 2 + 1

A parabola is a fundamental concept in mathematics, and understanding its equation is crucial for various applications in science, engineering, and other fields. The vertex of a parabola is a point that represents the minimum or maximum value of the function. In this article, we will explore the concept of the vertex of a parabola and determine which of the given equations could be its equation.

What is a Parabola?

A parabola is a quadratic function that can be represented in the form of $y = ax^2 + bx + c$ or $x = ay^2 + by + c$. The parabola has a unique shape, opening upwards or downwards, and has a single vertex. The vertex is the point where the parabola changes direction, and it represents the minimum or maximum value of the function.

The Vertex Form of a Parabola

The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this form, the vertex is the point $(h, k)$, and the parabola opens 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 Given Equations

We are given four equations, and we need to determine which one could be the equation of the parabola with its vertex at $(1, 3)$.

• A. $x = 3(y - 3)^2 - 1$
• B. $y = 3(x + 1)^2 - 3$
• C. $y = 3(x + 1)^2 + 3$
• D. $x = 3(y - 3)^2 + 1$

Analyzing the Equations

To determine which equation could be the equation of the parabola, we need to analyze each equation and compare it with the vertex form of a parabola.

Equation A: $x = 3(y - 3)^2 - 1$

This equation is in the form of $x = a(y - k)^2 + h$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(1, 3)$, and the equation is in the correct form. However, we need to check if the value of $a$ is positive or negative.

import sympy as sp
y = sp.symbols('y')

equation = sp.Eq(sp.sqrt((y - 3)**2), sp.sqrt(1/3))

solution = sp.solve(equation, y)

print(solution)

The solution to the equation is $y = 3 \pm \sqrt{1/3}$. Since the value of $a$ is positive, the parabola opens upwards.

Equation B: $y = 3(x + 1)^2 - 3$

This equation is in the form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(1, 3)$, and the equation is in the correct form. However, we need to check if the value of $a$ is positive or negative.

import sympy as sp

x = sp.symbols('x')

equation = sp.Eq(sp.sqrt((x + 1)**2), sp.sqrt(1/3))

solution = sp.solve(equation, x)

print(solution)

The solution to the equation is $x = -1 \pm \sqrt{1/3}$. Since the value of $a$ is positive, the parabola opens upwards.

Equation C: $y = 3(x + 1)^2 + 3$

This equation is in the form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(1, 3)$, and the equation is in the correct form. However, we need to check if the value of $a$ is positive or negative.

import sympy as sp

x = sp.symbols('x')

equation = sp.Eq(sp.sqrt((x + 1)**2), sp.sqrt(1/3))

solution = sp.solve(equation, x)

print(solution)

The solution to the equation is $x = -1 \pm \sqrt{1/3}$. Since the value of $a$ is positive, the parabola opens upwards.

Equation D: $x = 3(y - 3)^2 + 1$

This equation is in the form of $x = a(y - k)^2 + h$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(1, 3)$, and the equation is in the correct form. However, we need to check if the value of $a$ is positive or negative.

import sympy as sp

y = sp.symbols('y')

equation = sp.Eq(sp.sqrt((y - 3)**2), sp.sqrt(1/3))

solution = sp.solve(equation, y)

print(solution)

The solution to the equation is $y = 3 \pm \sqrt{1/3}$. Since the value of $a$ is positive, the parabola opens upwards.

Conclusion

Based on the analysis of the equations, we can conclude that the correct equation of the parabola with its vertex at $(1, 3)$ is:

• A. $x = 3(y - 3)^2 - 1$
• B. $y = 3(x + 1)^2 - 3$
• C. $y = 3(x + 1)^2 + 3$
• D. $x = 3(y - 3)^2 + 1$

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In our previous article, we explored the concept of the vertex of a parabola and determined which of the given equations could be its equation. In this article, we will answer some frequently asked questions related to the vertex of a parabola and its equation.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is a point that represents the minimum or maximum value of the function. It is the point where the parabola changes direction.

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

A: To find the vertex of a parabola, you can use the vertex form of a parabola, which is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. You can also use the fact that the vertex is the midpoint of the two x-intercepts of the parabola.

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful for finding the vertex of a parabola and for graphing the parabola.

Q: How do I determine the equation of a parabola with a given vertex?

A: To determine the equation of a parabola with a given vertex, you can use the vertex form of a parabola, which is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. You can also use the fact that the equation of a parabola is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is significant because it represents the minimum or maximum value of the function. It is also the point where the parabola changes direction.

Q: How do I graph a parabola with a given vertex?

A: To graph a parabola with a given vertex, you can use the vertex form of a parabola, which is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. You can also use the fact that the graph of a parabola is a U-shaped curve that opens upwards or downwards.

Q: What are some real-world applications of the vertex of a parabola?

A: The vertex of a parabola has many real-world applications, including:

• Projectile motion: The vertex of a parabola is used to model the trajectory of a projectile, such as a thrown ball or a rocket.
• Optimization: The vertex of a parabola is used to find the maximum or minimum value of a function, which is useful in optimization problems.
• Physics: The vertex of a parabola is used to model the motion of an object under the influence of gravity or other forces.

Conclusion

In conclusion, the vertex of a parabola is a fundamental concept in mathematics that has many real-world applications. Understanding the equation of a parabola with a given vertex is crucial for graphing the parabola and for solving optimization problems. We hope that this article has provided you with a better understanding of the vertex of a parabola and its equation.

Frequently Asked Questions

• Q: What is the vertex of a parabola? A: The vertex of a parabola is a point that represents the minimum or maximum value of the function.
• Q: How do I find the vertex of a parabola? A: To find the vertex of a parabola, you can use the vertex form of a parabola, which is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Q: What is the vertex form of a parabola? A: The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Q: How do I determine the equation of a parabola with a given vertex? A: To determine the equation of a parabola with a given vertex, you can use the vertex form of a parabola, which is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Related Topics

• Parabolas: Parabolas are quadratic functions that can be represented in the form of $y = ax^2 + bx + c$ or $x = ay^2 + by + c$.
• Vertex form: The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Graphing parabolas: Graphing parabolas involves plotting the points on a coordinate plane and drawing a smooth curve through the points.

Conclusion

In conclusion, the vertex of a parabola is a fundamental concept in mathematics that has many real-world applications. Understanding the equation of a parabola with a given vertex is crucial for graphing the parabola and for solving optimization problems. We hope that this article has provided you with a better understanding of the vertex of a parabola and its equation.