Which Equation Has A Graph That Is A Parabola With A Vertex At ( 5 , 3 (5,3 ( 5 , 3 ]?A. Y = ( X − 5 ) 2 + 3 Y=(x-5)^2+3 Y = ( X − 5 ) 2 + 3 B. Y = ( X + 5 ) 2 + 3 Y=(x+5)^2+3 Y = ( X + 5 ) 2 + 3 C. Y = ( X − 3 ) 2 + 5 Y=(x-3)^2+5 Y = ( X − 3 ) 2 + 5 D. Y = ( X + 3 ) 2 + 5 Y=(x+3)^2+5 Y = ( X + 3 ) 2 + 5

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Which Equation Has a Graph That is a Parabola with a Vertex at (5,3)?

A parabola is a type of quadratic equation that has a U-shaped graph. The vertex of a parabola is the point at which the graph turns, and it is typically represented by the point (h, k). In this case, we are looking for an equation that has a vertex at (5, 3).

Understanding 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 equation, a is the coefficient that determines the direction and width of the parabola.

Analyzing the Options

Let's analyze each of the options to see which one has a vertex at (5, 3).

Option A: y = (x - 5)^2 + 3

In this equation, the vertex is given by the point (5, 3), which matches the given vertex. The equation is in the vertex form, with a = 1. This means that the parabola opens upward, and the vertex is at the point (5, 3).

Option B: y = (x + 5)^2 + 3

In this equation, the vertex is given by the point (-5, 3), which does not match the given vertex. The equation is also in the vertex form, with a = 1. This means that the parabola opens upward, but the vertex is at the point (-5, 3).

Option C: y = (x - 3)^2 + 5

In this equation, the vertex is given by the point (3, 5), which does not match the given vertex. The equation is also in the vertex form, with a = 1. This means that the parabola opens upward, but the vertex is at the point (3, 5).

Option D: y = (x + 3)^2 + 5

In this equation, the vertex is given by the point (-3, 5), which does not match the given vertex. The equation is also in the vertex form, with a = 1. This means that the parabola opens upward, but the vertex is at the point (-3, 5).

Conclusion

Based on the analysis of each option, the correct answer is Option A: y = (x - 5)^2 + 3. This equation has a graph that is a parabola with a vertex at (5, 3).

Why is Option A the Correct Answer?

Option A is the correct answer because it has a vertex at (5, 3), which matches the given vertex. The equation is in the vertex form, with a = 1, which means that the parabola opens upward. This is consistent with the typical shape of a parabola.

What is the Significance of the Vertex Form?

The vertex form of a parabola is significant because it allows us to easily identify the vertex of the parabola. The vertex form is also useful for graphing parabolas, as it provides a clear and concise way to represent the shape of the parabola.

How Can We Use the Vertex Form to Graph Parabolas?

To graph a parabola using the vertex form, we can follow these steps:

  1. Identify the vertex of the parabola, which is given by the point (h, k).
  2. Determine the direction of the parabola, which is determined by the coefficient a.
  3. Plot the vertex of the parabola on the coordinate plane.
  4. Use the vertex form to determine the shape of the parabola.
  5. Graph the parabola using the vertex form.

What are the Advantages of Using the Vertex Form?

The vertex form has several advantages, including:

  • It allows us to easily identify the vertex of the parabola.
  • It provides a clear and concise way to represent the shape of the parabola.
  • It is useful for graphing parabolas.
  • It is useful for solving problems involving parabolas.

What are the Disadvantages of Using the Vertex Form?

The vertex form has several disadvantages, including:

  • It can be difficult to determine the vertex of the parabola if the equation is not in the vertex form.
  • It can be difficult to graph the parabola if the equation is not in the vertex form.
  • It may not be useful for solving problems involving parabolas that are not in the vertex form.

Conclusion

In conclusion, the vertex form of a parabola is a useful tool for graphing and solving problems involving parabolas. It allows us to easily identify the vertex of the parabola and provides a clear and concise way to represent the shape of the parabola. The vertex form is also useful for solving problems involving parabolas, and it has several advantages over other forms of quadratic equations.
Q&A: Understanding Parabolas and the Vertex Form

Q: What is a parabola?

A: A parabola is a type of quadratic equation that has a U-shaped graph. It is a quadratic function that can be represented by the equation y = ax^2 + bx + c, where a, b, and c are constants.

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 graphing and solving problems involving parabolas.

Q: What is the significance of the vertex form?

A: The vertex form is significant because it allows us to easily identify the vertex of the parabola. It also provides a clear and concise way to represent the shape of the parabola.

Q: How can we use the vertex form to graph parabolas?

A: To graph a parabola using the vertex form, we can follow these steps:

  1. Identify the vertex of the parabola, which is given by the point (h, k).
  2. Determine the direction of the parabola, which is determined by the coefficient a.
  3. Plot the vertex of the parabola on the coordinate plane.
  4. Use the vertex form to determine the shape of the parabola.
  5. Graph the parabola using the vertex form.

Q: What are the advantages of using the vertex form?

A: The vertex form has several advantages, including:

  • It allows us to easily identify the vertex of the parabola.
  • It provides a clear and concise way to represent the shape of the parabola.
  • It is useful for graphing parabolas.
  • It is useful for solving problems involving parabolas.

Q: What are the disadvantages of using the vertex form?

A: The vertex form has several disadvantages, including:

  • It can be difficult to determine the vertex of the parabola if the equation is not in the vertex form.
  • It can be difficult to graph the parabola if the equation is not in the vertex form.
  • It may not be useful for solving problems involving parabolas that are not in the vertex form.

Q: How can we convert a quadratic equation to the vertex form?

A: To convert a quadratic equation to the vertex form, we can follow these steps:

  1. Complete the square by adding and subtracting the square of half the coefficient of x.
  2. Factor the perfect square trinomial.
  3. Write the equation in the vertex form.

Q: What is the relationship between the vertex form and the standard form of a parabola?

A: The vertex form and the standard form of a parabola are related by the equation y = a(x - h)^2 + k = ax^2 + bx + c, where (h, k) is the vertex of the parabola.

Q: How can we use the vertex form to solve problems involving parabolas?

A: To solve problems involving parabolas using the vertex form, we can follow these steps:

  1. Identify the vertex of the parabola, which is given by the point (h, k).
  2. Determine the direction of the parabola, which is determined by the coefficient a.
  3. Use the vertex form to determine the shape of the parabola.
  4. Solve the problem using the vertex form.

Q: What are some common applications of the vertex form?

A: The vertex form has several common applications, including:

  • Graphing parabolas
  • Solving problems involving parabolas
  • Finding the vertex of a parabola
  • Determining the direction of a parabola

Q: How can we use the vertex form to find the vertex of a parabola?

A: To find the vertex of a parabola using the vertex form, we can follow these steps:

  1. Identify the vertex of the parabola, which is given by the point (h, k).
  2. Determine the direction of the parabola, which is determined by the coefficient a.
  3. Use the vertex form to determine the shape of the parabola.
  4. Find the vertex of the parabola using the vertex form.

Q: What are some common mistakes to avoid when using the vertex form?

A: Some common mistakes to avoid when using the vertex form include:

  • Not identifying the vertex of the parabola correctly
  • Not determining the direction of the parabola correctly
  • Not using the vertex form to determine the shape of the parabola
  • Not solving the problem using the vertex form.