Which Equation Has A Graph That Is A Parabola With A Vertex At ( − 2 , 0 (-2,0 ( − 2 , 0 ]?A. Y = − 2 X 2 Y = -2x^2 Y = − 2 X 2 B. Y = ( X + 2 ) 2 Y = (x+2)^2 Y = ( X + 2 ) 2 C. Y = ( X − 2 ) 2 Y = (x-2)^2 Y = ( X − 2 ) 2 D. Y = X 2 − 2 Y = X^2 - 2 Y = X 2 − 2

Which Equation Has a Graph That is a Parabola with a Vertex at (-2,0)?

Understanding Parabolas and Their Vertices

A parabola is a type of quadratic function that can be represented in various forms, including standard, vertex, and factored forms. The vertex of a parabola is the point at which the parabola changes direction, and it is represented by the coordinates (h, k). In this article, we will explore which equation has a graph that is a parabola with a vertex at (-2,0).

The Standard Form of a Parabola

The standard form of a parabola is given by the equation y = ax^2 + bx + c, where a, b, and c are constants. The vertex of the parabola is given by the coordinates (-b/2a, f(-b/2a)), where f(x) is the function represented by the equation.

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. This form is useful for identifying the vertex of the parabola, as the vertex is represented by the coordinates (h, k).

The Factored Form of a Parabola

The factored form of a parabola is given by the equation y = a(x-r)(x-s), where r and s are the roots of the equation. This form is useful for identifying the roots of the equation, as the roots are represented by the values of x that make the equation equal to zero.

Analyzing the Options

Now that we have a basic understanding of parabolas and their vertices, let's analyze the options given in the problem.

Option A: y = -2x^2

This equation is in the standard form of a parabola, y = ax^2 + bx + c. To find the vertex of this parabola, we need to find the values of h and k. Since the equation is in the standard form, we can use the formula h = -b/2a to find the value of h. In this case, a = -2 and b = 0, so h = -0/2(-2) = 0. To find the value of k, we need to substitute x = 0 into the equation. This gives us y = -2(0)^2 = 0. Therefore, the vertex of this parabola is (0, 0), not (-2, 0).

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

This equation is in the vertex form of a parabola, y = a(x-h)^2 + k. To find the vertex of this parabola, we can identify the values of h and k. In this case, h = -2 and k = 0. Therefore, the vertex of this parabola is (-2, 0), which matches the given vertex.

Option C: y = (x-2)^2

This equation is also in the vertex form of a parabola, y = a(x-h)^2 + k. To find the vertex of this parabola, we can identify the values of h and k. In this case, h = 2 and k = 0. Therefore, the vertex of this parabola is (2, 0), not (-2, 0).

Option D: y = x^2 - 2

This equation is in the standard form of a parabola, y = ax^2 + bx + c. To find the vertex of this parabola, we need to find the values of h and k. Since the equation is in the standard form, we can use the formula h = -b/2a to find the value of h. In this case, a = 1 and b = 0, so h = -0/2(1) = 0. To find the value of k, we need to substitute x = 0 into the equation. This gives us y = (0)^2 - 2 = -2. Therefore, the vertex of this parabola is (0, -2), not (-2, 0).

Conclusion

Based on our analysis of the options, we can conclude that the equation y = (x+2)^2 has a graph that is a parabola with a vertex at (-2, 0). This equation is in the vertex form of a parabola, and the vertex is represented by the coordinates (h, k), where h = -2 and k = 0.

Key Takeaways

• 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.
• The vertex of a parabola can be identified by the coordinates (h, k), where h = -b/2a and k = f(-b/2a).
• The standard form of a parabola is given by the equation y = ax^2 + bx + c, where a, b, and c are constants.
• The factored form of a parabola is given by the equation y = a(x-r)(x-s), where r and s are the roots of the equation.

Final Answer

The final answer is B.
Q&A: Understanding Parabolas and Their Vertices

Frequently Asked Questions

In this article, we will answer some frequently asked questions about parabolas and their vertices.

Q: What is a parabola?

A: A parabola is a type of quadratic function that can be represented in various forms, including standard, vertex, and factored forms.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the parabola changes direction, and it is represented by the coordinates (h, k).

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

A: To find the vertex of a parabola, you can use the formula h = -b/2a, where a and b are the coefficients of the quadratic function.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is given 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.

Q: How do I identify the vertex of a parabola in the vertex form?

A: To identify the vertex of a parabola in the vertex form, you can look at the equation and identify the values of h and k.

Q: What is the factored form of a parabola?

A: The factored form of a parabola is given by the equation y = a(x-r)(x-s), where r and s are the roots of the equation.

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

A: To find the roots of a parabola, you can set the equation equal to zero and solve for x.

Q: Can a parabola have more than one vertex?

A: No, a parabola can only have one vertex.

Q: Can a parabola be a straight line?

A: No, a parabola is a curved shape and cannot be a straight line.

Q: Can a parabola be a circle?

A: No, a parabola is a curved shape and cannot be a circle.

Q: Can a parabola be a hyperbola?

A: No, a parabola is a curved shape and cannot be a hyperbola.

Q: Can a parabola be a ellipse?

A: No, a parabola is a curved shape and cannot be an ellipse.

Q: Can a parabola be a rational function?

A: Yes, a parabola can be a rational function.

Q: Can a parabola be a polynomial function?

A: Yes, a parabola can be a polynomial function.

Q: Can a parabola be a trigonometric function?

A: No, a parabola is not a trigonometric function.

Q: Can a parabola be a logarithmic function?

A: No, a parabola is not a logarithmic function.

Q: Can a parabola be an exponential function?

A: No, a parabola is not an exponential function.

Q: Can a parabola be a power function?

A: Yes, a parabola can be a power function.

Q: Can a parabola be a root function?

A: No, a parabola is not a root function.

Q: Can a parabola be a constant function?

A: No, a parabola is not a constant function.

Q: Can a parabola be a linear function?

A: No, a parabola is not a linear function.

Q: Can a parabola be a quadratic function?

A: Yes, a parabola is a quadratic function.

Conclusion

In this article, we have answered some frequently asked questions about parabolas and their vertices. We have discussed the standard form, vertex form, and factored form of a parabola, as well as how to find the vertex of a parabola. We have also discussed the different types of functions that a parabola can be, including rational, polynomial, trigonometric, logarithmic, exponential, power, root, constant, linear, and quadratic functions.

Key Takeaways

• A parabola is a type of quadratic function that can be represented in various forms, including standard, vertex, and factored forms.
• The vertex of a parabola is the point at which the parabola changes direction, and it is represented by the coordinates (h, k).
• The standard form of a parabola is given by the equation y = ax^2 + bx + c, where a, b, and c are constants.
• 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.
• The factored form of a parabola is given by the equation y = a(x-r)(x-s), where r and s are the roots of the equation.
• A parabola can only have one vertex.
• A parabola is a curved shape and cannot be a straight line, circle, hyperbola, or ellipse.
• A parabola can be a rational, polynomial, power, or quadratic function, but not a trigonometric, logarithmic, exponential, root, constant, or linear function.

Final Answer

The final answer is that a parabola is a type of quadratic function that can be represented in various forms, including standard, vertex, and factored forms, and it has a vertex that is represented by the coordinates (h, k).