Which Pair Of Equations Generates Graphs With The Same Vertex?A. $y=-(x+4)^2$ And $y=(x-4)^2$ B. $y=-4x^2$ And $y=4x^2$ C. $y=-x^2-4$ And $y=x^2+4$ D. $y=(x-4)^2$ And $y=x^2+4$

by ADMIN 182 views

Introduction

In mathematics, particularly in algebra and geometry, understanding the properties of quadratic equations is crucial. One of the key aspects of quadratic equations is the concept of a vertex, which represents the turning point of the parabola. In this article, we will explore which pair of equations generates graphs with the same vertex.

What is a Vertex?

A vertex is a point on a parabola that represents the minimum or maximum value of the quadratic function. It is the point where the parabola changes direction, either from opening upwards to downwards or vice versa. The vertex form of a quadratic equation is given by:

y = a(x - h)^2 + k

where (h, k) represents the coordinates of the vertex.

Vertex Form of Quadratic Equations

The vertex form of a quadratic equation is essential in understanding the properties of the parabola. By comparing the given options with the vertex form, we can identify the pairs of equations that generate graphs with the same vertex.

Option A: y=−(x+4)2y=-(x+4)^2 and y=(x−4)2y=(x-4)^2

Let's analyze the first option:

y = -(x + 4)^2

To convert this equation to vertex form, we can expand the squared term:

y = -(x^2 + 8x + 16)

Comparing this with the standard form of a quadratic equation (y = ax^2 + bx + c), we can identify the vertex as (h, k) = (-4, 0).

Now, let's analyze the second equation in the pair:

y = (x - 4)^2

Expanding the squared term, we get:

y = x^2 - 8x + 16

Comparing this with the standard form of a quadratic equation, we can identify the vertex as (h, k) = (4, 0).

Since the vertices of the two equations are different ((-4, 0) and (4, 0)), this pair of equations does not generate graphs with the same vertex.

Option B: y=−4x2y=-4x^2 and y=4x2y=4x^2

Let's analyze the first equation in the pair:

y = -4x^2

Comparing this with the standard form of a quadratic equation, we can identify the vertex as (h, k) = (0, 0).

Now, let's analyze the second equation in the pair:

y = 4x^2

Comparing this with the standard form of a quadratic equation, we can identify the vertex as (h, k) = (0, 0).

Since the vertices of the two equations are the same (0, 0), this pair of equations generates graphs with the same vertex.

Option C: y=−x2−4y=-x^2-4 and y=x2+4y=x^2+4

Let's analyze the first equation in the pair:

y = -x^2 - 4

Comparing this with the standard form of a quadratic equation, we can identify the vertex as (h, k) = (0, -4).

Now, let's analyze the second equation in the pair:

y = x^2 + 4

Comparing this with the standard form of a quadratic equation, we can identify the vertex as (h, k) = (0, 4).

Since the vertices of the two equations are different ((0, -4) and (0, 4)), this pair of equations does not generate graphs with the same vertex.

Option D: y=(x−4)2y=(x-4)^2 and y=x2+4y=x^2+4

Let's analyze the first equation in the pair:

y = (x - 4)^2

Expanding the squared term, we get:

y = x^2 - 8x + 16

Comparing this with the standard form of a quadratic equation, we can identify the vertex as (h, k) = (4, 0).

Now, let's analyze the second equation in the pair:

y = x^2 + 4

Comparing this with the standard form of a quadratic equation, we can identify the vertex as (h, k) = (0, 4).

Since the vertices of the two equations are different ((4, 0) and (0, 4)), this pair of equations does not generate graphs with the same vertex.

Conclusion

In conclusion, the pair of equations that generates graphs with the same vertex is:

  • y=−4x2y=-4x^2 and y=4x2y=4x^2

This pair of equations has the same vertex (0, 0), indicating that the graphs of the two equations intersect at the same point.

Key Takeaways

  • The vertex form of a quadratic equation is essential in understanding the properties of the parabola.
  • By comparing the given options with the vertex form, we can identify the pairs of equations that generate graphs with the same vertex.
  • The pair of equations y=−4x2y=-4x^2 and y=4x2y=4x^2 generates graphs with the same vertex (0, 0).
    Frequently Asked Questions (FAQs) =====================================

Q: What is the vertex form of a quadratic equation?

A: The vertex form of a quadratic equation is given by:

y = a(x - h)^2 + k

where (h, k) represents the coordinates of the vertex.

Q: How do I identify the vertex of a quadratic equation?

A: To identify the vertex of a quadratic equation, you can compare it with the standard form of a quadratic equation (y = ax^2 + bx + c). The vertex is represented by the point (h, k), where h = -b/2a and k = c - b^2/4a.

Q: What is the significance of the vertex in a quadratic equation?

A: The vertex represents the turning point of the parabola, where the parabola changes direction. It is also the point where the parabola has its minimum or maximum value.

Q: How do I determine if two quadratic equations have the same vertex?

A: To determine if two quadratic equations have the same vertex, you can compare their vertex forms. If the values of h and k are the same, then the two equations have the same vertex.

Q: Can two quadratic equations have the same vertex but different equations?

A: Yes, two quadratic equations can have the same vertex but different equations. This occurs when the equations are equivalent, meaning they represent the same parabola.

Q: How do I find the vertex of a quadratic equation in standard form?

A: To find the vertex of a quadratic equation in standard form, you can use the formula:

h = -b/2a k = c - b^2/4a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between the vertex form and standard form of a quadratic equation?

A: The vertex form and standard form of a quadratic equation are two different ways of representing the same parabola. The vertex form highlights the vertex of the parabola, while the standard form highlights the coefficients of the quadratic equation.

Q: Can a quadratic equation have more than one vertex?

A: No, a quadratic equation can have only one vertex. The vertex represents the turning point of the parabola, and it is a unique point on the parabola.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the vertex form or standard form. Plot the vertex on the coordinate plane, and then use the equation to find the x-coordinates of the points on the parabola. Plot these points on the coordinate plane, and then draw a smooth curve through the points to form the parabola.

Q: What is the relationship between the vertex and the axis of symmetry of a quadratic equation?

A: The vertex and the axis of symmetry of a quadratic equation are related. 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 the axis of symmetry.

Q: Can a quadratic equation have an axis of symmetry that is not a vertical line?

A: No, a quadratic equation can have only a vertical axis of symmetry. The axis of symmetry is a line that passes through the vertex of the parabola, and it is always a vertical line.

Q: How do I find the axis of symmetry of a quadratic equation?

A: To find the axis of symmetry of a quadratic equation, you can use the formula:

x = -b/2a

where a, b, and c are the coefficients of the quadratic equation. This formula gives the x-coordinate of the axis of symmetry.