When we talk about the vertex of a graph, we are referring to the highest or lowest point of a parabolic function.
This point is also known as the maximum or minimum of the function, and it is located on the axis of symmetry of the parabola.
In this problem, we are given two equations and we need to determine which pair of equations generates graphs with the same vertex.
Find the vertex of a parabolic function
To find the vertex of a parabolic function in the form of y = ax² + bx + c, we can use the formula x = -b/2a to find the x-coordinate of the vertex. Once we have the x-coordinate, we can substitute it into the equation to find the y-coordinate.
Let’s consider the following pair of equations:
Equation1
y = 2x² + 4x + 1 y = 2(x + 1)² – 1
To find the vertex of the first equation, we can use the formula x = -b/2a. In this case, a = 2 and b = 4, so x = -4/(2*2) = -1. Substitute x = -1 into the equation to find the y-coordinate:
y = 2(-1)² + 4(-1) + 1 = -1
Therefore, the vertex of the first equation is (-1, -1).
To find the vertex of the second equation, we can see that it is already in vertex form, which is y = a(x – h)² + k. The vertex is located at (h, k), so in this case, the vertex is (-1, -1).
Therefore, the two equations generate graphs with the same vertex, which is (-1, -1).
Let’s consider another pair of equations:
Equation2
y = -3x² + 6x – 1 y = 3x² – 6x + 1
To find the vertex of the first equation, we can use the formula x = -b/2a. In this case, a = -3 and b = 6, so x = -6/(-2*3) = 1. Substitute x = 1 into the equation to find the y-coordinate:
y = -3(1)² + 6(1) – 1 = 2
Therefore, the vertex of the first equation is (1, 2).
To find the vertex of the second equation, we can use the same method as before. In this case, a = 3 and b = -6, so x = -(-6)/(2*3) = 1. Substitute x = 1 into the equation to find the y-coordinate:
y = 3(1)² – 6(1) + 1 = -2
Therefore, the vertex of the second equation is (1, -2).
Since the two equations have different vertices, they generate graphs with different vertices.
Conclusion
In conclusion, when we want to determine if two equations generate graphs with the same vertex, we can use the formula x = -b/2a to find the x-coordinate of the vertex.
If the two equations have the same x-coordinate for their vertices, we can substitute this value into the equations to find the y-coordinate.
If the two vertices have the same coordinates, then the equations generate graphs with the same vertex. If the two vertices have different coordinates, then the equations generate graphs with different vertices.