When graphing equations, it is important to be able to identify which equations will produce the same graph. This is especially important when trying to determine the vertex of a certain equation. The vertex of an equation is the point at which the graph reaches its peak or valley, and it can be used to identify certain properties of the equation.

The vertex of a graph is determined by the equation of the graph. Generally speaking, if two equations have the same vertex, then their graphs will be similar. This means that if you were to graph both equations, the two graphs will have the same vertex. To find out which pair of equations generates graphs with the same vertex, you will need to find the equation of the graph with the desired vertex.

One way to do this is to find the equation of the line that contains the vertex. To find this equation, you will need to use the slope-intercept form of the equation, which is y = mx + b where m is the slope and b is the y-intercept. The slope can be determined by using the rise over run formula. The y-intercept can be determined by the coordinates of the vertex.

Once the equation of the line is determined, you can then find the equation of the graph with the same vertex by shifting the line. Shifting the line requires changing the y-intercept of the equation. To do this, you must set the new y-intercept to the desired vertex. For example, if the desired vertex is (3, 9), then the new equation would be y = mx + 9.

The pair of equations that generate graphs with the same vertex are the line equation and the shifted equation. Using the example above, these two equations are y = mx + b and y = mx + 9. If both equations are graphed, then the vertex of the graph will be (3, 9).

In conclusion, to find a pair of equations that generates graphs with the same vertex, you must find the equation of the line that contains the vertex and then shift the line by changing the y-intercept to the desired vertex. The pair of equations will be the line equation and the shifted equation.