Any quadratic equation can be written in vertex form. The vertex form represents a transformation of the parent function.
y = a(x- h)^2 + k
In this form, the vertex is located at ( h, k) and a is a parameter that can stretch or compress the parabola vertically, as well as perform a reflection across the x-axis. In our case, the only requirement is that our parabola must have its vertex located at (3,2).
h= 3 and k= 2
Therefore, we can use any value for a. Let's use a= 12. As a result, we get the following equation.
y= 12(x- 3)^2 + 2
We can now graph the parabola.
The axis of symmetry for a parabola is the vertical line passing through the vertex. In this case, it is represented by the line x= 3. Notice that this line divides the parabola in two halves in such a way that one is the mirror image of the other.
Finally, we can locate two points lying on the parabola by looking at the graph. However, if we did not have the graph at hand, we could do this by evaluating the quadratic function for two different arbitrary values of x.
x
1/2(x-3)^2+2
y
1
1/2( 1-3)^2+2
4
5
1/2( 5-3)^2+2
4
Notice that we could have used any value for a to get our equation. Furthermore, we could have evaluated the function at any two x values to get the points. Therefore, there are infinitely many possible solutions fulfilling this exercise's requirements. This is just one example solution.
