Sure, I'll guide you through the process of finding the equation of the quadratic function that has a vertex at the point (1,3) and includes the point (2,1).
Firstly you should remember, the formula of a quadratic function in vertex form:
y = a(x - h)^2 + k
Much of the information you need is already given in the vertex point (1,3), because h and k correspond to x and y of the vertex, respectively. That's why you can verify that in our case h = 1 and k = 3, so our equation by now looks like this:
y = a(x - 1)^2 + 3
Next, we need to find the value of a. To do this, we use the point (2,1) which is also on the graph of our quadratic function. The x value from the point replaces x in our equation, and the y value, similarly, replaces y.
So, we substitute x = 2 and y = 1 into our equation. This will give us a correct numerical equation, allowing us to solve for a:
1 = a(2 - 1)^2 + 3
Solve this and we have:
1 = a(1)^2 + 3
1 = a + 3
Rearranging to find a gives us:
a = 1 - 3
a = -2
Final step: substitute a = -2 back into our original equation, now we have all the coefficients:
y = -2(x - 1)^2 + 3
That's it, we've found the equation of the quadratic function:
y = -2(x - 1)^2 + 3
It has a vertex at point (1,3) and includes the point (2,1).