To find the equation of a quadratic function given the vertex and a point, you can use the vertex form of the equation:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
Step 1: Use the given values for the vertex to determine the values of h and k:
h = 3
k = -4
Now we have:
y = a(x - 3)^2 - 4
Step 2: Substitute the coordinates of the given point (4,1) into the equation and solve for a:
1 = a(4 - 3)^2 - 4
1 = a(1)^2 - 4
1 = a - 4
a - 4 = 1
a = 5
Now we have the equation:
y = 5(x - 3)^2 - 4
To rewrite this equation in general form, we expand and simplify the equation:
y = 5(x - 3)(x - 3) - 4
y = 5(x^2 - 6x + 9) - 4
y = 5x^2 - 30x + 45 - 4
y = 5x^2 - 30x + 41
Therefore, the equation of the quadratic function in standard form is y = 5x^2 - 30x + 41.