Two distinct roots means two real solutions for x (the parabola needs to cross the x-axis twice)
Vertex form of a quadratic equation: (h,k) is vertex
y = a(x-h)^2 + k
The x of the vertex needs to equal 3
y = a(x-3)^2 + k
In order to have two distinct roots the parabola must be (+a) upward facing with vertex below the x-axis or (-a) downward facing with vertex above the x-axis. Parabolas are symmetrical so for an easy factorable equation make "a" 1 or -1 depending on if you want the upward/downward facing one.
y = (x-3)^2 - 1
Vertex (3,-1) upwards facing with two distinct roots 4 and 2
<span>The vertex lies on the axis of symmetry, so the axis of symmetry is x = 3. Find any two x-intercepts that are equal distance from the axis of symmetry. Use those x-intercepts to write factors of the function by subtracting their values from x. For example, 2 and 4 are each 1 unit from x = 3, so f(x) = (x – 2)(x – 4) is a possible function.</span>
