Explain how you could write a quadratic function in factored form that would have a vertex with an x-coordinate of 3 and two distinct roots.
Answers
Answer:
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.
→ 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
→ y = x^2 -6x + 9-1
→ y = x^2 -6x + 8
→ y = (x - 4)(x - 2)
Hope it helps ♡