If a quadratic function has a vertex at (1, -4) and passes through the point (0, -3), write the equation in factored form.

To find the quadratic function in factored form with a vertex at \((1, -4)\) and passing through the point \((0, -3)\), follow these steps: 1. **Start with the Vertex Form:** The vertex form of a quadratic function is: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex. Given the vertex \((1, -4)\), the equation becomes: \[ y = a(x - 1)^2 - 4 \] 2. **Use the Given Point to Find \(a\):** Substitute the point \((0, -3)\) into the equation to solve for \(a\): \[ -3 = a(0 - 1)^2 - 4 \\ -3 = a(1) - 4 \\ a = 1 \] So, the vertex form is: \[ y = (x - 1)^2 - 4 \] 3. **Convert to Standard Form:** Expand the vertex form to get the standard quadratic form: \[ y = (x - 1)^2 - 4 \\ y = x^2 - 2x + 1 - 4 \\ y = x^2 - 2x - 3 \] 4. **Factor the Quadratic:** Factor \(x^2 - 2x - 3\): \[ y = (x - 3)(x + 1) \] **Verification:** - The roots are \(x = 3\) and \(x = -1\). - The axis of symmetry is \(\frac{3 + (-1)}{2} = 1\), which matches the vertex's x-coordinate. - Substituting \(x = 1\) into the equation gives \(y = -4\), confirming the vertex's y-coordinate. **Final Factored Form:** \[ y = (x - 3)(x + 1) \] **Answer:** After simplifying, a suitable factored form is (x – 3) multiplied by (x + 1). Thus, y = (x−3)(x+1)