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

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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)
The quadratic function in factored form is $$ y = (x - 3)(x + 1) $$.

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

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