The form $y = a(x - p)(x - q)$ is called factored form (or intercept form). Thanks to the zero product property, it shows where the parabola crosses the $x$-axis.
Why it works
At $x = p$ the first factor is $0$, so $y = 0$; the same happens at $x = q$. These are the zeros of the function and the $x$-intercepts of the graph, and no other input gives $0$. By symmetry, the axis of the parabola lies halfway between the zeros, at $x = \frac{p + q}{2}$, so the vertex is there too. The number $a$ still controls direction and width.
Key idea
$y = a(x - p)(x - q)$ has zeros $p$ and $q$ (opposite in sign to the numbers in the factors), axis of symmetry $x = \frac{p + q}{2}$, and opens up when $a > 0$.
Worked examples
Worked example: Zeros and vertex
Find the zeros and the vertex of $y = 2(x + 1)(x - 5)$.
- Zeros: $x = -1$ and $x = 5$.
- Axis: $x = \frac{-1 + 5}{2} = 2$.
- Vertex height: $y = 2(2 + 1)(2 - 5) = 2 \cdot 3 \cdot (-3) = -18$.
Answer: Zeros $-1$ and $5$; vertex $(2, -18)$.
Worked example: Write from the zeros
A parabola has zeros $-3$ and $1$ and $y$-intercept $6$. Write it in factored form.
- Zeros give factors: $y = a(x + 3)(x - 1)$.
- At $x = 0$: $a(3)(-1) = -3a = 6$, so $a = -2$.
Answer: $y = -2(x + 3)(x - 1)$
Common mistake
Reading the zeros of $(x + 4)(x - 1)$ as $4$ and $-1$. A zero makes a factor equal $0$: $x + 4 = 0$ at $x = -4$. Check by substituting: $(-4 + 4)(-4 - 1) = 0$, but $(4 + 4)(4 - 1) = 24$.