Two points determine a line. For a parabola you need three pieces of information, because $y = ax^2 + bx + c$ has three unknown numbers. Which form you start from depends on what you are given, and the vertex and the zeros are the most useful facts to have.
Idea clave
Given the vertex $(h, k)$, start from $y = a(x - h)^2 + k$. Given the zeros $r$ and $s$, start from $y = a(x - r)(x - s)$. Either way, one more point pins down $a$. Given three ordinary points, substitute each into $y = ax^2 + bx + c$ and solve the system.
Por qué funciona
Each form already has the given facts built in. In vertex form, every value of $a$ gives a parabola with vertex $(h, k)$; the extra point picks out the one stretch that passes through it. In factored form, every value of $a$ gives zeros at $r$ and $s$, because a product is 0 when a factor is 0. With three plain points, each one turns into a true equation in $a$, $b$ and $c$, and three equations are enough to find three unknowns.
Worked examples
Ejemplo resuelto: From the vertex and a point
A parabola has vertex $(2, -3)$ and passes through $(4, 5)$. Find its equation.
- $y = a(x - 2)^2 - 3$.
- Substitute $(4, 5)$: $5 = a(2)^2 - 3$, so $4a = 8$ and $a = 2$.
Respuesta: $y = 2(x - 2)^2 - 3$
Ejemplo resuelto: From the zeros and a point
A parabola has zeros $-1$ and $5$ and $y$-intercept $10$. Find its equation.
- $y = a(x + 1)(x - 5)$.
- The $y$-intercept is the point $(0, 10)$: $10 = a(1)(-5) = -5a$, so $a = -2$.
Respuesta: $y = -2(x + 1)(x - 5)$
Ejemplo resuelto: From three points
Find $y = ax^2 + bx + c$ through $(0, 1)$, $(1, 4)$ and $(2, 11)$.
- $(0, 1)$ gives $c = 1$.
- $(1, 4)$ gives $a + b + 1 = 4$, so $a + b = 3$.
- $(2, 11)$ gives $4a + 2b + 1 = 11$, so $4a + 2b = 10$, or $2a + b = 5$.
- Subtract the first from the second: $a = 2$, then $b = 1$.
Respuesta: $y = 2x^2 + x + 1$
Error común
Writing the factor for the zero $-1$ as $(x - 1)$. A zero at $x = r$ makes the factor $(x - r)$ equal 0, so the zero $-1$ gives $(x + 1)$. Check by substituting: $(-1) + 1 = 0$. Also check your final equation against every given point.