Vertex Form for a Quadratic Equation
Vertex form for a quadratic equation (second instance) appears in Grade 7 math in Yoshiwara Intermediate Algebra. The form y = a(x-h)^2 + k immediately shows the parabola vertex at (h, k), the axis of symmetry x = h, and the direction of opening from the sign of a.
Key Concepts
Property A quadratic equation $y = ax^2 + bx + c$, $a \neq 0$, can be written in the vertex form $$y = a(x x v)^2 + y v$$ where the vertex of the graph is $(x v, y v)$. To convert from standard form, complete the square.
Examples The equation $y = 3(x 5)^2 + 1$ is in vertex form. By comparing it to $y = a(x x v)^2 + y v$, we can see the vertex is at $(5, 1)$.
For the equation $y = 4(x + 2)^2 7$, we can rewrite it as $y = 4(x ( 2))^2 7$. The vertex is at $( 2, 7)$.
Common Questions
What information does vertex form reveal directly?
Vertex form y = a(x-h)^2 + k directly reveals the vertex (h,k), the axis of symmetry x = h, and whether the parabola opens up (a>0) or down (a<0).
How do you convert vertex form to standard form?
Expand a(x-h)^2 + k by squaring the binomial and distributing a, then collect like terms to get ax^2 + bx + c.
How do you write vertex form if you know the vertex is (3, -2) and a = 1?
Substitute directly: y = 1(x-3)^2 + (-2) = (x-3)^2 - 2.
How do you find the x-intercepts from vertex form?
Set y = 0 and solve a(x-h)^2 + k = 0. Get (x-h)^2 = -k/a, then x = h ± √(-k/a).