Vertex Form for a Quadratic Equation
Vertex form for a quadratic equation is a Grade 7 math skill from Yoshiwara Intermediate Algebra expressing a parabola as y = a(x-h)^2 + k. This form directly reveals the vertex (h, k), the axis of symmetry x = h, and whether the parabola opens upward or downward.
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 $(x v, y v)$.
Examples The equation $y = 2(x 1)^2 + 5$ is in vertex form. The vertex is at $(1, 5)$, and because $a = 2$ is negative, the parabola opens downward.
To write $y = x^2 8x + 10$ in vertex form, find the vertex. $x v = \frac{ ( 8)}{2(1)} = 4$. Then $y v = 4^2 8(4) + 10 = 6$. The vertex form is $y = (x 4)^2 6$.
Common Questions
What is vertex form for a quadratic?
Vertex form is y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola and a determines opening direction and width.
How do you identify the axis of symmetry from vertex form?
The axis of symmetry is the vertical line x = h, passing through the vertex.
How do you convert from vertex form to standard form?
Expand a(x-h)^2 + k by squaring the binomial and distributing a, then combine constant terms to get ax^2 + bx + c.
How does the sign of a affect the parabola?
If a > 0, the parabola opens upward with a minimum at the vertex. If a < 0, it opens downward with a maximum at the vertex.