Vertex of a Parabola
Vertex of a parabola is a Grade 7 math skill from Yoshiwara Intermediate Algebra identifying the highest or lowest point on a parabola. The vertex coordinates are (h, k) from vertex form or calculated as h = -b/(2a) in standard form, serving as the axis of symmetry.
Key Concepts
Property For the graph of $y = ax^2 + bx + c$, the $x$ coordinate of the vertex is $$x v = \frac{ b}{2a}$$ To find the $y$ coordinate of the vertex, substitute the value of $x v$ into the equation for $y$.
Examples To find the vertex of $y = x^2 + 8x + 10$, we identify $a=1$ and $b=8$. The x coordinate is $x v = \frac{ 8}{2(1)} = 4$. The y coordinate is $y v = ( 4)^2 + 8( 4) + 10 = 6$. The vertex is $( 4, 6)$.
For the graph of $y = 2x^2 12x + 5$, we have $a= 2$ and $b= 12$. The x coordinate of the vertex is $x v = \frac{ ( 12)}{2( 2)} = 3$. The y coordinate is $y v = 2( 3)^2 12( 3) + 5 = 23$. The vertex is $( 3, 23)$.
Common Questions
What is the vertex of a parabola?
The vertex is the turning point of the parabola — the minimum if it opens upward or the maximum if it opens downward.
How do you find the vertex from y = ax^2 + bx + c?
The x-coordinate of the vertex is h = -b/(2a). Substitute this into the equation to find the y-coordinate k.
How do you find the vertex from y = a(x-h)^2 + k?
The vertex is directly given as (h, k) in vertex form.
What is the significance of the vertex in word problems?
The vertex gives the maximum or minimum value. In physics (projectile height) or business (maximum profit), the vertex is the optimal point.