Vertex form is an algebraic format used to express quadratic function rules.
y=a(x−h)2+k
In this form, a gives the direction of the parabola. When a>0, the parabola faces upward and when a<0, it faces downward. The vertex of the parabola lies at (h,k), and the axis of symmetry is x=h. Consider the graph of $f(x)=21 (x+4)_{2}+8.$
From the graph, we can connect the following characterisitcs to the function rule.When a quadratic function is written in vertex form, some characteristics of its graph can be identified.
y=a(x−h)2+k  

Direction  Vertex  Axis of Symmetry 
$a>0a<0 ⇒upward⇒downward $

(h,k)  x=h 
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. Since the axis of symmetry is x=h, here it is x=2.
Number  Function rule 

1  y=(x−1)2+3 
2  y=(x−1)2−3 
3  y=(x−1)2−3 
4  y=(x−1)2+3 
5  y=(x+1)2+3 
6  y=(x+1)2−3 
7  y=(x+1)2+3 
8  y=(x+1)2−3 
Number  Function rule 

2  y=(x−1)2−3 
3  y=(x−1)2−3 
5  y=(x+1)2+3 
7  y=(x+1)2+3 
There are two candidates remaining for each function. The rules with a=1 correspond to a downward parabola, whereas the rules with a=1 correspond to an upward parabola. To determine if any option is completely correct, we'd need more information about the parabola, specifically its direction or if its vertex is a minimum or a maximum.
Write the rule for the quadratic function that has the vertex (5,1) and passes through the point (10,11).
x=10, $y=11$
Subtract term
Calculate power
LHS+1=RHS+1
$LHS/25=RHS/25$
Simplify quotient
Rearrange equation
Since the parabola passes through the given points, the created rule is correct.