Axis of Symmetry and Vertices
Graphs of y = a ( x + d )^{ 2 }+ k
Remember that
• when a positive number k is added to a quadratic
function, y = f ( x ) + k , then the parabola shifts up k units,
and
• when k is subtracted from a quadratic function, y = f (
x )  k , then the parabola shifts down k units.
This is because if the graph of y = f ( x ) has a minimum
point (the parabola opens up), then the minimum value of y = f (
x) + k is k more than the minimum value of y = f ( x ). Moreover,
the minimum values of y = f ( x ) + k and y = f ( x ) occur at
the same x value. So, if the graph of y = f ( x ) has a minimum
point at ( x , y ), then the graph of y = f ( x ) + k has a
minimum point at ( x , y + k ). Since the x value of the minimum
point is unchanged, the equation of the axis of symmetry is also
unchanged.
In the same way, if the quadratic function y = f ( x ) has a
maximum point at ( x , y ) (the parabola opens down), then the
maximum point of the graph of y = f ( x ) + k is ( x , y + k ).
We can now apply this discussion to the functions of the form f (
x ) = a( x + d )^{ 2} that we have just been studying.
The graphs of quadratic functions of the form y = f ( x ) + k or
y = a ( x + d ) ^{2} + k have the following properties,
where a is nonzero, and d and k are any numbers.
1. The axis of symmetry is the line x = d .
2. The vertex is at (  d , k ). If a is positive, the
parabola opens up and the vertex is a minimum point. If a is
negative, the parabola opens down and the vertex is a maximum
point.
Now notice that if we write y = a( x + d )^{ 2} + k in
standard form y = ax^{ 2} + bx + c , we can see how to
get the formula for the axis of
symmetry.
Write this calculation on the chalkboard. Comparing the
coefficient of x , we have
Since the graph of y = a( x + d )^{ 2} + k has an axis
of symmetry x = d , this is equal to the equation . End this lesson by
providing the following summary. Explain that any quadratic
function y = ax^{2} + bx + c can be written in the form y
= a( x + d )^{2} + k by a process called completing
the square. Therefore, these justifications for the
formulas for the axis of symmetry and the vertex work for any
quadratic function.
Function 
Effect on Graph 
Axis of Symmetry 
Vertex 
y = ax^{ 2}

Graph widens ( a < 1) or narrows ( a
> 1). 

y = a (0)^{
2} = 0 Vertex is at (0, 0).

Graphs opens up ( a > 0) or down ( a
< 0) 
y = x^{ 2} + c 
Graph moves up. 

y = (0)^{ 2} + c =
c Vertex is at (0, c ).

y = x^{ 2}  c 
Graph moves down. 
y = (0)^{ 2}  c =
c Vertex is at (0,  c ).

y = ( x + d )^{ 2}

Graph moves left. 

y = (  d + d )^{ 2}
+ 0 Vertex is at (  d , 0).

y = ( x  d )^{ 2}

Graph moves right. 

y = ( d  d )^{ 2}
+ 0 Vertex is at ( d , 0).

y = a( x + d )^{ 2}
+ k 
Width and direction of graph depend on a. 
x = d 
y = a (  d + d )^{ 2}
+ k = 0 + k or k
Vertex is at (d , k ).

y = a( x  d )^{ 2}
+ k 
Width and direction of graph depend on a. 
x = d 
y = a [ d + (  d )]^{
2} + k = 0 + k or k
Vertex is at ( d , k ).

