 • Introducing the concept: Transformation of
the Graph of y = x2
4
fx = x2
-5
2
5
Graph of f(x) =
2
ax
and
2
a(x-h)
• Objective: Graph a function f(x)=a(x-h)2,
and determine its characteristics.
Definition: A QUADRATIC FUNCTION is a function that
can be described as f(x) = ax2 + bc + c  0.
Graphs of QUADRATIC FUNCTIONS are called
PARABOLAS.
Now let us see the graphs of
FUNCTION
4
fx = -x2
gx = -2x2
hx = -0.5x2
fx = x2
2
gx = 2x2
2
LINE , OR AXIS
OF SYMMETRY
-5
hx = 0.5x2
-5
5
VERTEX
5
-2
VERTEX
-2
LINE , OR AXIS
OF SYMMETRY
-4
• Thus the y-axis is the LINE SYMMETRY.
The point (0,0) where the graph crosses the
line of symmetry, is called VERTEX OF
THE PARABOLA
Next consider f(x) = ax2, we know the following about its
a
graph. Compared with the graph
of f(x) = x2.
2. If
a &gt; 1, the graph is stretched vertically.
a &lt; 1, the graph is shrunk vertically.
3. If
a &lt; 0, the graph is reflected across the x-axis.
1. If
EXAMPLE:
a. Graph f(x) =3x2
b. Line of Symmetry? Vertex?
fx = 3x2
4
2
-5
5
LINE OF SYMMETRY
The y-axis
VERTEX
-2
(0,0)
Exercise:
2
a. Graph f(x) = -1/4 x
b. Line of symmetry and Vertex?
fx = -
 
x2
4
LINE OF SYMMETRY
2
Y-AXIS
VERTEX
(0,0)
-5
5
-2
-4
In f(x) = ax2, let us replace x by x – h. if
h is positive, the graph will be translated
to the right. If h is negative the
translation will be to the left. The line, or
axis of symmetry and the vertex will also
be translated the same way. Thus f(x) =
a(x-h)2, the axis of symmetry is x = h
and the vertex is (h, 0).
Compare the Graph of f(x) = 2(x+3)2 to the graph
of f(x) = 2x2.
4
4
fx = 2x+32
LINE OF
SYMMETRY,
X = -3
VERTEX (0,0),
fx = 2x2
2
2
SYMMETRY,
Y-AXIS
VERTEX
(0,3)
-5
5
-5
5
EXAMPLE:
2
a. Graph f(x) = - 2(x-1)
b. Line of Symmetry and Vertex?
fx = -2x-12
2
VERTEX (h, 0) = (1,0)
-5
LINE OF
SYMMETRY, X=1
5
-2
-4
-6
EXERCISES:
a. Graph f(x) = 3(x-2)2
b. Line of Symmetry and Vertex?
6
fx = 3x-2
2
4
2
-5
LINE OF
SYMMETRY, X=2
VERTEX (2,0)
5
Graph of f(x) =
2
a(x-h) +k
• Objective: Graph a function f(x) = a(x-h)2 +
k, and determine its characteristics.
In f(x) = a(x-h)2, let us replace f(x) by f(x) – k
f(x) – k = a(x-h)2
Adding k on both sides gives f(x) = a(x-h)2 + k.
The Graph will be translated UPWARD if k is Positive
and DOWNWARD if k is NEGATIVE. The Vertex will
be translated the same way. The Line of Symmetry will
NOT be AFFECTED
Guidelines for Graphing
f(x)=a(x-h)2 + k
•
1.
2.
3.
4.
When graphing quadratic function in the form
f(x)=a(x-h)2+k,
The line of symmetry is x-h=0, or x = h.
The vertex is (h,k).
If a &gt; 0, then (h,k) is the lowest point of the graph, and
k is the MINIMUM VALUE of the function.
If a &lt; 0, then (h,k) is the highest point of the graph,
and k is the MAXIMUM VALUE of the function.
Example:
a. Graph f(x) = 2(x+3)2 – 2
b. Line of Symmetry, Vertex?
c. is there a min/max value? If so, what is it?
fx = 2x+3
2-2
4
LINE OF SYMMETRY, X=-3
2
-5
5
VERTEX: ( -3,-2)
-2
MINIMUM:
-2
Exercises:
for each of the following, graph the function,
find the vertex, find the line of symmetry, and
find the min/ max value.
• 1. f(x) = 3(x-2)2 + 4
• 2. f(x) = -3(x+2)2 - 4
6
fx = 3x-2
2+4
VERTEX: (2,4)
4
2
MIN: 4
LINE OF SYMMETRY:X =2
-5
5
fx = -3x+22-1
4
VERTEX: (-2,-1)
2
MAX: -1
LINE OF SYMMETRY:X = -2
-5
5
-2
-4
ANALYZING f(x) = a(x-h)2+k
• Objective: Determine the characteristics of
a function f(x) = a(x-h)2+k
EXAMPLE:Without graphing, find the
vertex,line of symmetry, min/max value.
Given:
1. f(x) = 3(x-1/4)2+4
2. g(x) = -4x+5)2+7
a. What is the
Vertex?
b. Line of
Symmetry?
c. Is there a Min /
Max Value?
d. What is the
min / max value?
a. What is the
Vertex?
b. Line of
Symmetry?
#1. (1/4, -2)
#2. ( -5, 7)
X=&frac14;
X = -5
c. Is there a Min / Minimum. The
Maximum. The
Max Value?
graph extends
graph extends
upward since 3&gt;0 downward since
–4&lt;0.
d. What is the
Min.Value is –2 Max.Value is 7
min / max value?