Vertex Form
November 10, 2014
Page 34-35 in Notes
Objective
relate representations of quadratic
functions, such as algebraic, tabular,
graphical, and verbal descriptions[6.B]
Essential Question
• What parts of a quadratic function can I
determine from the vertex form?
Vocabulary
• parabola: the shape of a quadratic function
• vertex: the highest or lowest point on a
parabola
• y-intercept: the point where the graph crosses
the y-axis
• x-intercepts: the points where the graph
crosses the x-axis
• axis of symmetry: the vertical line that divides
a parabola in two equal parts
Vertex Form
• f(x) = a(x – h)2 + k
– “a” reflection across the x-axis and/or
vertical stretch or compression
– “h” horizontal translation
– “k”: vertical translation
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
h
k
Vertex:
(h, k)
Axis of
symmetry:
x=h
y-intercept:
(0, y)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
h
k
Vertex:
(h, k)
Axis of
symmetry:
x=h
y-intercept:
(0, y)
y=(x–2)2
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
k
Vertex:
(h, k)
Axis of
symmetry:
x=h
y-intercept:
(0, y)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
k
0
Vertex:
(h, k)
Axis of
symmetry:
x=h
y-intercept:
(0, y)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
k
0
Vertex:
(h, k)
(2, 0)
Axis of
symmetry:
x=h
y-intercept:
(0, y)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
k
0
Vertex:
(h, k)
(2, 0)
Axis of
symmetry:
x=h
x=2
y-intercept:
(0, y)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
k
0
Vertex:
(h, k)
(2, 0)
Axis of
symmetry:
x=h
x=2
y-intercept:
(0, y)
(0, 4)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
k
0
Vertex:
(h, k)
(2, 0)
Axis of
symmetry:
x=h
x=2
y-intercept:
(0, y)
(0, 4)
y = (x+3)2 – 1
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
y = (x+3)2 – 1
h
2
-3
k
0
Vertex:
(h, k)
(2, 0)
Axis of
symmetry:
x=h
x=2
y-intercept:
(0, y)
(0, 4)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
y = (x+3)2 – 1
h
2
-3
k
0
-1
Vertex:
(h, k)
(2, 0)
Axis of
symmetry:
x=h
x=2
y-intercept:
(0, y)
(0, 4)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
y = (x+3)2 – 1
h
2
-3
k
0
-1
Vertex:
(h, k)
(2, 0)
(-3, -1)
Axis of
symmetry:
x=h
x=2
y-intercept:
(0, y)
(0, 4)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
y = (x+3)2 – 1
h
2
-3
k
0
-1
Vertex:
(h, k)
(2, 0)
(-3, -1)
Axis of
symmetry:
x=h
x=2
x = -3
y-intercept:
(0, y)
(0, 4)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
y = (x+3)2 – 1
h
2
-3
k
0
-1
Vertex:
(h, k)
(2, 0)
(-3, -1)
Axis of
symmetry:
x=h
x=2
x = -3
y-intercept:
(0, y)
(0, 4)
(0, 8)
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
k
0
-1
Vertex:
(h, k)
(2, 0)
(-3, -1)
Axis of
symmetry:
x=h
x=2
x = -3
y-intercept:
(0, y)
(0, 4)
(0, 8)
y = (x+3)2 – 1 y= -3(x+2)2+4
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
k
0
-1
Vertex:
(h, k)
(2, 0)
(-3, -1)
Axis of
symmetry:
x=h
x=2
x = -3
y-intercept:
(0, y)
(0, 4)
(0, 8)
y = (x+3)2 – 1 y= -3(x+2)2+4
-2
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
k
0
-1
4
Vertex:
(h, k)
(2, 0)
(-3, -1)
Axis of
symmetry:
x=h
x=2
x = -3
y-intercept:
(0, y)
(0, 4)
(0, 8)
y = (x+3)2 – 1 y= -3(x+2)2+4
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
k
0
-1
4
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
Axis of
symmetry:
x=h
x=2
x = -3
y-intercept:
(0, y)
(0, 4)
(0, 8)
y = (x+3)2 – 1 y= -3(x+2)2+4
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
k
0
-1
4
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
y-intercept:
(0, y)
(0, 4)
(0, 8)
y = (x+3)2 – 1 y= -3(x+2)2+4
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
k
0
-1
4
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
y-intercept:
(0, y)
(0, 4)
(0, 8)
(0, -8)
y = (x+3)2 – 1 y= -3(x+2)2+4
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
k
0
-1
4
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
y-intercept:
(0, y)
(0, 4)
(0, 8)
(0, -8)
y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
k
0
-1
4
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
y-intercept:
(0, y)
(0, 4)
(0, 8)
(0, -8)
y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
-3
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
-3
k
0
-1
4
1
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
y-intercept:
(0, y)
(0, 4)
(0, 8)
(0, -8)
y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
-3
k
0
-1
4
1
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
(-3, 1)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
y-intercept:
(0, y)
(0, 4)
(0, 8)
(0, -8)
y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
-3
k
0
-1
4
1
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
(-3, 1)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
x = -3
y-intercept:
(0, y)
(0, 4)
(0, 8)
(0, -8)
y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
What can we determine from the Vertex Form?
Vertex Form:
y=a(x-h)2 + k
y=(x–2)2
h
2
-3
-2
-3
k
0
-1
4
1
Vertex:
(h, k)
(2, 0)
(-3, -1)
(-2, 4)
(-3, 1)
Axis of
symmetry:
x=h
x=2
x = -3
x = -2
x = -3
y-intercept:
(0, y)
(0, 4)
(0, 8)
(0, -8)
(0, 19)
y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
To Graph from Vertex Form:
1. Identify the vertex and axis of symmetry and
graph.
2. Find the y-intercept and graph along with its
reflection.
3. Make a table (with the vertex in the middle)
to calculate at least 5 points on the parabola.
Example 1 (Left Side): y = (x + 3)2 - 1
-3
-1 vertex:__________
(-3, -1)
h:_______
k:_______
x = -3
(0, 8)
axis of symmetry: ___________
y-int: ________
x
y
-5
-4
-3
-2
-1
3
0
-1
0
3
y = (x + 3)2 – 1
y = (0 + 3)2 – 1
y = (3)2 – 1
y=9–1
y=8
Example 2 (Left Side): y = -3(x – 2)2 + 4
2
4
(2, 4)
h:_______
k:_______
vertex:__________
x=2
(0, -8)
axis of symmetry: ___________
y-int: ________
x
y
0
1
2
3
4
-8
1
4
1
-8
y = -3(x – 2)2 + 4
y = -3(0 – 2)2 + 4
y = -3(-2)2 + 4
y = -3 • 4 + 4
y = -8
Assignment
1. f(x) =
2
x
–2
2. g(x) = -(x – 4)2
3. h(x) = (x +
2
1)
–3
4. j(x) = (x + 2)2 + 2
Reflection
1. How do you know if a parabola will open
upward or downward?
2. When does the parabola have a maximum
point?
3. When does the parabola have a minimum
point?