Standard 9.0 Determine how the graph of a parabola changes as a, b, and c
vary in the equation
Students demonstrate and explain the effect that changing a coefficient has on the graph of
quadratic functions
GRAPHING A QUADRATIC FUNCTION
A quadratic function has the standard form
y = ax 2 + bx + c where a 0.
GRAPHING A QUADRATIC FUNCTION
The graph is “U-shaped” and is called a parabola.
GRAPHING A QUADRATIC FUNCTION
The highest or lowest point on the parabola is called the ver tex.
What is another word for highest?What is another word for lowest?
In your notes, draw and label the maxima of a parabola.
CORRECT!! The highest point is called maxima And the lowest point is called the minima?
GRAPHING A QUADRATIC FUNCTION
In general, the axis of symmetry for
the parabola is the vertical line
through the vertex.
GRAPHING A QUADRATIC FUNCTION
These are the graphs of y = x 2
and y = x 2.
GRAPHING A QUADRATIC FUNCTION
The y-axis is the axis of symmetryfor both graphs.
Graphing a Quadratic Function
Graph y = 2 x 2 – 8 x + 6
x = – = – = 2 b2 a
– 82(2)
y = 2(2)2 – 8 (2) + 6 = – 2
So, the vertex is (2, – 2).
(2, – 2)
The x-coordinate is:
The y-coordinate is:
Find and plot the vertex.
VERTEX FORM OF A QUADRATIC FUNCTION
GRAPHING A QUADRATIC FUNCTION
CHARACTERISTICS OF GRAPH
Vertex form: y = a (x – h)2 + k
For this form, the graph opens up if a > 0 and opens down if a < 0.
The vertex is (h, k ).
Open up “ + a”Open down “ - a”.
Regular, narrow, wide. “a”
Centered, horizontal shift left, horizontal shift right “h”
Centered, vertical shift up, vertical shift down “k”
Is h positive or negative?
positive move h units to the right
negative move h units to the left
Is a positive or negative?
positive open up
negative open down
Is k positive or negative?
positive move k units up
negative move k units down
Transforming the Graph of a Quadratic Function y=a(x-h)2+k
Quadratic Function8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
f x = x2
Graphing a Quadratic Function
y = a(x-h)2 + k8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2-1f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2-2f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2-3f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2-4f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2-5f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2-6f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2+0.8
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2+73
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2+4
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x2+5.3
f x = x2
Graphing a Quadratic Function
Things to Notice
• Where is the constant term located?
• Outside of the power of 2
• Inside the power of 2
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x-1 2f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x-2 2f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x-3 2f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x-4 2f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x-5 2f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x-6 2f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x+1 2
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x+2 2
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x+3.5 2
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x+5 2
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x+142
2
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = x+283
2
f x = x2
Graphing a Quadratic Function
-10 -5 5 10
8
6
4
2
-2
-4
-6
-8
f x = x2
Graphing a Quadratic Function
-10 -5 5 10
8
6
4
2
-2
-4
-6
-8
h x = -x2f x = x2
Graphing a Quadratic Function
Describe the change from blue to red.
-10 -5 5 10
8
6
4
2
-2
-4
-6
-8
h x = -x2
f x = x2
g x = - x-3 2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = -x2
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = -x2-1
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = -x2-3
f x = x2
Graphing a Quadratic Function
Parabola in foci in motion8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = -x2-5
f x = x2
Graphing a Quadratic Function
8
6
4
2
-2
-4
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-10 -5 5 10
f x = x-2 2-3
Graphing a Quadratic Function
Graphing a Quadratic Function
Graph y = – (x + 3)2 + 412
SOLUTION The function is in vertex form
y = a (x – h)2 + k.
a = – , h = – 3, and k = 4
12
First graph y=x2
4
2
-2
-4
-5 5
Graphing a Quadratic Function
Graph y = – (x + 3)2 + 412
SOLUTION The function is in vertex form
y = a (x – h)2 + k.
a = – , h = – 3, and k = 4
12
First graph y=x2
4
2
-2
-4
-5 5
4
2
-2
-4
-5 5
Then graph y x2
Graphing a Quadratic Function
Graph y = – (x + 3)2 + 412
SOLUTION The function is in vertex form
y = a (x – h)2 + k.
a = – , h = – 3, and k = 4
12
First graph y=x2
4
2
-2
-4
-5 5
4
2
-2
-4
-5 5
Then graph y x2
Then graph y (x 3)2
Graphing a Quadratic Function
Graph y = – (x + 3)2 + 412
SOLUTION The function is in vertex form
y = a (x – h)2 + k.
a = – , h = – 3, and k = 4
12
First graph y=x2
4
2
-2
-4
-5 5
4
2
-2
-4
-5 5
Then graph y x2
Then graph y (x 3)2 Then graph y (x 3)2 4
Graphing a Quadratic Function
Graph y = – (x + 3)2 + 412
SOLUTION The function is in vertex form
y = a (x – h)2 + k.
a = – , h = – 3, and k = 4
12
First graph y=x2
4
2
-2
-4
-5 5
Finaly graph y 1
2(x 3)2 4
Then graph y x2
Then graph y (x 3)2 Then graph y (x 3)2 4
4
2
-2
-4
-5 5
Graphing a Quadratic Function
Graph y = – (x + 3)2 + 412
The graph of y 1
2(x 3)2 4 is...
4
2
-2
-4
-5 5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
6
4
2
-2
-4
-6
-5 5
y x2
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
6
4
2
-2
-4
-6
-5 5
y (x 4)2
y x2
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x26
4
2
-2
-4
-6
-5 5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x2 3
y x26
4
2
-2
-4
-6
-5 5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x2
10
8
6
4
2
-2
-5 5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x2
10
8
6
4
2
-2
-5 5
y (x 2)2 4.5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x22
-2
-4
-6
-8
-10
-5 5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x2
y x2 6
2
-2
-4
-6
-8
-10
-5 5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x24
2
-2
-4
-6
-8
-5 5
Graphing a Quadratic Function
Write an equation that could describe the red Parabola
y x2
y (x 3)2 1.5
4
2
-2
-4
-6
-8
-5 5
Graphing a Quadratic Function
What can you conclude about the constantsinside the parenthesis and outside the parenthesis?
2( )h ky x