5.1 Stretching/Reflecting Quadratic Relations Every quadratic

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Kerr
MPM2D
Applying Quadratic Models
5.1 Stretching/Reflecting Quadratic Relations
Every quadratic relation begins as y = x2. This is your basic parabola.
x
y
-2
4
-1
1
0
0
1
1
2
4
Today we are going to transform the parabola in two different ways by
changing the shape and the direction of the opening.
Reflection in the x-axis
 Examine the value of a in y = ax2.
 If a is positive, the parabola opens up.
 If a is negative, the parabola opens down because it been
reflected in the x-axis.
Example #1:
Describe the transformation applied to the graph of y = x2.
a)
y = -x2
reflection in the x-axis
Vertical Stretch
 If a < -1 (less than -1) or if a > +1 (more than +1), then the
parabola undergoes a vertical stretch by a factor of a.
 A vertical stretch means the parabola becomes more narrow.
 Every y-coordinate on the parabola is multiplied by a.
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MPM2D
Example #2:
a)
y = 3x2
vertical stretch by a factor of 3
2
b)
y = -5x
vertical stretch by a factor of 5, reflection in
the x-axis
7 2
x
2
41
y = - x2
5
c)
7
2
41
vertical stretch by a factor of
, reflection in
5
y=
d)
vertical stretch by a factor of
the x-axis
Examine in more detail with a diagram.
x
y = x2
y
-2
-1
0
y = (-2)2
y = (-1)2
y = (0)2
4
1
0
1
2
y = (1)2
y = (2)2
1
4
x
y = 3x2
y
2
-2
-1
0
y = 3(-2)
y = 3(-1)2
y = 3(0)2
12
3
0
1
2
y = 3(1)2
y = 3(2)2
3
12
Vertical Compression
 If -1 < a < +1 (between -1 and +1) then the parabola undergoes a
vertical compression by a factor of a.
 A vertical compression means the parabola becomes more wide.
 Every y-coordinate on the parabola is multiplied by a.
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MPM2D
Example #3:
a)
y = -0.5x2
vertical compression by a factor of 0.5,
reflection in the x-axis
1 2
x
4
b)
y=
vertical compression by a factor of
c)
y = 0.1x2
vertical compression by a factor of 0.1
d)
2
y = - x2
11
in the x-axis
1
4
2
, reflection
11
Examine in more detail with a diagram.
x
y = x2
y
-2
-1
0
y = (-2)2
y = (-1)2
y = (0)2
4
1
0
1
2
y = (1)2
y = (2)2
1
4
x
y = 0.5x2
y = 0.5(-2)2
y
-2
-1
0
2
y = 0.5(-1)
y = 0.5(0)2
2
0.5
0
1
2
y = 0.5(1)2
y = 0.5(2)2
0.5
2
More Examples:
4.
The graph of y = x2 is transformed to y = ax2 (a ≠ 1). For each
point on y = x2, determine the coordinates of the transformed
point for the indicated value of a.
a)
(5, 7) when a = -2
(5, -14)
The value of a only affects the ycoordinate of the vertex. The xcoordinate remains the same and the
y-coordinate is multiplied by a.
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MPM2D
b)
(-2, -20) when a =
5.
Describe the transformation(s) that were applied to the graph
of y = x2. Write the equation of the transformed graph.
1
10
(-2, -2)
Examine the y-coordinate of each graph when x = 1.
(1, 1)
(1, 2)
The y-coordinate was multiplied by 2.
vertical stretch by a factor of 2
y = 2x2
6.
Identify the transformation(s) that must be applied to the graph
of y = x2 to create a graph of each equation. Then state the
coordinates of the image of the point (-3, 5).
a)
y = - x2
1
6
5
(-3, - )
6
the x-axis
1
, reflection in
6
The x-coordinate of the point stays the same.
We must multiply the y-coordinate by the vertical
compression factor. (5 x -
b)
y = 2.5x2
(-3, 12.5)
1
)
6
vertical stretch by a factor of 2.5
The x-coordinate of the point stays the same.
We must multiply the y-coordinate by the
vertical stretch factor. (5 x 2.5)
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MPM2D
7.
Determine the equation for the parabola below and describe the
transformation.
Substitute this point (2.5, 5)
into the equation y = ax2 to
solve for a.
y = ax2
5 = a(2.5)2
a = 0.8
y = 0.8x2
vertical compression by a factor by 0.8
Hints for the homework:
 Question #2 is just like example #4.
 Question #6 and 7 are just like example #7.
Homework: Page 256 #1-8, 10, 11

5.1 Stretching/Reflecting Quadratic Relations Every quadratic

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