1.5 g raphing q uadratic f unctions by u sing t ransformations
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1.5 GRAPHING QUADRATIC FUNCTIONS BY USING TRANSFORMATIONS
Graphusing the graph of
2)3()( xxf2)( xxf
You move the key points of
To shift to the right 3 spaces you add 3 to all of the x values!
2)( xxf x y-2 4
-1 1
0 0
1 1
2 4
+31
2
3
4
5
And then graph the new set of points!
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
2)3()( xxf
(1 , 4)
(2 , 1)
(3 , 0)
(4 , 1)
(5 , 4)
-7
-6
-5
-4
-3
Graph
This is a shift to the left 5 spaces. To shift to the left 5 spaces you subtract 5 from all of the x values!
x y-2 4
-1 1
0 0
1 1
2 4
-5
And then graph the new set of points!
2)5()( xxf
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y 2)5()( xxfShift LEFT 5 Units
(-7 , 4)
(-6 , 1)
(-5 , 0)
(-4 , 1)
(-3 , 4)
Graph
This is a shift down 4 spaces. To shift down 4 spaces you subtract 4 from all of the y values!
0
-3
-4
-3
0
x y-2 4
-1 1
0 0
1 1
2 4
-4
And then graph the new set of points!
42)( xxf
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
42)( xxf
Shift DOWN 4 Units
(-2 , 0)
(-1 , -3)
(0, -4)
(1, -3)
(2, 0)
Graph
This is a shift up 6 spaces. To shift up 6 spaces you add 6 to all of the y values!
10
7
6
7
10
x y-2 4
-1 1
0 0
1 1
2 4
+6
And then graph the new set of points!
62)( xxf
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
62)( xxf
Shift UP 6 Units(-2 ,10)
(-1 , 7)
(0, 6)
(1, 7)
(2, 10)
Graph
This is a vertical stretch by a factor of 2. To stretch the parabola you multipy all of the y values by 2!
8
2
0
2
8
x y-2 4
-1 1
0 0
1 1
2 4
x2
And then graph the new set of points!
22)( xxf
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
22)( xxf Strectch by a factor of 2
(-2 ,8)
(-1 , 2)
(0, 0)
(1, 2)
(2, 8)
Graph
This is a vertical compression by a factor of one half. To compress the parabola you multipy all of the y values by 0.5!
(or divide them all by 2!)
2
0.5
0
0.5
2
x y-2 4
-1 1
0 0
1 1
2 4
÷2
And then graph the new set of points!
221)( xxf
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
221)( xxf Compress by a factor of
2
1
(-2 , 2)(-1 , 0.5)
(0, 0)(1, 0.5)
(2, 2)
Graph
This is a reflection in the x-axis. To reflect the parabola you multipy all of the y values by -1!
-4
-1
0
-1
-4
x y-2 4
-1 1
0 0
1 1
2 4
x -1
And then graph the new set of points!
2)( xxf
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
2)( xxf Reflect in the x-axis
(-1 , -1)
(-2 , -4)
(0, 0)
(2, -4)
(1, -1)
Graph
-8
-2
0
-2
-8
x y-2 4
-1 1
0 0
1 1
2 4
x -2
•reflect in the x-axis and stretch by a factor of 2
522)( xxf
•shift the parabola up 5
+5-3
3
5
3
-3
And then graph the new set of points!
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
522)( xxf
(-1 , 3)
(-2 , -3)
(0, 5)
(2, -3)
(1, 3)
Graph
-9
-8
-7
-6
-5
x y-2 4
-1 1
0 0
1 1
2 4
-7
•shift the parabola left 7
32)7()( xxf
•shift the parabola down 3
-31
-2
-3
-2
1
And then graph the new set of points!
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
32)7()( xxf
(-9 , 1)
(-8 , -2)
(-7, -3)
(-6, -2)
(-5, 1)
Graph
6
7
8
9
10
x y-2 4
-1 1
0 0
1 1
2 4
+8
•Compress by a factor of 0.25
2)8(41)( xxf
•shift the parabola right 8
÷41
0.25
0
0.25
1
And then graph the new set of points!
-10
-5
5
10
15
-15 -10 -5 5 10 15
X
Y
2)8(41)( xxf
(6, 1)(7, 0.25)
(8, 0)(9, 0.25)
(10, 1)
HOMEWORK:
PAGE 47 #5 – 12
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