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Transformation of graphsRemember: inside the brackets affects the x-coordinate, but the opposite way.
outside the brackets affects the y-coordinate.
Describing transformations examples
Graph 1: y = f(x)Graph 2: the graph has moved 2 to the right, so we have added 2 to the x-coordinate. inside the brackets affects the x-coordinate, but the opposite wayso graph 2 f(x – 2)
Graph 1: y = f(x)Graph 2: the y-coordinates are now all negative – they have been multiplied by -1. outside the brackets affects the y-coordinate.so graph 2 -1f(x)
Graph 1: y = f(x)Graph 2: the graph has moved 1 down,
Graph 1: y = f(x)Graph 2: the y-coordinates have been multiplied by 3.
1 2 1
2
1
2
so we have subtracted 1 from the y-coordinate. outside the brackets affects the y-coordinateso graph 2 f(x) - 1
outside the brackets affects the y-coordinateso graph 2 3f(x)
1
1
1
1
Affec
t x
or y
? (-12, 5) becomes
(18, -8) becomes
f(x + 3) x (-15, 5) (15, -8)f(x – 4)f(x) + 2f(x) – 3f(3x)f(½x) 2f(x)½ f(x)f(-x)-f(x)
Performing transformations This is the graph of f(x). Sketch each transformation on the grids provided,
stating the new coordinates of the marked point in each case.
Equation
Sketch Marked Coordinat
e1. example
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y Outside the brackets affects the y coordinate, so the graph will move down by 4.The new coordinate is (1, -3)
y = f(x) - 4
2.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
3.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
4.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
5.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
y = f(x + 3)
y = 2f(x)
y = f(x) - 1
y = f(3x)
6.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
This is the graph of f(x). Sketch each transformation on the grids provided, stating the new coordinates of the marked point in each case.
Equation
Sketch Marked Coordinat
e1. example
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y Inside the brackets affects the x coordinate, the opposite way, so the graph will move left by 2.
y = f(x + 2)
y = f(x + 2)
The new coordinate is (0, 0)
2.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
3.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
4.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
y = 2f(x)
y = f(x + 1)
y = f(x) + 1
5.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
6.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
y = f(3x)
y = f(x - 2)
Sketch the following curves giving the coordinates of the marked point in each caseEquati
onSketch Marked
Coordinate
1. example
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y Outside the brackets affects the y-coordinate, so the y-coordinates will be multiplied by 2 The new coordinate is (0, 2)
2.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
3.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
y = f(x) + 1
y = f(x - 2)
y = 2f(x)
4.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
5.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
6.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
7.
–6 –4 –2 2 4 6
–4
–3
–2
–1
1
2
3
4
x
y
y = 3f(x)
y = f(x + 1)
y = f(x) - 3
y = f(x + 2)
AnswersDescribing transformationsMoved left 1, so subtracted 1 from the x-coordinate f(x + 1)
Moved up 1, so added 1 to the y-coordinate f(x) + 1
Moved left 2, so subtracted 2 from the x-coordinate f(x + 2)
y-coordinate doubled f(x) + 1
Performing transformations
Equation Marked Coordinate
2. y = f(x + 3) (-2,1)3. y = 2f(x) (1,2)4. y = f(x) - 1 (1,0)5. y = f(3x) (1/3 ,0)6. y = f(x + 2) (-1,0)
Affec
t x
or y
? (-12, 5) becomes
(18, -8) becomes
f(x + 3) x (-15, 5) (15, -8)f(x – 4) X (-11, 5) (19, -8)f(x) + 2 Y (-15, 7) (15, -6)f(x) – 3 Y (-15, 2) (15, -11)f(3x) X (-5, 5) (5, -8)f(½x) X (-30, 5) (30, -8)2f(x) Y (-15, 10) (15, -16)½ f(x) Y (-15, 2.5) (15, -4)f(-x) X (15, 5) (-15, -8)-f(x) y (-15, -5) (15, 8)
Equation Marked Coordinate
2. y = 2f(x) (2, 0)3. y = f(x + 1) (1, 0)4. y = f(x) + 1 (2, 1)5. y = f(3x) (2/3 , 0)6. y = f(x - 2) (4,0)
Equation Marked Coordinate
2. y = f(x) + 1 (0, 2)3. y = f(x - 2) (2, 1)4. y = 3f(x) (0, 3)5. y = f(x + 1) (-1, 1)6. y = f(x) – 3 (0, -2)7. y = f(x + 2) (-2, 1)
