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Shifting Curves II
Shifting Curves II y kf x 1, curve is steeperk
0 1, curve is shallowerk (curve is stretched vertically)
ORy f xk
Shifting Curves II y kf x 1, curve is steeperk
0 1, curve is shallowerk (curve is stretched vertically)domain unchanged, range altered
ORy f xk
Shifting Curves II y kf x 1, curve is steeperk
0 1, curve is shallowerk (curve is stretched vertically)domain unchanged, range altered
y f kx 1, curve is steeperk 0 1, curve is shallowerk (curve is stretched horizontally)
ORy f xk
Shifting Curves II y kf x 1, curve is steeperk
0 1, curve is shallowerk (curve is stretched vertically)domain unchanged, range altered
y f kx 1, curve is steeperk 0 1, curve is shallowerk (curve is stretched horizontally)domain altered, range unchanged
ORy f xk
Shifting Curves II y kf x 1, curve is steeperk
1y
f x intercepts asymptotesx
0 1, curve is shallowerk (curve is stretched vertically)domain unchanged, range altered
y f kx 1, curve is steeperk 0 1, curve is shallowerk (curve is stretched horizontally)domain altered, range unchanged
asymptotes interceptsx1 1y y 1 1y y
ORy f xk
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
2y x
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
2y x2 2 21) , , 2
2b y x y x y x
y
x
2y x 22y x
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
2y x2 2 21) , , 2
2b y x y x y x
y
x
2y x
212
y x
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
2y x2 2 21) , , 2
2b y x y x y x
y
x
2y x
212
y x
22y x
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
2y x2 2 21) , , 2
2b y x y x y x
y
x
2y x
212
y x
22y x
22( ) Sketch 1
4yii x
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
2y x2 2 21) , , 2
2b y x y x y x
y
x
2y x
212
y x
22y x
22( ) Sketch 1
4yii x y
x
2 21. : 1basic curve x y
1
1
–1
–1
y
x
e.g. on one graph drawi1) , , 22
a y x y x y x
y x12
y x
2y x2 2 21) , , 2
2b y x y x y x
y
x
2y x
212
y x
22y x
22( ) Sketch 1
4yii x y
x
2 21. : 1basic curve x y
1
1
–1
–1
22
2. , 24 2y y k
stretch vertically by a factor of 2
21
2iii y
x
y
x
21. : basic curve y x
21
2iii y
x
y
x
21. : basic curve y x
21
2iii y
x
2. reflect in x axis
y
x
21. : basic curve y x
21
2iii y
x
3. shift up 2 units
2. reflect in x axis
2
y
x
21. : basic curve y x
21
2iii y
x
3. shift up 2 units
2. reflect in x axis2
4. x intercepts become asymptotes
22
y
x
21. : basic curve y x
21
2iii y
x
3. shift up 2 units
2. reflect in x axis2
4. x intercepts become asymptotes
22
5. y >1 become y <112
y
x
21. : basic curve y x
21
2iii y
x
3. shift up 2 units
2. reflect in x axis2
4. x intercepts become asymptotes
22
5. y >1 become y <112
6. y <1 become y >1
y
x
21. : basic curve y x
21
2iii y
x
3. shift up 2 units
2. reflect in x axis2
4. x intercepts become asymptotes
22
5. y >1 become y <112
6. y <1 become y >1
y
x
21. : basic curve y x
21
2iii y
x
3. shift up 2 units
2. reflect in x axis2
4. x intercepts become asymptotes
22
5. y >1 become y <112
6. y <1 become y >1
1iv y xx
1iv y xx
y
x
1. :1 and
draw the basic curves
y x yx
1iv y xx
y
x
1. :1 and
draw the basic curves
y x yx
2. add the y values together* choose key points first
- x interceptsnote: vertical asymptotes remain
1iv y xx
y
x
1. :1 and
draw the basic curves
y x yx
2. add the y values together* choose key points first
- x interceptsnote: vertical asymptotes remain
- points of intersection
1iv y xx
y
x
1. :1 and
draw the basic curves
y x yx
2. add the y values together* choose key points first
- x interceptsnote: vertical asymptotes remain
- points of intersection - as many other points as youneed to work out the shape
1iv y xx
y
x
1. :1 and
draw the basic curves
y x yx
2. add the y values together* choose key points first
- x interceptsnote: vertical asymptotes remain
- points of intersection - as many other points as youneed to work out the shape
1iv y xx
y
x
1. :1 and
draw the basic curves
y x yx
2. add the y values together* choose key points first
- x interceptsnote: vertical asymptotes remain
- points of intersection - as many other points as youneed to work out the shape
1iv y xx
y
x
1. :1 and
draw the basic curves
y x yx
2. add the y values together* choose key points first
- x interceptsnote: vertical asymptotes remain
- points of intersection - as many other points as youneed to work out the shape
Exercise 2J: 1, 2a, 3b, 4c, 5ac, 6b, 7ac, 8, 9