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