P. 431P. 431

7.5 Graphing 7.5 Graphing Square Root & Square Root &

Cube Root Cube Root FunctionsFunctions

First, let’s look at the parent graphs.

xy 3 xy

0, 0

1, 1

4, 2

9, 3

16, 4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2 4 6 8 10 12 14 16 18

-27, -3

-8, -2

-1, -1

0, 0

1, 1

8, 2

27, 3

-4

-3

-2

-1

0

1

2

3

4

-30 -20 -10 0 10 20 30

Now, what happens when there is a number in front of the radical?

xy 2 33 xy

0, 0

1, 2

4, 4

9, 6

16, 8

25, 10

0

2

4

6

8

10

12

0 5 10 15 20 25 30

-27, 9

-8, 6

-1, 3

0, 0

1, -3

8, -6

27, -9-10

-8

-6

-4

-2

0

2

4

6

8

10

-30 -20 -10 0 10 20 30

* Notice the graph goes thru the points (0,0) and (1,2).

* Notice the graph goes thru the points (-1, 3), (0,0), & (1,-3).

GeneralizationGeneralization

xay

3 xay

Always goes thru the points (0,0) and (1,a).

Always goes thru the points (-1,-a), (0,0), and (1,a).

If a > 1, Then the graph stretches.If 0< a < 1, Then the graph shrinks.

Ex: Graph xy 4

Goes thru the points (0,0) and (1,a).

Since a=-4, the graph will pass thru (0,0) and (1,-4)

0, 0

1, -4

9, -12

16, -16

25, -20

-25

-20

-15

-10

-5

0

0 5 10 15 20 25 30

The NegativeReflects the graphAbout the x axis.

Now, what happens when there are numbers added or subtracted inside and/or

outside the radical?

khxaykhxay 3or

Step 1: Find points on the parent graph

Step 2: Shift these points h units horizontally (use opposite sign) and k units vertically (use same sign).

3or xayxay

Ex: Describe how to obtain the graph of from the graph of

123 xy

3 xy

3 xy

Shift all the points from

To the right 2 and up 1.

Ex: Graph142 xy

-4, -1

-3, 1

0, 3

5, 5

-2

-1

0

1

2

3

4

5

6

-6 -4 -2 0 2 4 6

x y

0 0

1 2

4 4

9 6

xyfortable 2 Now, shift these points to the left 4 and down 1.

x y

-4 -1

-3 1

0 3

5 5

(x-value – 4)

(y-value -1)

Ex: Graph2323 xy

-24, 8

-5, 6

2, 4

3, 2

4, 0

11, -2

30, -4

-6

-4

-2

0

2

4

6

8

10

-30 -20 -10 0 10 20 30 40

32 xyfortable

x y

-27 6

-8 4

-1 2

0 0

1 -2

8 -4

27 -6

Now, shift these points to the right 3 and up 2.

(x-value + 3)(y-value + 2)

x y

-24 8

-5 6

2 4

3 2

4 0

11 -2

30 -4

Ex: State the domain and range of the functions in the last 2 examples.

142 xy

-4, -1

-3, 1

0, 3

5, 5

-2

-1

0

1

2

3

4

5

6

-6 -4 -2 0 2 4 6

-24, 8

-5, 6

2, 4

3, 2

4, 0

11, -2

30, -4

-6

-4

-2

0

2

4

6

8

10

-30 -20 -10 0 10 20 30 40

numbersrealall

2323 xyx-values y-values

Domain:Domain:

Range:Range:

Domain:Domain:

Range:Range:

4x1y

numbersrealall

The graph doesn’t have a beginning or ending point.

(Meaning all x & y-values are possible.)

The graph has a beginning point of (-4,-1).

Assignment