2.1 radical functions and transformations - · pdf fileunit iii – radical functions math...

Unit III – Radical Functions Math 3200 1

Unit 3 – Radical Functions

2.1 Radical Functions and Transformations

(I) Investigating the graph of a radical function

Using a table of values, graph the radical function .

x

0

1

4

9

16

x- 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

y

- 2

- 1

1

2

3

4

5

6

State the:

Domain:___________________

Range:____________________

Objectives:

Investigating the function

Graphing radical functions using

transformations

Identifying the domain & range of a radical

function


Given that is the base graph for radical functions, use graphing

technology (www.desmos.com) to:

(i) sketch the graph

(ii) state the type of transformation

(HT, VT, HS, VS and reflection)

(iii) state the domain and range

Function

Graph

Describe the

transformation

Domain &

Range

1.

2.

x

y

x

y


Function

Graph

Describe the

transformation

Domain &

Range

3.

4.

5.

6.

x

y

x

y

x

y

x

y


Characteristics of a transformed radical function of the form

or

(i) represents a _______________

If a < 0 the graph is reflected through the _____________

(ii) represents a _______________

If b < 0 the graph is reflected through the _____________

(iii) represents a _______________

If h > 0 the graph is translated ________________

If h < 0 the graph is translated ________________

(iv) represents a _______________

If k > 0 the graph is translated ________________

If k < 0 the graph is translated ________________


(II) Graphing Radical Functions using Transformations

Example: For each function:

(i) sketch the graph using transformations

(ii) and state the domain and range

1.

Domain:__________________

Range:___________________

x- 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12

y

- 13- 12- 11- 10

- 9- 8- 7- 6- 5- 4- 3- 2- 1

123456789

101112

y = f(x)

NOTE: What parameter(s) from the function can

help identify the domain and range?


2.

Domain:__________________

Range:___________________

x- 12 - 11 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12

y

- 13- 12- 11- 10

- 9- 8- 7- 6- 5- 4- 3- 2- 1

123456789

101112

y = f(x)

NOTE: What parameter(s) from the function can

help identify the domain and range?


(III) Identifying the domain & range of a

radical function from an equation

Example:

Determine the domain and range for:

(i)

Domain:__________________

Range:___________________

(ii)

Domain:__________________

Range:___________________

P.72 – P.73 #2, #3, #4, #5, #6, #9, #20, C1, C2


2.2 Square Root of a Function

(I) Sketching the graph of given the

graph of y = f(x)

Consider the graph of y = f(x), which in this case is the graph of

y = x2 – 1.

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 4

- 3

- 2

- 1

1

2

3

4

Generate the graph for .

Objectives:

Sketching the graph of given the graph of y = f(x)

Graphing Strategies for given the graph of y = f(x)

Comparing the domains & ranges of the functions y = f(x) and


Complete the table of values and sketch the graph of

or on the grid below.

x f(x)

–2

–1

0

1

2

Observations based on the graphs:

1. Why is the graph or undefined from x є (–1, 1) ?

2. Are there any invariant points? If so, what are they?

3. Where is the graph of above y = f(x)? Below y = f(x)?

4. State the domain and range for both functions.

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 4

- 3

- 2

- 1

1

2

3

4

y= x2 – 1

Note:

Invariant points occur when y = ___

and y = ___


(II) Graphing Strategies for given the

graph of y = f(x)

Example:

Given the graph of y = f(x) create the graph of .

Function y = f(x)

x-intercepts

y-intercept

Max. value

Min. value

State the domain and range of .

Domain:_____________ Range:______________

Method – Analyze Key Points

is undefined where f(x) < 0 (Identify undefined

regions)

Identify values of f(x) to predict values for

is above the graph of y = f(x) where 0 < f(x) < 1

Consider invariant points where y = 0 and y = 1 and

intercepts

x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10

y

- 10

- 8

- 6

- 4

- 2

2

4

6

8

10


(III) Comparing the domains & ranges of the

functions y = f(x) and

Example:


(i) y = 4x – 8 and

Note:

is undefined where f(x) < 0 and

defined where f(x) > 0


Example:


(ii) y = x2 + 4x – 5 and

Function y = x2 + 4x – 5

x-intercepts

y-intercept

Max. value

Min. value

Domain:____________ Domain:____________

Range:___________ Range:___________

REMEMBER:

Use knowledge of quadratics to determine x and y-intercepts

and vertex of the parabola. This can be used to determine key

points on the graph of .

P.86 – P.89 #2, #3, #5, #6, #8, #11,#13, #17, C3


2.3 Solving Radical Equations

Review–Sketching the graph of a radical function

Example: Sketch the graph of

Remember to sketch graphs

(I) Use transformations of

Develop a mapping rule

(II) Analyze Key Points

is undefined where f(x) < 0 (Identify undefined

regions)

Identify values of f(x) to predict values for

is above the graph of y = f(x) where 0 < f(x) < 1

Consider invariant points where y = 0 and y = 1 and

intercepts

Objectives:

Relating the roots of a radical equation and the x – intercepts

of the graphs of the corresponding radical function.

Determine, graphically, an approximate solution of a radical

equation.


Sketch the graph of .

(I) Solving Radical Equations Graphically

How can we solve the radical equation ?

(A) Graphically

Method 1

Graph the corresponding function _______________

x- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14

y

- 5- 4- 3- 2- 1

12345678

x- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14

y

- 5- 4- 3- 2- 1

12345678


Method 2

Graph a system of functions that corresponds to the

expression on both sides of the equal sign and identify

the x– coordinate at the point of intersection.

Example: To solve graph the functions

_____________ and __________.

(Use www.desmos.com)

x- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14

y

- 5- 4- 3- 2- 1

12345678

Solving Radical Equations Graphically

What aspect of the graph above provides a solution for the

radical equation ? ____________________


Solving a Radical Equation Involving an Extraneous Solution

Example:

Solve the equation algebraically and graphically.

Algebraically Solve

Graphically Solve

By graphing

__________ and ___________

x- 2 - 1 1 2 3 4 5 6 7 8

y

- 2

- 1

1

2

3

4

5

6


Observations to Algebraic and Graphical Solutions above

1. What is the difference between the two methods concerning the

number of solutions?

2. Why is there a difference in the number of solutions?

Verifying a solution graphically and algebraically

Example:

The equation has no solutions.

a) Verify that this is correct, using both a graphical and

an algebraic approach.

b) Is it possible to tell that this equation has no solutions simply by

examining the equation? Explain.

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 2

- 1

1

2

3

4

5

6


(II) Graphically determining an approximate

solution of a radical equation

Approximate Solutions to Radical Equations

Example 1:

Solve the equation graphically.

Verify solution algebraically.

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 2

- 1

1

2

3

4

5

6


Example 2:

Solve the equation graphically. Verify the solution

algebraically.

Verify solution algebraically.

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 2

- 1

1

2

3

4

5

6

P.96 – P.98 #3, #5, #7, #9, C1, C3

2.1 radical functions and transformations - · pdf fileunit iii – radical functions math...

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