Page 1
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
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Unit III – Radical Functions Math 3200 2
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
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Unit III – Radical Functions Math 3200 3
Function
Graph
Describe the
transformation
Domain &
Range
3.
4.
5.
6.
x
y
x
y
x
y
x
y
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Unit III – Radical Functions Math 3200 4
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 ________________
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Unit III – Radical Functions Math 3200 5
(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?
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Unit III – Radical Functions Math 3200 6
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?
Page 7
Unit III – Radical Functions Math 3200 7
(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
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Unit III – Radical Functions Math 3200 8
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
Page 9
Unit III – Radical Functions Math 3200 9
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 = ___
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Unit III – Radical Functions Math 3200 10
(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
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Unit III – Radical Functions Math 3200 11
(III) Comparing the domains & ranges of the
functions y = f(x) and
Example:
Determine the domain and range for:
(i) y = 4x – 8 and
Note:
is undefined where f(x) < 0 and
defined where f(x) > 0
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Unit III – Radical Functions Math 3200 12
Example:
Determine the domain and range for:
(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
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Unit III – Radical Functions Math 3200 13
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.
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Unit III – Radical Functions Math 3200 14
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
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Unit III – Radical Functions Math 3200 15
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 ? ____________________
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Unit III – Radical Functions Math 3200 16
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
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Unit III – Radical Functions Math 3200 17
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
Page 18
Unit III – Radical Functions Math 3200 18
(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
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Unit III – Radical Functions Math 3200 19
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