Standards for Radical Functions
• MM1A2a. Simplify algebraic and numeric
expressions involving square root.
• MM1A2b. Perform operations with square
roots.
• MM1A3b. Solve equations involving
radicals such as y  x  b , using
algebraic techniques.
Radical Functions
• Essential questions:
1. What is a radical function?
2. What does the graph look like and how does
it move?
3. How are they used in real life applications?
Real Life Applications
• Pythagorean Theorem
• Distance Formula
• Solving any equation that includes a
variable with an exponent, such as:
A  r 2
4 3
V  r
3
Radical Expressions
• Index
Radical Sign
3
2x  4
Radicand
General Radical Equation
Vertical stretch or compression
by a factor of |a|; for a < 0,
the graph is a reflections across
the x-axis
Vertical translation k units
up for k > 0 and |k| units
down for k < 0
y  a b( x  h)  k
Horizontal stretch or
compression by a factor of
|1/b|; for b < 0, the graph is a
reflection across the y-axis
(b = 1 or -1 for this course)
Horizontal translation
h units to the right for
h > 0 and |h| units to
the left if h < 0.
(h = 0 for this course)
Radical Functions
• Make a (some) table(s), graph the following
functions and describe
f ( x)  x
the transformations for
x = 0, 1, 4, 9, 16 & 25. g ( x )  2 x
• What transformation
h( x)  2 x  3
Rules do you see from
i ( x )  ( 2 x  3)
Your graphs?
X
0
1
f ( x)  x
y
0
0
1
1
What value for x
That’s our
gives us a zero
smallest value
under the
in our t-chart.
radical?
10
8
6
4
4
2
4
2
0
9
9
3
-2-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-4
-6
16
16
4
25
25
5
-8
-10
X
0
1
f ( x)  2 x
y
2 0
0
2 1
2
What value for x
That’s our
gives us a zero
smallest value
under the
in our t-chart.
radical?
10
8
6
4
2 4
4
4
2
0
9
2 9
6
-2-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-4
-6
16
2 16
8
25
2 25
10
-8
-10
X
0
1
f ( x)  2 x  3
y
2 0 3
-3
2 13
-1
What value for x
That’s our
gives us a zero
smallest value
under the
in our t-chart.
radical?
10
8
6
4
2 4 3
1
4
2
0
9
2 9 3
3
-2-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-4
-6
16
2 16  3
5
25
2 25  3
7
-8
-10
X
0
1
f ( x)  1(2 x  3)
y
2 0
3
2 1
1
What value for x
That’s our
gives us a zero
smallest value
under the
in our t-chart.
radical?
10
8
6
4
2 4
-1
4
2
0
9
2 9
-3
-2-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-4
-6
16
2 16
-5
25
2 25
-7
-8
-10
Radical Functions
• State an equation that would make the
square root function shrink vertically by a
factor of ½ and translate up 4 units.
y  0.5 x  4
• How would we reflect the above equation
across the y-axis?
• Make the “x” negative
Domain & Range: Radical Functions
• State the domain,
range, and intervals
of increasing and
decreasing for each
function.
f ( x)  x
g ( x)  2 x
h( x)  2 x  3
i ( x )  ( 2 x  3)
Graphing Radical Functions Summary
• Transformations for radical functions are the
same as polynomial functions.
• The domain of the parent function is limited
to {x | x  0} (the set of all x such that x  0)
• The range of the parent function is limited
to {y | y  0} (the set of all y such that y  0)
• The domain and range may change as a
result of transformations.
• The parent radical function continuously
increases from the origin.
Simplifying Radical Expressions
x 2  x, if _ x  0
• Square and square root
are inverse functions, but x 2   x, if _ x  0
the square root has to be therefore
x n  x _ if _ n _ is _ even
positive
x n  x _ if _ n _ is _ odd
25  5
5 5
( 5) 2  5
36  6
62  6
(6) 2  6
2
Simplifying Radical Expressions
ab  a * b
or
a * b  ab
• “Simplify” a radical means to:
1. Take all the perfect squares out of the
radicand.
Simplify:
32
150
6 * 24
6* 3
Simplifying Radical Expressions
• “Simplify” a radical means to:
2. Combine terms with like radicands
• Must have the same radicand to be able to
add or subtract radials
• Simplify:
2 2 3 8
2 2  3(3 32  3 )
Simplifying Radical Expressions
Quotient Property of Square Roots: If a ≥ 0 and
b > 0:
a
a
b

b
• “Simplify” a radical means to:
3. Do not leave a radical in the denominator
• Simplify:
2
7
6
2
2
8
2
10
Simplify Expressions via Conjugates
•Remember: (a+ b) is the conjugate of (a – b)
•We get conjugates when we factor a perfect
square minus a perfect square.
•We also get conjugates other times.
Simplify:
(5  7 )(5  7 )
( 3  2 )( 3  2 )
Simplify Expressions via Conjugates
Simplify:
3
5 8
3 5
5 5
Summary Important Operations
• Square and square root
are inverse functions, but
the square root has to be
positive
x 2  x, if _ x  0
x 2   x, if _ x  0
• Product Property of Square Roots: If a ≥ 0 and b
≥ 0:
ab  a * b
• Quotient Property of Square Roots: If a ≥ 0 and
b > 0:
a
a
•
b

b
Summary Simplification Rules
• To “simplify” a radical means to:
1. Take all the perfect squares out of the
radicand.
2. Combine terms with like radicands
3. Do not leave a radical in the denominator
Simplifying Radical Expressions
• Homework page 144, # 3 – 24
by 3’s and 25 & 26
Warm-up
Simplify
1. 4 7 
2.
4 7 5
125  80
3 (3 2  3 )
3 6 3
Solve.
3.
4.
x 2  0
4
3x  2  3  7
6
Standards for Radical Functions
• MM1A2a. Simplify algebraic and numeric
expressions involving square root.
• MM1A2b. Perform operations with square
roots.
• MM1A3b. Solve equations involving
radicals such as y  x  b , using
algebraic techniques.
Radical Functions
• Today’s essential questions:
1. How do we find the solution of a radical
function?
2. How are they used in real life applications?
12.3 Solving Radical Equations
1. Get the radical on one side.
2. Square both sides of the equals sign.
3. Solve for the variable.
4. Check your answer. IF the answer doesn’t
check, then “no solution.”
EXAMPLE 3:
10  6x  2
100  6x  2
102  6x
x  17
?
10  (6(17)  2)
10  10
EXAMPLE 1:
3 x  21  0
3 x  21
x  7
x  49
?
3 49  21 
0
?
3(7)  21 0
?
21  21 0
42  0
Your Turn – Solve with your
neighbor:
3 x  2  14
3 x  12
x 4
x  16
?
3 16  2 
14
?
3(4)  2  14
?
12  2  14
14  14
What if > One Radical?
•
1.
2.
3.
4.
5.
The process is the same:
Get a radical on one side.
Square both sides of the equal sign.
Solve for the variable.
Repeat as necessary
Check your answer. IF the answer does
not check, then there is NO SOLUTION
for that answer
Example 3:
7 x  5  3x  19
7 x  5  3x  19
4x  24
?
7(6)  5  3(6)  19
?
42  5  18  19
37  37
x6
Example 3:
x  2 x  24
x  2 x  24
2
x  2 x  24  0
2
x  6x  4  0
x  6 _ or
x  4
?
6  2(6)  24
?
6  12  24
66
?
 4  2(4)  24
?
 4   8  24
 4  16
Your Turn – Solve
Individually:
2 x  21  x  3
2 x  21  x  6 x  9
2
x  4 x  12  0
2
x  6x  2  0
x  6 _ or
x2
?
2(6)  21  6  3
?
12  21  3
9  3
?
2(2)  21  2  3
?
4  21  5
55
Practice with Tic-Tac-Toe
• The object is to get three in a row.
• Work together in designated pairs.
• Notice: Different problems have
different points.
• Your score will be the three in a row
you solve with the most points
Practice
• Page 148, # 3 – 30 by 3’s and 31 & 32