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For Educational Use Only © 2010
12.1 Functions Involving Square
Roots Brian Preston
Algebra 1 2009-2010
For Educational Use Only © 2010
Lesson Objectives1) Evaluate and graph a function involving square roots.
2) Use functions involving square roots to model real-life problems, such as the length of a cycle of a pendulum.
For Educational Use Only © 2010
Graph y = 2x + 1
-1
-3
-2
-4-5
-1-3 -2-4-5
1
2
3
4
5
2 3 4 51
+2 +1
(0,1)
slope = 2
y-int = 1
1
1
Regular line
Review
For Educational Use Only © 2010
-1-3-5 3 51
3
1
7
5
9
New line y = ax2 + bx + c
(0,0)
(1,1)
(2,4)
(3,9)
(-1,1)
(-2,4)
(-3,9)
y = x21
Parabola line
Review
For Educational Use Only © 2010
22y= +3
x–(–2)
Center-1
-3
-1-3
1
3
5
1 3-5 5
( h ,k)
–2
– 2 2
+
Review
For Educational Use Only © 2010
y= + kx – ha
Graphing a Square Root
a =The bigger a is the steeper the curved line is. The smaller a is the flatter the curve line is.
(h,k)= Starting point (Center)
k = Translates (shifts) up or down.
a + kh k
Definition
For Educational Use Only © 2010
y= +k
Can you take the square root of a negative number?
Graphing a Square Root
Definition
x – ha
-1
-3
-2
-4-5
-1-3 -2-4-5
1
2
3
4
5
2 3 4 51
X is
neg
ativ
e
Y is negative
No
For Educational Use Only © 2010
y = +k
We only use the positive
quadrants.
Graphing a Square Root
Definition
x – ha
2-1 6 84
4
2
8
6
-1
For Educational Use Only © 2010
y = +k
Square Root
Graph.
Graphing a Square Root
Starting point(h,k)
Definition
x – ha
For Educational Use Only © 2010
y =
Sketch the graph. Find the domain & range.
Example
2-1 6 84
4
2
8
6
-1
1) x
Not the right form
For Educational Use Only © 2010
00y= +
Sketch the graph. Find the domain & range.
Example
x – 0
2-1 6 84
4
2
8
6
-1
Center
(h ,k)
0
0 0
1) 1
Normal
For Educational Use Only © 2010
y = +0
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
1)
How can you get an undefined answer or error?
1
0or – 2
No negative radicals
For Educational Use Only © 2010
y = +0
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
x – 01)
x – 0 ≥ 0
x ≥ 0
The smallest number the radical can
become is 0.
Range
For Educational Use Only © 2010
y = +0
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
1)
x – 0 ≥ 0
x ≥ 0
y = +0x – 00
This is the lowest y value.
Range
For Educational Use Only © 2010
00y = +
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
1)
x – 0 ≥ 0
x ≥ 0
Range
y ≥ 0
y ≥ 0
For Educational Use Only © 2010
2) The period T (in seconds) of a pendulum is the time it takes for the pendulum to swing back and forth. The period is related to the length L (in inches) of the pendulum by the
model T = 2π . Find the period
of a pendulum with a length of eight inches. Give your answer to the nearest tenth.
period
eighteight
period
Real World Application
82π
L
384
L
384=T
For Educational Use Only © 2010
8
How long is the pendulum period?
Real World Application
2) 2π384
=T
2π=T 0.20…
2π=T 0.144…
0.9=T seconds
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Real World Application
How long does it take a cycle of the pendulum to occur?
0.9 seconds
For Educational Use Only © 2010
y = – 1
Sketch the graph. Find the domain & range.
Example
2-1 6 84
4
2
8
6
-1
3) x
Not the right form
For Educational Use Only © 2010
y= –1
Sketch the graph. Find the domain & range.
Example
x – 0
2-1 6 84
4
2
8
6
-1
– 1
Center
(h , k)
0
0 – 1
3) 1
Normal
For Educational Use Only © 2010
y = – 1
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
x – 03)
x – 0 ≥ 0
x ≥ 0
The smallest number the radical can
become is 0.
Range
For Educational Use Only © 2010
y = – 1
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
3)
x – 0 ≥ 0
x ≥ 0
y = – 1x – 00
This is the lowest y value.
Range
For Educational Use Only © 2010
y = – 1
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
3)
x – 0 ≥ 0
x ≥ 0
Range
y ≥ – 1
y ≥ – 1
– 1
For Educational Use Only © 2010
y = + 4
Sketch the graph. Find the domain & range.
Example
2-1 6 84
4
2
8
6
-1
4) x
Not the right form
For Educational Use Only © 2010
44y= +
Sketch the graph. Find the domain & range.
Example
x – 0
2-1 6 84
4
2
8
6
-1
Center
(h ,k)
0
0 4
4) 1
Normal
For Educational Use Only © 2010
y = +4
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
x – 04)
x – 0 ≥ 0
x ≥ 0
The smallest number the radical can
become is 0.
Range
For Educational Use Only © 2010
y = +4
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
4)
x – 0 ≥ 0
x ≥ 0
y = +4x – 00
This is the lowest y value.
Range
For Educational Use Only © 2010
44y = +
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
4)
x – 0 ≥ 0
x ≥ 0
Range
y ≥ 4
y ≥ 4
For Educational Use Only © 2010
00y= +
Sketch the graph. Find the domain & range.
Example
x – 3
2-1 6 84
4
2
8
6
-1
Center
(h ,k)
3
3 0
5) 1
Normal
For Educational Use Only © 2010
+ 3+ 3
x – 3x – 3y = +0
Sketch the graph. Find the domain & range.
Example
Domain
5)
x – 3 ≥ 0
x ≥ 3
The smallest number the radical can
become is 0.
Range
+ 3
For Educational Use Only © 2010
00y = +
Sketch the graph. Find the domain & range.
Example
x – 3
Domain
5)Range
y ≥ 0
y ≥ 0
x – 3 ≥ 0
+ 3 + 3
x ≥ 3
For Educational Use Only © 2010
1y= +
Sketch the graph. Find the domain & range.
Example
x + 1
2-1 6 84
4
2
8
6
-1
6)
Wrong form or think the
opposite sign.
For Educational Use Only © 2010
11y= +
Sketch the graph. Find the domain & range.
Example
x–(–1)
2-1 6 84
4
2
8
6
-1
Center
( h ,k)
–1
– 1 1
6) 1
Normal
For Educational Use Only © 2010
– 1– 1
x–(–1)x–(–1)y = +1
Sketch the graph. Find the domain & range.
Example
Domain
6)
x–(–1)≥ 0
x ≥ – 1
The smallest number the radical can
become is 0.
Range
– 1
For Educational Use Only © 2010
11y = +
Sketch the graph. Find the domain & range.
Example
Domain
6)Range
y ≥ 1
y ≥ 1– 1
x–(–1)≥ 0
x ≥ – 1
– 1
x–(–1)
For Educational Use Only © 2010
y = + 0
Sketch the graph. Find the domain & range.
Example
2-1 6 84
4
2
8
6
-1
7) x
Not the right form
2
For Educational Use Only © 2010
00y= +
Sketch the graph. Find the domain & range.
Example
x – 0
2-1 6 84
4
2
8
6
-1
Center
(h , k)
0
0 0
7) 2
Steeper curved
line
For Educational Use Only © 2010
y = +0
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
x – 07)
x – 0 ≥ 0
x ≥ 0
The smallest number the radical can
become is 0.
Range2
For Educational Use Only © 2010
y = +0
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
7)
x – 0 ≥ 0
x ≥ 0
y = +0x – 00
This is the lowest y value.
Range2
For Educational Use Only © 2010
00y = +
Sketch the graph. Find the domain & range.
Example
x – 0
Domain
7)
x – 0 ≥ 0
x ≥ 0
Range
y ≥ 0
y ≥ 0
2
For Educational Use Only © 2010
33y= +
Sketch the graph. Find the domain & range.
Example
x – 4
2-1 6 84
4
2
8
6
-1
Center
(h ,k)
4
4 3
8) 2
Steeper curved
line
For Educational Use Only © 2010
+ 4+ 4
x – 4x – 4y = +3
Sketch the graph. Find the domain & range.
Example
Domain
8)
The smallest number the radical can
become is 0.
Range2
x – 4 ≥ 0
x ≥ 4
+ 4
For Educational Use Only © 2010
33y = +
Sketch the graph. Find the domain & range.
Example
Domain
8)Range
y ≥ 3
y ≥ 3
x – 42
+ 4x – 4 ≥ 0
x ≥ 4
+ 4
For Educational Use Only © 2010
99y =
Evaluate the function for the given value of x.
Example
9) 3
9
x ;
y = 3 x
For Educational Use Only © 2010
9
9y =
Evaluate the function for the given value of x.
Example
9) 3 x ;
y = 3
3y = 3
y = 9
For Educational Use Only © 2010
– 2– 2
x
y =
Evaluate the function for the given value of x.
Example
10)
(– 2)
21 – 2x ;
y = 21 – 2
y = 21
y =
+ 4
= 525
For Educational Use Only © 2010
x
y =
Evaluate the function for the given value of x.
Example
11)
( )
36x – 2 ;
y = 36
y = 18
y =
– 2
= 416
– 2
1
21
2
For Educational Use Only © 2010
1) Don’t forget the negative signs.
2) Make sure you have the right center.
Key Points & Don’t Forget
y= +kx – ha(h,k) not (– h,k)
h
For Educational Use Only © 2010
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