standards for radical functions mm1a2a. simplify algebraic and numeric expressions involving square...
TRANSCRIPT
![Page 1: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/1.jpg)
Standards for Radical Functions
• MM1A2a. Simplify algebraic and numeric expressions involving square root.
• MM1A2b. Perform operations with square roots.
• MM1A3b. Solve equations involvingradicals such as , using algebraic techniques.
bxy
![Page 2: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/2.jpg)
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?
![Page 3: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/3.jpg)
Real Life Applications
• Pythagorean Theorem• Distance Formula• Solving any equation that includes a
variable with an exponent, such as:
3
2
3
4rV
rA
![Page 4: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/4.jpg)
Radical Expressions
• Index Radical Sign
Radicand
3 42 x
![Page 5: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/5.jpg)
General Radical Equation
khxbay )(
Vertical stretch or compressionby a factor of |a|; for a < 0, the graph is a reflections across the x-axis
Vertical translation k unitsup for k > 0 and |k| unitsdown for k < 0
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)
![Page 6: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/6.jpg)
Radical Functions
• Make a (some) table(s), graph the followingfunctions and describe the transformations forx = 0, 1, 4, 9, 16 & 25.
• What transformationRules do you see fromYour graphs? )32()(
32)(
2)(
)(
xxi
xxh
xxg
xxf
![Page 7: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/7.jpg)
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 0
1 1
4 2
9 3
16 4
25 5
0
1
4
9
16
25
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
xxf )(
![Page 8: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/8.jpg)
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 0
1 2
4 4
9 6
16 8
25 10
02
12
42
92
162
252
xxf 2)(
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
![Page 9: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/9.jpg)
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 -3
1 -1
4 1
9 3
16 5
25 7
302
312
342
392
3162
3252
32)( xxf
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
![Page 10: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/10.jpg)
What value for x gives us a zero under the radical?
That’s our smallest value in our t-chart.
X y
0 3
1 1
4 -1
9 -3
16 -5
25 -7
)32(1)( xxf
02
12
42
92
162
252
-1 1 3 5 7 9 11 13 15 17 19 21 23 25
-10
-8
-6
-4
-2
0
2
4
6
8
10
![Page 11: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/11.jpg)
Radical Functions
• State an equation that would make the square root function shrink vertically by a factor of ½ and translate up 4 units.
• How would we reflect the above equation across the y-axis?
• Make the “x” negative
45.0 xy
![Page 12: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/12.jpg)
Domain & Range: Radical Functions
• State the domain,range, and intervalsof increasing anddecreasing for each function.
)32()(
32)(
2)(
)(
xxi
xxh
xxg
xxf
![Page 13: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/13.jpg)
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.
![Page 14: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/14.jpg)
Simplifying Radical Expressions• Square and square root
are inverse functions, butthe square root has to bepositive
oddisnifxx
evenisnifxx
therefore
xifxx
xifxx
n
n
____
____
0_,
0_,
2
2
525 5)5( 2 552
636 6)6( 2 662
![Page 15: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/15.jpg)
Simplifying Radical Expressions
baab * abba *or
• “Simplify” a radical means to:1. Take all the perfect squares out of the
radicand.Simplify:
32 150 24*6 3*6
![Page 16: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/16.jpg)
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:
8322 )3323(322
![Page 17: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/17.jpg)
Simplifying Radical Expressions Quotient Property of Square Roots: If a ≥ 0 and
b > 0:
• “Simplify” a radical means to:3. Do not leave a radical in the denominator
• Simplify:
2
6
8
2
b
a
b
a
10
27
2
![Page 18: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/18.jpg)
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:
)75)(75( )23)(23(
![Page 19: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/19.jpg)
Simplify Expressions via Conjugates
Simplify:
85
3
55
53
![Page 20: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/20.jpg)
Summary Important Operations • Square and square root
are inverse functions, butthe square root has to bepositive
• Product Property of Square Roots: If a ≥ 0 and b ≥ 0: • Quotient Property of Square Roots: If a ≥ 0 and
b > 0:
• b
a
b
a
baab *
0_,
0_,
2
2
xifxx
xifxx
![Page 21: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/21.jpg)
Summary Simplification Rules• To “simplify” a radical means to:1. Take all the perfect squares out of the
radicand.2. Combine terms with like radicands3. Do not leave a radical in the denominator
![Page 22: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/22.jpg)
Simplifying Radical Expressions
• Homework page 144, # 3 – 24 by 3’s and 25 & 26
![Page 23: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/23.jpg)
Warm-upSimplify1.
2.
Solve.
3.
4.
4 7 125 80
3 3 2 3( )
x 2 0
4 7 5
3 6 3
4
3 2 3 7x 6
![Page 24: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/24.jpg)
Standards for Radical Functions
• MM1A2a. Simplify algebraic and numeric expressions involving square root.
• MM1A2b. Perform operations with square roots.
• MM1A3b. Solve equations involvingradicals such as , using algebraic techniques.
bxy
![Page 25: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/25.jpg)
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?
![Page 26: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/26.jpg)
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.”
![Page 27: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/27.jpg)
EXAMPLE 3:
10 6 2 x100 6 2 x
10 6 17 2 ?
( ( ) )
102 6 x
x 17
10 10
![Page 28: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/28.jpg)
EXAMPLE 1:
3 21 0x
3 21x 3 49 21 0
?
3 7 21 0( )?
42 0x 7x 49
21 21 0 ?
![Page 29: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/29.jpg)
Your Turn – Solve with your
neighbor:
3 2 14x 3 12x
3 16 2 14 ?
3 4 2 14( )?
x 4x 16
12 2 14 ?
14 14
![Page 30: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/30.jpg)
• The process is the same:1. Get a radical on one side.2. Square both sides of the equal sign.3. Solve for the variable.4. Repeat as necessary5. Check your answer. IF the answer does
not check, then there is NO SOLUTION for that answer
What if > One Radical?
![Page 31: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/31.jpg)
Example 3:
19357 xx
19357 xx
244 x
6x
19)6(35)6(7
3737
1918542
?
?
![Page 32: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/32.jpg)
Example 3:
242 xx
2422 xx
02422 xx
4
_6
x
orx
24)6(26
66
24126
?
?
046 xx24)4(24
?
?2484
164
![Page 33: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/33.jpg)
Your Turn – Solve
Individually:
3212 xx
96212 2 xxx
01242 xx
2
_6
x
orx
3621)6(2
39 32112
?
?
026 xx ?
?
55
3221)2(2
5214
![Page 34: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/34.jpg)
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
![Page 35: Standards for Radical Functions MM1A2a. Simplify algebraic and numeric expressions involving square root. MM1A2b. Perform operations with square roots](https://reader036.vdocuments.mx/reader036/viewer/2022062321/56649eaa5503460f94baf7f4/html5/thumbnails/35.jpg)
Practice• Page 148, # 3 – 30 by 3’s and 31 & 32