Name: Block:

Unit 6: Radical Functions

Day 1 Simplifying nth Roots, Operations, and Rationalizing

Day 2 Solving Power & Radical Equations

Day 3 Review Days 1 & 2 for Quiz

Day 4 Quiz: Days 1 & 2 Intro to Inverse Functions

Day 5 Square and Cube Root Functions & Characteristics

Day 6 Review for Unit Test

Day 7 Unit 6 Test

Tentative Schedule of Upcoming Classes

Day 1 B Tues 2/23 Day 1 Notes:

Simplifying, Operations A Wed 2/24 (as HW)

Day 2 B Thurs 2/25 Day 2 Notes:

Solving Power & Radical Equations A Fri 2/26

Day 3 B Mon 2/29 Review: Days 1 & 2 Skills Check #2 A Tues 3/1

Day 4 B Wed 3/2 Quiz: Days 1 & 2

Day 4 Notes: Intro to Inverses A Thurs 3/3

Day 5 B Fri 3/4 Day 5 Notes: Square & Cube Root Functions A Mon 3/7

Day 6 B Tues 3/8 Review: Unit 6 Skills Check #3 A Wed 3/9

Day 7 B Thurs 3/10 Unit 6 Test A Fri 3/11

Absent?

See Ms. Huelsman AS SOON AS POSSIBLE to get work and any help you need.

Notes are always posted online on the calendar. (If links are not cooperative, try changing to “list” mode)

Handouts and homework keys are posted under assignments

You may also email Ms. Huelsman at Kelsey.huelsman@lcps.org with any questions!

____

Need Help?

Ms. Huelsman and Mu Alpha Theta are available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:10.

Ms. Huelsman is in L402 on Wednesday mornings.

Need to make up a test/quiz?

Math Make Up Room schedule is posted around the math hallway & in Ms. Huelsman’s classroom

Day 1: Simplifying nth Roots, Operations, and Rationalizing

Today we will learn how to simplify nth roots so that we can later apply that knowledge to solve power equations.

Getting to know rational exponents:

Practice: Rewrite the following. If it is written in exponential notation, rewrite in radical notation; if it is written in radical notation, rewrite in exponential notation –

1. 2. 3.

4. 5. 6.

Let’s fill in these tables together to better help us simplify square roots, cube roots and 4th roots: 12 =

22 = 32 = 42 = 52 = 62 = 72 = 82 = 92 = 102 =

112 =

122 = 132 = 142 = 152 = 162 = 172 = 182 = 192 = 202 =

13 =

23 = 33 = 43 = 53 = 63 = 73 = 83 = 93 = 103 =

14 = 24 = 34 = 44 = 54 =

15 = 25 = 35 =

Exponential Notation Radical Notation

12x 12 x or x

13x 3 x

14x 4 x

23x ( 3 x )2 or 3 2x

34x ( 4 x )3 or 4 3x

am/n ( ) nm mn a a=

a-m/n 1 , 0n m

aa

≠

Practice Questions:

Write 3

29 in radical notation

Write 2

3125 in radical notation

Write ( )53 64 in exponential notation

Write 3 9x in exponential notation

Though we will start out using the calculator to help us simplify radicals, eventually there will be a no calculator section on tests – start memorizing and recognizing perfect squares and cubes Based on the tables we filled in on the previous page, evaluate the following.

1. −100 =

2. −83 =

3. −81 =

4. −273 =

5. −25 =

6. −1253 =

7. −16 =

8. −325 =

Discuss: Why can we cube root a negative number, but not square root it? Let’s evaluate these expressions without a calculator.

144 225

1236

4 16

3 125− 3 227

34( 81)

5 243 5 432−

Hint: for the following it may help to rewrite in radical form, then evaluate. It is often easier to take a root first (if you can), then raise to a power (go small before big!)

1 38− 239

3264

3216

3532

−

14256

Writing Radicals in Simplest Form – use your Exponent Chart for perfect nth Roots

45 48

3 48 Find perfect 3rd Root that goes into 48 4 48 Find perfect 4th Root that goes into 48

Operations with Radicals: Addition and Subtraction: Remember – the “radicands” (numbers under the radical) AND the “index” (root) must match!

3 4 3+

3 2 8−

5 57 12 12−

33 81 24−

Multiplication & Division: For multiplication & division, “radicands” do not need to match – just indexes (same root)!

7 6 2− •

3 32 12 5 18•

520

3

3

189

5

5

963

Sometimes we are left with radicals in the denominator…when that happens we ____________________! With square roots:

52

9 23

NOW with other roots …The goal is to get a perfect nth root in the denominator to get rid of the radical

3

34

We need a perfect cube in the denominator…

Four goes into which perfect cube? _______ So I need to multiply by… _______

3

153

We need a perfect cube in the denominator.

3 goes into which perfect cube? _______ So I need to multiply by… _______

3

3

52

We need a perfect cube in the denominator…

2 goes into which perfect cube? _______ So I need to multiply by… _______

4

28

We need a 4th root in the denominator…

8 goes into which x4 from our table? _______ So I need to multiply by… _______

Simplify Expressions with Variables – We have done this before with square roots…Now we are factoring out perfect nth Roots. Leave all answers in radical notation.

4 5 1116x y z

3 1012x z

5 6 123 27x y z

We are taking out groups of ______

2 4 143 16w x y We are taking out groups of ______

8 14 54 16a b c We are taking out groups of ______

5 10 732a b We are taking out groups of ______

Day 2: Solving Power & Radical Equations

Today we will learn how to solve power & radical equations Using what we learned last class about simplifying radicals and

What we already know about solving linear & quadratic equations!

Solving power equations: A power equation is an equation that involves a variable raised to a power.

1. Isolate the exponent / power. 2. Take the nth root. 3. Check all solutions by plugging them back into the original equation for x

Recall from quadratics…the square root method: (x + 3) 2 = 64

3 (x – 5)2 = - 27

Whenever we solve by square rooting a number, we must put a ______ in front! This is true for ALL even roots (ie: 4th root, 6th root), but NOT true for odd roots. Why is this? ( - 2) 3 vs. ( 2 ) 3 ( - 2) 2 vs. ( 2 ) 2

Examples: x3 = 8 x3 = – 8 x2 = – 8 x2 = 8 When solving, we must follow the reverse order of operations. 1. Add or subtract 2. Multiply or divide 3. Exponents 4. Parentheses (follow same order) 2x4 = 162

(x – 2) 3 = -125

6x3 = 384

(x – 3) 4 = 625

31 24

x = −

x4 = 32

2x3 = x3 + 54

(x + 1) 5 = 100

(x – 1)5 – 3 = –35

2(x – 9)3 = 250

Solving radical equations: A radical equation is an equation with a variable in the “radicand” (under the radical).

1. Isolate the radical on one side of the equation 2. Raise each side of the equation (not each term) to the power that would eliminate the radical.

You will be left with a linear, quadratic, or other polynomial equation to solve. 3. Solve the remaining equation (using knowledge from previous units) 4. Check all solutions – There may be extraneous solutions!

2 4 8x+ =

6 1 9 16x+ + =

2 6 83

x + =

5 7 3 3x x− = +

How to check for extraneous solutions: 1. Take solution(s), x = 2. Plug solution into ORIGINAL equation. See if the statement is TRUE.

3 2 1 3x + =

3 5 1 2 2x − − =

34 5 13 3x − − =

4 5 2x − =

7 15 1x x+ = +

21 1 5x x+ = +

Day 3: Review Days 1 & 2

1. 1001

2 = 2. 161

4 = 3. 100,0001

5 = 4. 1

327 =

5. 2251

2 = 6. 2161

3 = 7. 1,0001

3 = 8. 11

4 =

1. 1003

2 = 2. 163

4 = 3. 10002

3 = 4. 3

225 =

5. 84

3 = 6. 2

364 = 7. 3

264 = 8. 3

532 =

1. 10−2 = 2.

1216

−= 3. 1000

−23 =

4.

5. 36− = 6. 32−3

5 = 7. 27− = 8.

916

−12

=

QUICK QUESTIONS RADICALS Question A B

1 Which expression reciprocates? x1/2 x–1/2

2 A fractional exponent: Reciprocates Turns into a radical

3 Which is equal to: ½ 2

4 A number to the 1/3 power means: Divide by 3 Cube Root

5 Which formula is correct?

6 Which expression will be imaginary?

7 Which expression will be negative?

8 Simplify: x4 x8

9 Which will need a ? x2 = 16

10 Which will need a ? x2 = 25

11 Which will need a ? x2 = 64 x3 = 64

12 Which solution will be imaginary? x2 = –128 x3 = –128

13 Which will need a ? x2 = –128 x3 = –128

14 Which equation will have two solutions? x2 = 81 x3 = 81

15 What is the next step: 4x2 + 5 = 8 Divide by 4 Subtract 5

16 What is the next step: –2 + 9x2 = 14 Add 2 Divide by 9

17 What is the next step: 2x2 + 1 = –5x2 + 3 Square Root Add 5x2

18 What is the next step: 3 (x + 2)2 + 13 = 8 Subtract 13 Divide by 3

19 What is the next step: (x + 4)2 = 49 x + 4 = 7 x + 4 = 7

20 What is the next step: (x – 1)3 = –64 x – 1 = 4i x – 1 = –4

21 What is the next step: (3x – 2)2 = –16 3x – 2 = –4i 3x – 2 = 4i

22 What is the next step: (x – 1)2 = –81 x – 1 = 9i x – 1 = 9i

23 What is the next step: (x + 5)3 = 8 x + 5 = 2 x + 5 = 2

Operations with Nth Roots Simplify

1. 2.

3. 4.

Add/Subtract. 5. 6.

7. 8.

Exponent properties with nth roots. Rewrite your final answer in radical form.

9. 10. 11.

Simplifying & Rationalizing the Denominator Rationalize the denominator.

1. 2.

3. 4.

5. 6.

Different Ways to Rationalize the Denominator

Solving Power Equations

1. 31 2

4x = −

2. (x + 5)4 = 25

3. 6x4 = 486

4. (x – 1)3 + 3 = –122

Solving Radical Equations

1. 2.

3. 4.

When do we need a + / - ? When do we need to check for extraneous solutions?

Day 4: Inverse Functions In these notes we will learn what an inverse function is and how to find it algebraically.

Recall from Unit 2: Inverse functions map output values back to their original input values. (switch x, or input, & y, or output, values in ordered pairs) Graph both functions. What is their relationship? __________________________ 1. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2, but limit the domain x > 0 𝑔𝑔(𝑥𝑥) = √𝑥𝑥

Ordered pairs: 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 𝑔𝑔(𝑥𝑥) = √𝑥𝑥

We saw graphically that the functions are inverses because…. 1. In the ordered pairs, _____ and ______ were switched. 2. The graphs were reflected across _______

Verifying that Functions are Inverses of each other: 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 and 𝑔𝑔(𝑥𝑥) = √𝑥𝑥 Show that f(g(x)) = x AND g(f(x)) = x

Verify algebraically that f(x) = 4x + 2 and g(x) = 1 14 2

x − are inverses of each other.

Show that f(g(x)) = x AND Show that g(f(x)) = x

Finding an Inverse Relation from an equation 1. Switch x and y in equation 2. Solve for y. 1. y = 4x + 2

2. f(x) = -2x + 5 Substitute y for f(x) before you start!

3. f(x) = x2 + 2 , x ≥ 0

4. y =

12

x 3 − 2

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

Day 5: Graphing Square & Cube Root Functions

In these notes we will ANALYZE the graphs of Square Root and Cube Root Functions The Square Root Function: The “parent function” is y x= Let’s look at the table of values using our calculator: x y = √𝑥𝑥

** You can always get “nice” values off the table

X-intercept: Y-intercept: Domain: Range:

Increasing: Decreasing:

The Cube Root Function: The “parent function” is 3y x= Let’s look at the table of values using our calculator: x y = √𝑥𝑥

X-intercept: Y-intercept: Domain: Range:

Increasing: Decreasing:

“Vertex”, (h, k) at _______ From the vertex… Over ______, up ______ Over ______, up ______

“Vertex”, (h, k) at _______ From the vertex… Right ______, up ______ Right ______, up ______ AND Left ______, down ______ Left ______, down ______

These two functions have the same transformations as…

absolute value function, y = a|x – h| + k , quadratic function, y = a (x – h)2 + k

and the cubic function, y = a (x – h)3 + k

if 1a >

Vertically Stretched by a factor of a (ignore negative)

if 1a <

Vertically Shrunk by a factor of a

If a < 0 (a is negative) Vertical flip

reflected across x-axis

(h, k): Translates the graph horizontally h units and vertically k units

Remember, think ___________ when C is with the x (horizontal shift), and ________ when C is outside (vertical shift)

Transformations of Square Root Function: Remember, over _____, up _____; over _____, up _____.

( ) 2h x x= + (h,k): _________

x y

( ) 2h x x= − + (h,k): _________

x y

When you are completing the table of values with SQUARE ROOT FUNCTIONS…you will have x-values on ONE side of the initial point (h,k)…not both, like other graphs so far!

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

Let c be a positive real number. Let ( )y f x= .

Vertical shift c units upward: ( ) ( )h x f x c= + Horizontal shift right c units: ( ) ( )h x f x c= −

Vertical shift c units downward: ( ) ( )h x f x c= − Horizontal shift left c units: ( ) ( )h x f x c= +

Transformations of Cube Root Function: Remember, over _____, up _____; over _____, up _____ (& backward)

3( ) 2 3h x x= + + (h,k): _________

x y

33y x= (h,k): _________

x y

When you are completing the table of values with CUBE ROOT FUNCTIONS…you will have x-values on BOTH sides of the initial point (h,k)! (So put (h, k) in the middle of your table !)

Domain Restrictions based on an equation: 1. Dividing by zero is undefined: a denominator can NEVER be equal to zero.

(We will discuss this later on in Unit 7) 2. The square root of a negative number does not exist - When it comes to graphing, we NEVER put a negative number under a square root (unless we are dealing in complex numbers). Domain Restrictions - Case #2 above: No Negatives Under the Radical Sign!!

x ≥ ____ Notice the inequality! Or Equal to 0 (because we can square root 0!)

Do you have a square root? Do you have a rational power with a denominator of 2?

(If not, then you don’t have to worry about this restriction!)

f(x) = x f(x) = 21

x Domain: The set of all real numbers x ≥ 0

5y x= − What’s under the radical

must be ≥ 0

3y x= +

What’s under the radical must be ≥ 0

( ) 2 3f x x= + What’s under the radical

must be ≥ 0

________ What’s under the radical

must be ≥ 0

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

Practice: Graph the following and determine the domain without a calculator. Then verify with your calculator and use to find intercepts if necessary. Round to the nearest tenth.

y = −2 x − 3 (h,k): __________ x-int: ___________ y-int: ___________ Domain: _________ Range: __________ How we know the domain without graphing:

( ) 4r x x= + (h,k): __________ x-int: ___________ y-int: ___________ Domain: _________ Range: __________

2 5y x= + −

(h,k): __________ x-int: ___________ y-int: ___________ Domain: _________ Range: __________

3 4 1y x= + −

(h,k): __________ x-int: ___________ y-int: ___________ Domain: _________ Range: __________ How we know the domain without graphing:

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

Day 6: Solving Equations Graphically

So far we have learned how to solve equations algebraically, but we can also solve equations or check solutions graphically!

For example: Solve algebraically:

Solve graphically: This means:

where does the graph of cross ?

Solve algebraically:

Solve graphically: This means:

where does the graph of cross ?

Even though we found a solution algebraically, the solution was extraneous. Why?

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y