Radical Functions

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
Day 5 / Intro to Inverse Functions
Day 6 / Square and Cube Root Functions & Characteristics
Day 7 / Review for Unit Test
Day 8 / Unit 6 Test

TentativeSchedule of Upcoming Classes

Day 1 / Fri 2/3 / Day 1 Notes:
Simplifying nth roots, Operations, and Solving Power Equations
Day 2 / Tues 2/7 / Day 2 Notes:
Solving Radical Equations
Day 3 / Thur 2/9 / Day 3
Review: Days 1 & 2
Day 4 / Mon 2/13 / Day 4
Quiz: Days 1 & 2
Day 5 / Wed 2/15 / Day 5 Notes:
Intro to Inverse Functions and Solving Equations Graphically (w/ a calculator)
Day 6 / Fri 2/17 / Day 6 Notes:
Graphing Square & Cube Root Functions
Day 7 / Wed 2/22 / Day 7
Review: Unit 6
Day 8 / Fri 2/24 / Day 8
Unit 6 Test

Need Help?

Mu Alpha Theta (math honor society) is available for homework help on

Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:10.

Ms. Raschiatore is in L404starting at 8:30 each morning.

Need to make up a test/quiz?

Math Make Up Room is open Tuesday and Thursday after school and Wednesday before school. The schedule with room locations is posted around the math hallway & in the math classrooms

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.

WARM-UP: Fill in the table, use your calculator only when needed:

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 =

Getting to know rational exponents:

Exponential Notation / Radical Notation
/ ()2 or
/ ()3 or
am/n /
a-m/n /

You try: 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.

Although we will start by using the calculator to help us simplify radicals, eventually we will need to simplify withouta calculator – start memorizing and recognizing perfect squares and cubes now 

Evaluate the following.(Use the tables from the previous page as necessary)

1. = / 2. =
3. = / 4. =

Discuss: Why can we take the cube root of a negative number and get a real answer, but when we take the square root of a negative number we get an imaginary answer?

Let’s evaluate these expressions without a calculator.

Writing Radicals in Simplest Form– Refer to your exponent chart for perfect nth rootsif needed

Find a perfect 3rd root that is a factor of 482. Find a perfect 4th root that is a factor of 48

Operations with Radicals:

Addition and Subtraction:

Remember – the “radicands” (numbers under the radical) AND the “index” (root) must match!

3. / 4.

Multiplication & Division:

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

5. / 6.
7. / 8.

Sometimes we are left with radicals in the denominator…when that happens we ______!

Recall how to rationalize with square roots:

NOW with otherroots …The goal is to get a perfect nth root in the denominator to eliminate the radical

We need a perfect cubein the denominator,
4is a factor of what perfect cube? ______
So multiply by… ______/ We need a perfect cubein the denominator,
3 is a factor of what perfect cube? ______
So multiply by… ______
We need a perfect 5th power in the
denominator…
8 is a factor of what perfect power of 5?_____
So multiply by… ____ / We need a perfect 4th power in the
denominator…
8 is a factor of what perfect power of 4?_____
So multiply by… ____

Simplify Expressions with Variables (factor out perfect nth roots) Write answers in radical notation.

/ /
We are taking out groups of ______

We are taking out groups of ______/
We are taking out groups of ______/
We are taking out groups of ______

Solving power equations:

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

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

Solve the following equations. (Hint: Recall from quadratics…. the square root method)

(x + 3) 2 = 64 / 3(x – 5)2 = - 27

Whenever we solve by taking the square root of 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, solve each equation:

x3 = 8x3 = – 8 x2 = – 4x2 = 4

When solving, follow the reverse order of operations.

1. Add or subtract 2. Multiply or divide 3. Exponents 4. Parentheses (follow same order)

Solve over the real numbers (Find the REAL solutions only)

2x4 = 162 / (x – 2) 3 = -125
6x3 = 384 / (x – 3) 4 = 625
/ (x + 1) 5 = 100
(x – 1)5 – 3 = –35 / 2(x – 9)3 = 250

Day 2: Solving Radical Equations

Today we will learn how tosolve radical equations

Solving radical equations:

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

Isolate the radical on one side of the equation
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.
Solve the remaining equation (using knowledge from previous units)
Check all solutions – There may be extraneous solutions!

Find the REAL solutions

1. / 2.
3. / 4.

How to check for extraneous solutions:

1. Take solution(s), x =
2. Plug solution into ORIGINAL equation. See if the statement is TRUE.

5. / 6.
7. / 8.
9. / 10.
11. / 12.
13. / 14. Review…. Simplify:
15. / 16.

Day3: Review Days 1 & 2

Evaluate the following:

1. / 2. / 3. / 4.
5. / 6. / 7. / 8.
1. / 2. / 3. / 4.
5. / 6. / 7. / 8.
1. / 2. / 3. / 4.
5. / 6. / 7. / 8.

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.

Solving Power Equations

1. / 2. (x + 5)4 = 25
3. 6x4 = 486 / 4. (x – 1)3 + 3 = –122

Solving Radical Equations

1. / 2.
3. / 4.
5. / 6.

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

Day 5: Inverse Functions

In these notes we will learn what an inverse function is and how to find it algebraically, and you will learn how to solve an equation graphically (with a calculator).

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. ,
Ordered pairs:

/ We saw graphically that the functions are inverses because….
1. In the ordered pairs, _____ and ______
were switched.

The graphs were reflected across

______
2. , but limit the domainx 0
Ordered pairs:

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

Why must we restrict the domain of f(x) in example 2?

How must the domain of the following be restricted (if at all) to have an inverse function?

4. 5. 6.

Verifying that Functions

are Inverses of each other:

7. Verifyalgebraically that the 2 functions are inverses.

and

This means to show thatf(g(x)) = xANDShow that g(f(x)) = x

8. Verifyalgebraically that the 2 functions are inverses.

f(x) = 4x + 2 and g(x) = .

Show thatf(g(x)) = xANDShow that g(f(x)) = x

Finding an Inverse Function from an equation

1. Restrict the domain of the original function if needed

2. Switch x and y in equation

3. Solve for y

9. y = 4x + 2 / 10. f(x) = -2x + 5 Substitute y for f(x) before you start!
11. f(x) = x2 + 2 / 12. y =
11. f(x) = (x – 1)2 / 12. f(x) = (x + 2)2 + 5

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: / Use your calculator to solve graphically:

This means:
where does the graph of cross?

Solve algebraically: / Use your calculator to solve graphically:

This means:
where does the graph of cross?

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

Day 6: 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

Let’s look at the table of values using our calculator:

x / y =

** 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

Let’s look at the table of values using our calculator:

X-intercept: / Y-intercept: / Domain: / Range:
Increasing: / Decreasing:

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 thecubic function, y = a (x – h)3 + k

if
Vertically Stretched
by a factor of (ignore negative)
/ if
Vertically Shrunk
by a factor of
/ 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, right _____, up _____; right _____, up _____.

(h,k): ______
Horizontal Vertical
/ (h,k): ______
Horizontal Vertical

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), therefore (h, k) is an endpoint.

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

(h,k): ______

Horizontal Vertical
/ (h,k): ______

Horizontal Vertical

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), therefore (h, k) is a point of inflection.

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, 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 with even index!!

Radicand ____

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

Do you have a square root? Do you have a rational powerwith a denominator of 2?

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

f(x) = Domain: Radicand ≥ 0, therefore x ≥ 0

f(x) =Domain:Radicand ≥ 0, therefore x ≥ 0

Radicand ≥ 0 /
Radicand ≥ 0 /
Radicand ≥ 0 / ______
Radicand ≥ 0

Practice: Graph the following and determine the domainwithout a calculator. Then verify with your calculator and use to find intercepts if necessary. Round to the nearest tenth.

(h,k): ______
x-int: ______
y-int: ______
Domain: ______
Range: ______
How we know the domain without graphing: /

(h,k): ______
x-int: ______
y-int: ______
Domain: ______
Range: ______

(h,k): ______
x-int: ______
y-int: ______
Domain: ______
Range: ______/

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

Use this exponent table to help you simplify nth roots!

Familiarize yourself with these numbers!

x / x2 / x3 / x4 / x5
1 / 1 / 1 / 1 / 1
2 / 4 / 8 / 16 / 32
3 / 9 / 27 / 81 / 243
4 / 16 / 64 / 256 / 1024
5 / 25 / 125 / 625 / 3125
6 / 36 / 216 / 1296
7 / 49 / 343
8 / 64 / 512
9 / 81 / 729
10 / 100 / 1000
11 / 121
12 / 144
13 / 169
14 / 196
15 / 225
16 / 256
17 / 289
18 / 324
19 / 361
20 / 400
21 / 441
22 / 484
23 / 529
24 / 576
25 / 625

44 = 256 = 4

43 = 64 = 4