MATH 2 - UNIT 10 – NOTE PACKET

MATH 2 Unit 10 Notes: DAY 1 – Exponent RulesWarm-up

1. Write the following in expanded form and then evaluate.a. 34

b. 28

c. 42

d. 53

Day 1 Notes: Exponent Rules

What is a POWER?

When it comes to mathematics, we refer to a ___________________________ as an exponential expression used for indicating the numbers of factors involved in multiplication.

There are special rules for powers that will make solving questions in this course easier if we apply them correctly. You may have been exposed to these rules way back in the day....

What is a BASE?

What is an EXPONENT?

The Exponent Rules

Rule #1: Multiplying Powers with the Same Base

When multiplying powers with the same base you _________________ the exponents.

Example: a3 x a2 = a3 + 2 = a5

You try:

1.

2.

3.

Rule #2: Dividing Powers with the Same Base

When dividing powers with the same base you ______________________ the exponents. Example: a5 ¿ a3 = a5 – 3 = a2

You try:

1.

2.

3.

Rule #3: When Raising Powers to Another Power

When raising a power to another power you __________________________ the exponents.

Example: (a4)2 = a4 x 2 = a8

You try:

1.

2.

3.

Rule #4: Powers with a Negative ExponentPowers with a negative exponent can be written as a FRACTION with a _______________________________ exponent.

Example: a-5 = 1a5

You try:

1.

2.

3.

Rule #5: A Power with an Exponent of One

When evaluating a power with an exponent of one, the answer will be the base.

Example: a1 = aYou try:

1.

2.

3.

Rule #6: A Power with an Exponent of ZeroWhen evaluating a power with an exponent of zero, the answer will be one.

Example: a0 = 1You try:

1.

2.

3.

Day 1 Classwork

FIND THE VALUE OF EACH EXPRESSION:

1) 2) 3) 4)

5) 6) 7)

SIMPLIFY EACH PRODUCT:

8) 9) 10)

11) 12)

SIMPLIFY EACH PRODUCT:

13) 14)

15) 16)

17)

SIMPLIFY EACH EXPRESSION:

18) 19) 20)

21) 22) 23)

24) 25) 26)

27) 28)

SIMPLIFY EACH EXPRESSION:

29) 30) 31)

32) 33) 34)

MATH 2 Unit 10 Notes: DAY 2 – Simplifying RadicalsWarm-up

1) 6a5 y3

12a4 y6

2) 2 x−2 y3

3) ¿

Day 2 Notes

________________________________ is the inverse function of exponents.

n√ x=r

An _____________________ in a radical tells you how many times you have to multiply the root times itself to get the radicand.

Ex) √81=¿______________________________________

81 = ____________________________________________

9 = _____________________________________________

2 = _____________________________________________

When a radical is written without an index, there is an understood index of 2.

Examples:

3√64 = _____________________________________________

Radicand: _____________________________________________

Index: ________________________________________________

Root is ___________ because ____________________________=________=_________

5√32x5 = __________________________________________

Radicand: _____________________________________________

Index: ________________________________________________

Root is ___________ because ____________________________=________=_________

Yes….. you can use a calculator to do this, but for some of the more simple problems, you should be able to figure them out in your head.

To use the calculator:

1) Math 2) Select #5: x√❑3) Type in index and radicand4) Enter

Discuss with your neighbor if you’ve forgotten the shortcuts to find square roots and cube roots using your calculator.

You try:

1) 5√243 y5

2) 4√1296m4n8

3) √144 v8

Not every problem will work out that nicely! Try using our calculator to find an exact answer

for 3√24.

The calculator will give us an estimation, but we CAN’T write down an irrational number like this exactly – it can’t be written as a fraction and the decimal never repeats or terminates. The best we can do for an exact answer is use simplest radical form.

Simplifying Radical Steps:

1) Factor Tree (Both Numbers and Variables)2) Group according to the index3) Simplify ( Negative numbers under radical are okay for odd index’s but create imaginary

numbers( _________ ) for even index’s.

Ex) √12

Ex)3√24

Ex) 4√48

MATH 2 Unit 10 Notes: DAY 3 – Rational Exponents

Warm-up

Quincy has been confused about simplifying algebraic expressions. His brother Oscar prepared this quiz for him to practice. Look at Quincy’s responses below. If the response is correct, write correct. If it is incorrect, write the correct answer.

1. 4y – 5y -y2. 3x + 2x 5x2

3. 2n2 + 4n2 6n4

4. 2x • 3y 5xy5. 2a – (a – b) a – b

6.n+5n 5

7. 4 – (n + 7) -3 – n

8.2x6

x3 2x2

9. (2xy)2 4x2y2

10. 2a + 3b + 3a + 4b 6a2 + 12b2

Day 3 Notes

Rational Exponents are another way to write radicals.

Raising a number to the power of ½ is the same as performing a familiar operation. Let’s take a

look at the graph of y = x12 to discover that operation.

Step 1: Type x12 into y = screen on your graphing calculator.

Step 2: Look at the table of values generated by this function.

Step 3: Discuss with your classmates what you believe to be the relationship between the x and y values in the table. Where have you seen this relationship before?

Rational Exponents:

i√ xe _____________

x = ________________________

e = ________________________

i = _________________________

MATH 2 Unit 10 Notes: DAY 4 – Solving Radicals

Warm up:

Day 4 Notes

Solving Radicals are very similar to the square root method in Unit 2.

Steps:

1) Get the base alone (aka. get the √❑ by itself!!!)2) Take the inverse (the exponent should match the index)3) Solve for the variable.4) Check Answers

Special Cases:

- If index is EVEN the you must check both the positive and negative solutions.- If you have an EVEN index and the equation is equal to a negative, then the answer is

NO SOLUTION.- Always check solutions for extraneous case (solution that doesn’t work).

Ex) 4√ x−2=3

Ex) 3 ( 3√ x2+5 )=207

Ex) 3√ x−2=4

Ex) √2x−5=9

Ex) √a+2−2=a

Ex) √33 x−2=−5

Ex) √3 x−5=√5 x+5

MATH 2 Unit 10 Notes: DAY 5 – Graphing RadicalsWarm-up

Elizabeth was going to complete her algebra assignment using her graphing calculator. Her older sister Carolyn decided to change the equations that she had copied into her notebook to tease her. Carolyn tells her that the equations are still equivalent to what her teacher had given her. Help Elizabeth by making each equation below quicker and easier to type into her calculator.

1.y = 2x-1

2. y = (2x)-1

3.y=7 .2 x

7

3 .6x 4

4. y = (-4)0

5. y = x-4.2x6.2

6. y = 33 • 32

7.y= 15−4

Day 5 Notes

Graphing Square Root Functions

Make a table for each function.

F(x) = x2 f(x) = √ x

0 01 12 23 34 45 56 67 78 89 9

Ignore the points with decimals. What do you notice about the other points?______________________________________________________________________________

These functions are _______________ of each other. By definition, this means the

_____________ and the _____________ ______________.

Plot the points from the tables above.

As a result, the graphs have the same numbers in their points but the _____ and the _________

coordinates have ___________ _______________.

x f(x) x f(x)

This causes the graphs to have the _____________ _______________ but to be

__________________ over the line ____________.

The Square Root Function

Reflect the function f(x) = x2 over the line y = x.

Problems? _________________

We have to define the Square Root ______________ as ________________. This means that

we will only use the _________________ side of the graph.

The result: f(x) = √ x Characteristics of the graph

Vertex

End Behavior

Domain

Range

Symmetry

Pattern

Transforming the GraphsNow that we know the shapes we can use what we know about transformations to put that shape on the coordinate plane. Remember:Translate Reflect Dilate

1) f(x) = √ x−3

2) f(x) = √ x+4

3) f(x) = −√ x

4) f(x) = √−x

5) f(x) = 2√x+3

6) f(x) = 12 √ x

Sometimes the functions are not in graphing form. We may have to use some of our algebra skills to transform the equations into something we can use.

Ex: f(x) = √4 x−12 This is not in graphing form.

Ex: f(x) = √9 x+36−5 This is not in graphing form.

MATH 2 Unit 10 Notes: DAY 6 – Radical ApplicationsWarm-up

Find any roots or zeros of each quadratic function below. Then explain what roots or zeros mean graphically.

1. f(x) = 2x2 – 5x – 3

2. f(x) = x2 + 2x + 1

3. f(x) = x2 + 2x + 3

4. f(x) = 2x2 + 3x - 1

MATH 2 Unit 10 Notes: DAY 7 – VariationWarm-up

Gwen and Rita are discussing the characteristics of certain functions. Rita claims that the equation y = x2 does not model a function because the line y = 5 intersects the graph of the equation in two points. Gwen disagrees. How might she present her case to Rita?

Day 7 Notes

Direct Variation: As x gets ________________________________________________________

A direct variation is represented by the equation ___________________ where k is the

________________________________________________________.

1) Real world examples of direct variation include wages varying directly to hours worked or circumference of a circle varies directly as the diameter. Can you think of others?

2) If y varies directly as x and x=10 when y=9, then what is y when x=4?

3) The refund (r) you get varies directly as the number of cans (c) you recycle. If you get a $3.75 refund for 75 cans, how much should you receive for 500 cans?

Inverse Variation: _____________________________________________________________.

Can be represented by the equation ____________________ where k is the constant of variation.

1) Real Word Examples of Inverse Variation – For a trip to Myrtle Beach, the greater your car

speed, the ________________time it would take you to get there. Also, if a rectangle has an

area of 15 square units, then as the length increases the width ________________________.

2) If y varies inversely as x and x=3 when y=9, then what is x when y=27?

1) The amount of resistance in an electrical circuit required to produce a given amount of power varies inversely with the square of the current. If a current of .8amps requires a resistance of 50 ohms, what resistance will be required by a current of .5 amps?

Classwork

Find the Missing Variable:

1) y varies directly with x. If y = -4 when x = 2, find y when x = -6.

2) y varies inversely with x. If y = 40 when x = 16, find x when y = -5.

3) y varies inversely with x. If y = 7 when x = -4, find y when x = 5.

4) y varies directly with x. If y = 15 when x = -18, find y when x = 1.6.

5) y varies directly with x. If y = 75 when x =25, find x when y = 25.

Classify the following as: a) Direct b) Inverse c) Neither

6) m = -5p 9) c =

e−4 12) c = 3v

7) r =

9t 10) n = ½ f 13) u =

i18

8) d = 4t 11) z =

−.2t

What is the constant of variation for the following?

14) d = 4t 15) z =

−.2t 16) n = ½ f 17) r =

9t

Answer the following questions.

18) If x and y vary directly, as x decreases, what happens to the value of y?

19) If x and y vary inversely, as y increases, what happens to the value of x?

20) If x and y vary directly, as y increases, what happens to the value of x?

21) If x and y vary inversely, as x decreases, what happens to the value of y?

MATH 2 Unit 10 Notes: DAY 8 – System of EquationsWarm-up

Lily found the price of table tennis balls listed on the Internet at $4.75 for a package of 6 balls. Shipping and handling was listed at $1.00 per package.

1. Write an equation that represents the total cost for different numbers of packages of table tennis balls.

2. Sketch a graph of this relationship.

3. If you shift your graph up a value of $0.50, does this mean the price per package increased, or the shipping price increased?

Write a new equation for the situation.

Day 8 Notes

Solving Systems Steps:

1) Put 1st equation under y12) Put 2nd equation under y23) Graph4) 2nd Trace #5: Intersection5) Enter 3 times6) Repeat steps 4-5 for 2nd point

**** The points where they intersect are the solutions!

Ex)

What does it look like to solve a system with other functions?

1. {y=x2−24y=x−12

2. {y=x2+3x−5y=x+3

3. {y=x2−4 x+6y=x+2