Chaper 9 Notes

Day 10 9.1 – Solving Quadratic Equations by Finding Square Roots 9.2 – Simplifying Radicals 9.1 Solving Quadratic Equations Recall: “Squaring a Number” 52 =

(-5)2 =

We say: The square of ……… and ……… both equal ……… Today: Reverse it! “Square Root of a Number” If b2 = a, then b is a ……………………… of a. Ex: If b2 = 25, then b = ……………… All positive numbers have two square roots: a …………………… and a …………………… Caution: Do not get confused… 1)

Evaluate: 1) 36

9=

2) − 9 =

2) − 36

4) b 2 =9

3) ± 9 =

3)

0

4) ± 0.64

5)

−1

If the square root of a number a is an integer or can be written as a quotient of integers, then a is called a ……………………………. Evaluate. Is the number underneath the radical symbol a perfect square? 1) − 81

2) − 1.21

3)

0.04

4)

2

5)

5

Evaluate the Radical Expression

b 2 −4ac where a=2, b=1, and c=-3

I. Solving a Quadratic Equation Standard Form:

If b=0, then the equation becomes: To solve this form,   Solve the equation. 1) x 2 =64

2) x 2 =0

3) x 2 =− 4

4) 3x 2 =6

5) 5x 2 + 5= 20

6) x 2 + 49 = 0

Modeling: A Falling Object When an object is dropped, the speed at which it falls continues to increase. Its height h can be approximated by the model: h = − 16t 2 + s where h = height (feet) t = time fallen (seconds) s = initial height (feet) An egg is dropped from a building 32 feet tall. a) Write an equation to model the height of the egg. b) How long will it take for the egg to splat on the ground? (Use both methods) Method 1 (I/O Table) time (sec) height (ft) 0.0 0.5 1.0 1.2 1.4 1.5

Method 2 (Solve Quadratic Equation)

9.2 Simplifying Radicals Investigating Properties of Radicals 1. Fill in the table. a) a = 4, b = 9

ab

a• b

a b

a b

b) a = 64, b = 100 c) a = 25, b = 4 d) a = 36, b = 16 e) a = 100, b = 625

2. Fill in the table. a) a = 4, b = 49 b) a = 16, b = 64 c) a = 25, b = 36 d) a = 225, b = 4 e) a = 144, b = 100

Conclusion: Properties of Radicals 

Product Property



Quotient Property

A radical is completely simplified when… 1. No perfect squares under the radical:

3 = 4 1 = 3. No radicals in denominator: 9 2. No fractions under radical:

8=

Practice! Yay! Simplify.

3 25

20 4

1.

40

5.

1 112 2

6. 8

13 9

7. 2

5 16

9.

1 32 • 2 2

10.

9 •4 25

11.

7•

2.

3.

HW 10: p.508: 55-65 (odd), 78-84 p. 514: 17, 19, 21, 27, 29, 31, 39-49 (odd), 50, 51

18 2

4.

32 50

8.

32 49

12.

10 • 16 5

Day 11 Rational Exponents Tidbit: Exponent comes from the Latin word meaning “place outside”. Rational Exponents - 2 forms… Radical form

Exponential form

25 3

27

4

16

I. Simplify the Rational Exponent Expression. a.

125

1 3

b.

16

1 4

c.

1 2

2 •2

1 2

d.

Definition: Rational Exponent If the nth root of a is a real number and m is an integer, then 1 n

and

a =

m n

=

a =

II. Converting between Exponential and Radical Form Convert to radical form. a.

x

3 5

b.

y

5 − 2

Convert to exponent form. c.

a

3

d.

( ) 5

b

2

1 2

2 •8

1 2

Summary: Properties of Rational Exponents Let m and n represent rational numbers. Assume that the denominator ≠ 0. Property

1 3

Example 2 3

a m •a n =

8 •8 =

2.

(a m ) n =

 1 4 5 2  =  

3.

(ab) m =

1 2

4.

a−m

5.

a = an

1.

(4 •5) = 9

m

π π

4

1.

3 2 1 2

= =

 5 3   =  27 

Simplify. Use 2 different methods. Method 1

−

1 2

1

 a m 6.   = b

(−32)

−

Method 2

3 5

7 2

−

25

3 2

2.

32

3 5

3.

(−32)

4 5

Homework 11 – Exponents, Radicals and Rational Exponents Simplify each expression. −9a 2b−2 3 −2 4 •2−5 1. 2.   (−2−4 )  3ab 

Simplify. 4. 72

5.

3

−6x 2 y 2 −2 (2xy −1 ) 2 3.   • xy  3x 

6. 12x 2 y 3 z 4

54

7.

3

4 x 3 y •3 2xy 2

Rewrite the expression in radical form. Simplify fully. 1

8. 32 2

1

9. (−8x) 3

2

10. (4a 3b 2 ) 3

Simplify the expression. 12. 8

−2 3

12. 16

1

11. (x 2 +y 2 ) 2

1

3 4

13.

3(4 x 4 ) 2 (3x 2 )

−

1 2

15.

( 3

6 •4 6

) 12

Day 12 9.3 – Graphing Quadratic Functions 9.4 – Solving Quadratic Equations by Graphing 9.3 Graphing Quadratic Functions Quadratic Function (standard form):

picture:



The graph of a quadratic function is U-shaped, called a ……………………



If a is positive, the parabola opens ……………………



If a is negative, the parabola opens ……………………



The vertex has an x-coordinate of ……………………



To find the y-coordinate of the vertex, ……………………



The axis of symmetry is a vertical line that passes through the …………………… and has the equation ……………………

Directions: Graphing a Quadratic Function a) State whether the parabola opens up or down. b) Find the vertex. c) Write an equation of the axis of symmetry d) Make a table of values. Use x-values that are to the L and R of the vertex. e) Sketch a smooth U-shaped parabola. ex1: y = x2 + 4x + 1

ex2: y = -2x2 - x + 2

ex3: y = -7x2 + 2x

ex4: At the bank of the Chicago River, there is a water cannon that sprays recycled water across the river. The path of the water’s arc is modeled by: y = -0.006x2 + 1.2x + 10 where x is the distance (feet) across the river, y is the height of the arc, and 10 is the height of the cannon above the river. a) What is the maximum height of the Water Arc? b) How far across the river does the water land?

9.4 Solving Quadratic Equations by Graphing The solution(s) to ax2 + bx + c = 0 are called the x-intercepts where …………. X-intercepts are a.k.a ………………………… or …………………… Solve Algebraically. Check graphically. 1. x2 + 2x = 3

2. –2x2 – 4x = –30

HW 12:

1. Worksheet on next 3 pages 2. p.521: 29-33(o), 47, 50, 53, 65, 66, 72-74 3. p.529: 19, 20, 24, 27, 30, 33, 36, 42

HW 12 - 9.3 Worksheet I. Complete the table and graph each quadratic equation. 1. y = x2 x -2 -1 0 1 2 y

3. y = 2x2 x -2 y

2. y = –x2 x -2 -1

0

1

2

0

1

2

y

-1

0

1

2

4. y = –½ x2 x -2 -1 y

Graph 5 and 6 on your calculator. Sketch what you see in your window. 5. y = ¼x2 6. y = –4x2

II. Fill in the Blank Given the equation y = ax2, predict what will happen in each of the following cases. (Look at 1-6 and graph more examples if necessary. Choose between the words “up or down and wider or narrower for the blanks below.)

7. When a < –1, the graph of y=ax2 will open _________________ and will be ____________________ than y=x2. 8. When –1 < a < 0, the graph of y=ax2 will open _________________ and will be ____________________ than y=x2. 9. When 0 < a < 1, the graph of y=ax2 will open _________________ and will be ____________________ than y=x2.

10. When a > 1, the graph of y=ax2 will open _________________ and will be ____________________ than y=x2.

III. Graph each of the following on your calculator. Sketch what you see & label each vertex.

1. y = x2 – 1

2. y = –x2 + 2

3. y = x2 + 3

4. y = –x2 – 4

IV. Fill in the blank. Given the equation y = ± x2 + c, predict what will happen in each of the following cases. (Look at 1-4 and graph more examples if necessary.)

5. When c < 0, the graph of y = ± x2 + c will move ___________________________________. 6. When c > 0, the graph of y = ± x2 + c will move ___________________________________. V.  Quadratic Equations in Standard Form Given the following quadratic equations in standard form y = ax2 +bx + c, (Calculators OK for 2-4) a.) Calculate −

b 2a

1. NO CALCULATOR y = x2 +2x + 1 x y a.)_________________ c.)_________________ d.)_________________

b.) Sketch the graph

c.) Find the axis of symmetry

d.) Find the vertex

2. y = –x2 +3x

3. y = x2 – 2x + 1

4. y = –x2 + 2x – 3

a.)_______________

a.)_______________

a.)_______________

c.)_______________

c.)_______________

c.)_______________

d.)_______________

d.)_______________

d.)_______________

5. Given a quadratic equation in standard form y = ax2 +bx + c, −

b is the value of the ________2a

coordinate of the ________________. 6. Explain how to find the axis of symmetry.

VI.  A word problem  Joseph did a front dive off of a diving board (assume the board is 3 feet higher than the surface of the water). His path is described by the quadratic equation: y = -0.5x2 +3x where y is the height above the diving board and x is the horizontal distance away from the diving board (x and y are both measured in feet). What was Joseph’s maximum height above the water? Sketch a picture of the situation.

Day 13 9.5 – Solving Quadratic Equations by the Quadratic Formula 9.6 – Applications of the Discriminant Before: Solved quadratic equations with forms: 1. ax2 + c = 0 by finding square roots 2. ax2 + bx + c = 0 by graphing Today: Solve any quadratic equation algebraically 9.5 The Quadratic Formula To solve any quadratic equation in the form of ax2 + bx + c = 0 Use the quadratic formula: ……………………………….. where a ≠ 0 and b2 – 4ac ≥ 0. Why does the formula work? Don’t just take my word for it! Let’s prove it!

Solve: 1) 4x2 - 13x + 3 = 0

2) -10x2 + 3x + 2 = 0

3) -1 +3x2 = 2x

The x-intercept(s) aka ……………… of a graph is when ……………… Find the x-intercepts of the quadratic function. 4) y = 2x2 - 6x – 8 5) y = -2x2 + 6x + 9

6) y = -4x2 + 8x - 2

Q: What is the difference between… quadratic equation quadratic formula

quadratic function

Vertical Motion Models (p. 535) Object is dropped: Object is thrown: where h =

t=

s=

v=

ex: An acorn falls from a tree that is 45 feet tall. How long will it take the acorn to hit the ground?

9.6 The Discriminant Discriminant - The expression inside the radical of the quadratic formula: ………………….. Q: What is the discriminant used for?

When solving a quadratic equation ax2 + bx + c = 0, 

If b2 – 4ac > 0, then there are ………………………



If b2 – 4ac = 0, then there are ………………………



If b2 – 4ac < 0, then there are ………………………

ex: a) Find the discriminant. b) State the number of solutions for the quadratic equation. 1) 2x 2 − 4x + 3= 0

1 3

2 2) − x +x +4 =0

1 2

2 3

2 ex: Use the quadratic function y = x + x −3.

a) Evaluate the discriminant.

b) How many roots does the parabola have?

c) Find the roots of the parabola.

3) 3x 2 − 6x + 3= 0

Homework 13 – Vertical Motion Funsheet  Calculator okay to computer final answer. Show all work! 1. At t=0 seconds, a projectile has an initial velocity v0 and an initial height h0. Write an equation for the motion of the projectile using the values for v0 and h0. Assume gravity is the only force acting on the projectile. a) v0 = 128 feet per second, h0 = 0 feet

_________________________________________

b) v0 = 96 feet per second, h0 = 124 feet

_________________________________________

2. For each of the following equations describing projectile motion, give the initial upward velocity and the height above the ground at t=0 seconds. a) h(t) = -16t2 + 48t + 7

Initial upward velocity: __________

Height above ground at t=0: __________

b) h(t) = -16t2 + 32t

Initial upward velocity: __________

Height above ground at t=0: __________

3. You retrieve a football from a tree 25 feet above ground. You throw it downward with an initial speed of 20 feet per second. Use a vertical motion model to find how long it will take for the football to reach the ground.

4. You throw a ball from 6 ft high to a friend 40 ft up in the stands at 49 ft/sec. Write and solve a quadratic equation for how long the ball takes to reach your friend.

5. From the top of a 40-foot cliff, you throw a stone downward at 20 ft/sec into the water below. How long will it take the stone to hit the water?

6. You jump off a 15-foot diving board with an initial upward speed of 3 ft/sec. How long will it take you to hit the water?

7. Big Bird, standing at the edge of a canyon 2500 feet above the canyon floor, throws a rock into the air with an initial upward velocity of 96 feet per second. a) When will the rock return to the same height as Big Bird? b) When will the rock stroke the floor of the canyon?

8. The height in feet of a basketball thrown from the free-throw line after t seconds is given by: h(t) = -16t2 + 30t + 7. a) From what height is the basketball thrown? b) What is the basketball’s greatest height above the floor? c) When does the ball reach its greatest height? d) The basket is missed. When will the ball hit the floor?

3 Models:

Day 14 (Quiz Today on 9.1-9.6) 9.8 – Comparing Linear, Exponential, and Quadratic Models 1. Linear 2. Exponential 3. Quadratic

General Equation:

Graph:

ex1: Make a scatter plot using the data. Decide which model is the best fit. 1. (0, 3), (8, 3), (-4, -1), (4, 4), (-6, -3), (10, 1)

2. (1, 3), (0.5, 1.5), (-2, 0.1), (0, 1), (1.5, 5)

3. (-3, 4), (-2, 7/2) (-1, 3), (0, 5/2)

Homework 14 – 9.8 Comparing Linear, Exponential, and Quadratic Models In the table below, fill in the general formula for each of the following models. Then sketch 2 different graphs of what that model could look like. State below each sketch what it is in the formula that would make your sketch look the way it does (e.g. the linear model goes down because the “m” or slope can be negative). Linear Model Exponential Model Quadratic Model General Formula Sketch #1

Sketch #2

For the following application problems, a) State which model (linear, exponential, or quadratic) it is describing b) Write a “y=” equation with the appropriate constants that can model the situation c) Solve for the appropriate variable 1. The tennis team wants to purchase t-shirts for its members. A local company charges a $20 set-up fee and $6 per shirt. The total cost of the t-shirt can be written as a function of the number of t-shirts purchased. (At this point, you should be able to do parts a and b.) If the tennis team only has $250 to spend, how many shirts can they purchase?

2. A computer that originally cost $1500 depreciates in value at a rate of 10% per year. The value of the computer can be written as a function of how much time passes. (Do parts a and b.) After 4 years how much will the computer be worth? You may use a calculator to do part c.

3. A physics student is in an “egg dropping contest”. The goal is to create a container for an egg so it can be dropped from a height of 64 feet without breaking the egg. The egg container’s height can be written as a function of how much time passes (in seconds). (Do parts a and b.) How long will it take for the egg to reach the ground?

4. After graduating from high school you put some money you received from family members in a savings account that accrues 3.5% interest compounded annually. The amount of money in your account can be written as a function of how much time passes. (Hint: At this point, there is one constant that is not given to you. For now, put a variable where that constant should go.) (Do parts a and b.) After 5 years of not touching that money you have $2500. How much money did you put in the account originally? (Use a calculator for part c.)

5. A batter hits a pitched baseball when it is 3 feet above the ground with an upward initial velocity of 80 feet per second. The height of the ball can be written as a function of how much time passes (in seconds). (Do parts a and b.) After 3 seconds, is the ball still in the air? If no, justify how you know. If yes, how high is the ball? (Use a calculator for part c.)

6. Pablo wants to start his own carwash business so he buys car-cleaning supplies for $115. He plans to charge $8 per car. Pablo’s profit can be written as a function of the number of cars he washes. (Do parts a and b.) How many cars does Pablo have to wash in order to break even?

7. A species of wild bunnies is put in a protected reserve. When they were first put in the reserve, there were 25 bunnies. Each year the population tripled. The number of bunnies can be written as a function of how much time passes. (Do parts a and b.) After 6 years how many bunnies were in the reserve? (Use calculator for part c).

8. A car traveling at a constant rate of 60 mph is 300 miles from its destination. The miles from the destination can be written as a function of how much time is spent traveling. (Do parts a and b.) How far away from the destination is the car after 3 hours?

9. You are on the 3rd floor throwing a whiffle ball to your friend who is on the ground 30 feet below. You throw the ball with an initial downward speed of 10 feet per second. The height of the ball can be written as a function of how much time passes (in seconds). (Do parts a and b.) How long will it take for the ball to reach the ground? (Use calculator for c.)