Notes Ch. 9 Alg. 1H 1

Chapter 9 Notes Alg. 1H

9-A1 (Lesson 9-3) “Solving Quadratic Equations by Finding the Square Root and

Completing the Square” p. 486 *Calculator

Find the Square Root: take the square root of ___________ __________.

Ex: 2 9x

2 13x 2

3 16x

Solve by finding square roots. Round to the nearest tenth.

√1) 2 18 81 90m m 2)

2 6 9 25b b 3) 2 14 49 20m m

Steps for Completing the Square:

1. If necessary, divide both sides by _______________________.

2. Isolate the _____________ terms.

3. Divide the coefficient of ____ by ____; ____________ it.

4. Add that number to _________ _______________of the equation.

5. ____________ the trinomial as ____________ or ____________.

6. Take the _______________ _____________ of each side.

7. ____________ for x.

Notes Ch. 9 Alg. 1H 2

Examples:

Solve by completing the square. Leave answers in radical form where necessary.

4) 2 8 4x x 5)

25 20 60 0x x

Solve by completing the square. Round answers to the nearest tenth where necessary.

6) 23 18 30n n 7)

23 4 4 3x x

Notes Ch. 9 Alg. 1H 3

9-A3/4 (Lesson Heath 12.4 and Glencoe 9-1) “Graphing Quadratic Functions” (Vertex Form) Heath p. 639-642

Standard Form:

Completed Square Form (Vertex Form):

Parabola:

Vertex:

o Minimum:

o Maximum:

Axis of symmetry: the line that divides a _____________________ into 2 halves

o Equation:

To sketch a parabola:

1) Find the ______________________.

Use the Completed Square/Vertex to find the values of h and k

Write the Vertex as an ordered pair:

2) Make a _______________ of values.

Choose at least _______ additional values of ______.

2 values __________ than the x-value of the vertex and 2 values that are _______________.

__________________ to find y values and complete the table.

3) Plot the ________________ and draw a __________________.

1. 2 2 3y x x

vertex: ( , )

x

y

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Notes Ch. 9 Alg. 1H 4

2. 22 4 1y x x

vertex: ( , )

3. 25 2 6y x x

vertex: ( , )

4. Physics Problem Example: (Use graphing to solve)

Miranda throws a set of keys up to her brother, who is standing on a balcony 38 ft. above the ground.

She throws with a velocity of 40 ft/sec. and her hand is 5 ft. off the ground. How long does it take the

keys to reach their highest point? Will her brother be able to catch the keys?

x

y

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Notes Ch. 9 Alg. 1H 5

9-A5 (Lesson 9-2) “Solving Quadratic Equations by Graphing & Finding Roots”

p. 480-483

Quadratic Equation: (standard form)

Roots: ________________________

Zeros: ________________________

Check: by _____________________

1. 2 5 4 0c c

vertex: ( , )

2. 20 6 9x x

vertex: ( , )

3. 2 3 5t t

vertex: ( , )

x

y

x

y

x

y

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Notes Ch. 9 Alg. 1H 6

Use Factoring first to determine how many times the graph ________________ the

_________________.

4. 2 10 25f x x x

Factor:

Intersections: __________ Roots: _________

Integral Roots:

If roots aren’t integers, __________________; write solution as a compound

_____________________

5. 22 6 3 0a a

x

y

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Notes Ch. 9 Alg. 1H 7

y

y

x

9-A6 Notes Quadratic Word Problems *calculator Alg. 1H

EX 1: The path of a ball kicked against a wall (the y axis is the wall) follows the path y = 1

9 x2 + x + 4,

where y is the height of the ball in feet, and x is the horizontal distance (in feet) from the wall.

A) How high is the ball at its maximum height?

B) How far above the ground does the ball hit

the wall? (The y axis represents the wall.)

EX 2: In the diagram below, the backboard is located on the y-axis and the hoop is located at the point

(1,10). A basketball thrown toward the hoop follows the path y = .45־x2 + 3.2x + 6.2 where x and y

are measured in feet.

A) When the ball was at its highest point, what

was its horizontal distance from the backboard?

(Round to two decimal places.)

B) At its highest point, how far off the ground

was the basketball?

EX 3: How deep is the pond given by the equation y = 1

3x2 + 2x – 5?

Pythagorean Theorem:

Notes Ch. 9 Alg. 1H 8

9-A8 (Lesson 9-4A) “Solving Quadratic Equations by Using the Quadratic

Formula” CALCULATOR p. 493-497

Quadratic Formula:

2 4

2

b b acx

a

Used to solve Quadratic Equations:

Read Ex. 1

√1A. ..

a =

b =

c =

√1B. 24 2 17x x

a =

b =

c =

C.) A roofer tosses a piece of roofing tile from a roof onto the ground 30 ft below.

He tosses the tile with an initial velocity of 10 ft. per second. How long does it

take the tile to hit the ground?

216h t vt s

Notes Ch. 9 Alg. 1H 9

9-A9 (Lesson 9-4B) “Using the Discriminant” p. 496-497

Discriminant: the _______________ (expression inside the radical

symbol); part of the __________________ formula

2 4

2

b b acx

a

Three possibilities for the discriminant:

No Solution 0 One Solution Two Solutions

(Does not intersect the x-axis) (Vertex is on the x-axis) (Intersects x-axis at two points)

042 acb 042 acb 042 acb

acb 42 acb 42 acb 42

Read Ex. 3 p. 496

√3A. 24 20 25 0n n B.

25 3 8 0x x C. 22 11 15 0x x

When the discriminant is a perfect square, the solutions will be

______________________ numbers

Notes Ch. 9 Alg. 1H 10

9-A11 “Derivation of Quadratic Formula and Choosing a Method”

Solve the following by completing the square: 2 0ax bx c

This is called “the Derivation of the _______________________________”

Which method should you choose to solve 2 0ax bx c ?

(in order of preference and efficiency!)

Choice Method When to Use Lesson

1

When b = 0

To solve ____________

2 When easily ___________

3 Best when ______ and b is

an ________number

4 Any quadratic equation;

gives _________ solutions

Visual

model

Any quadratic equation;

gives _______________

solutions

Notes Ch. 9 Alg. 1H 11

9-A12 “Exponential Functions” p. 502-505

Standard Form:

The variable is:

1. Graph: 3 xy and 1

3

x

y

2. Graph: 2 xy 2 3 xy 2 4 xy

3. Graph: 2 xy 52 xy

42 xy

x y

x y

x y

x y

x y

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Notes Ch. 9 Alg. 1H 12

4. Graph: 2 xy 3 xy 4 xy

Transforming a Graph of an Exponential Function:

Is it necessary to make a table when you already know the transformation?

x y

x y

x y

Change

in

Function Type of Change

Positive

or

Negative

Change in Graph

b

x

a

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