Chapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving … 1H Chapter 9...Notes Ch. 9 Alg. 1H 1 Chapter 9...
Transcript of Chapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving … 1H Chapter 9...Notes Ch. 9 Alg. 1H 1 Chapter 9...
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?
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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: ( , )
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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
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Notes Ch. 9 Alg. 1H 7
y
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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
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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
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