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Math 2 Notes
Day 2 – Section 12-1
Name___________________
Goal:__________________________________________________________________________
I. Quadratic Equations
Form_________________________________________ where __________________________
__________ is the _________________________________________.
__________ is the _________________________________________.
__________ is the _________________________________________.
Graph is a ____________________________ called a ____________________________.
Parts:
1. Leading Coefficient _____________
Positive_______________________
Negative______________________
2. Vertex
3. Axis of Symmetry
II. Graphing
1. Find x:
Together you have ______________
2. Find y:
3. Graph Vertex
4. Use chart to determine the width of the parabola
Over
Up/Down
III. Examples
1. y  2 x 2  4 x
2. y   x 2  2 x  8
3. y  x 2  4 x  5
Math 2 Notes
Day 2 – Section 12-2
Name_______________________
Goal:_________________________________________________________________________
I. Rules for Graphing a Parabola in Vertex Form
1. Constant on the End of the Problem_____________________________________________
Positive ____________________
Negative _______________________
2. Constant Inside Parentheses____________________________________________________
Positive_____________________
Negative________________________
3. Negative in the Front___________________________________________________________
Examples:
A. y  4( x  3)2
B. y  3( x  1)2  2
C. y  2 x 2  4
D. y  2( x  3)2
II. Find the maximum or minimum.
A. y  x 2  8x  3
B. y  3x 2  18x  11
C. y  x 2  8 x  5
III. Find the maximum or minimum VALUE.
VALUE means_______________________________________.
A. y  20 x  5 x 2  9
B. y  7  3x 2  12 x
IV. Find the value of the statement.
A. What is three more than twice the maximum value of y   x 2  14 x  57 ?
B. What is seven less than the minimum value of y  x 2  6 x  14?
V. Find the quadratic regression and the correlation coefficient for each set of data.
A.
X
2
5
-5
-4
Y
4
12
39
19
12
B.
X
-6
-3
0
1
4
Y
-84
-27
-6
-7
-34
0
Algebra 2 Notes
Section Chapter 12 Optimization
Name___________________
Goal:__________________________________________________________________________
Steps:
1. Set up two equations (do not use x and y).
2. Solve the equation without the work max/min in it for any variable.
3. Substitute that expression into the other equation.
4. Know what your variables represent.
5. Enter into the calculator for y= and find the maximum or minimum.
6. Answer the question.
Examples:
1. Find two positive numbers whose sum if 36 and whose product is a maximum.
2. find two numbers whose difference is 8 and whose product is a minimum.
3. A rectangle has a perimeter of 40 meters. Find the dimensions of the rectangle with the
maximum area.
4. Loni has 48 feet of fencing to make a rectangular dog pen. If a house is used for one side of
the pen, what would be the length and width for maximum area?
5. The circulation of the Charlotte Observer is 50,000. Due to increased production costs, the
council must increase the current price of 50 cents a copy. According to a recent survey, the
circulation of the newspaper will decrease 5000 for each 1- cent increase in price. What price
per copy will maximize the income from the newspaper?
6. Marsha is making a box to collect toys for the school toy drive. She cuts a 5 centimeter
square from each corner of a rectangular piece of cardboard and folds the sides up to make a
box. If the perimeter of the bottom of the box must be 50 centimeters, what should the length,
width, and height of the box be for a maximum volume?
7. An object is thrown into the air with an initial velocity of 128 feet per second. The formula
h(t )  128t  16t 2 gives its height above the ground after t seconds. What is the height after 2 seconds?
What is the maximum height reached? At what time is the maximum height reached?
Math 2 Notes
Section 12-4 Solving by graphing
Name_________________
Goal:_________________________________________________________________________
When a quadratic function (y=) is set equation to a value, the result is a __________________.
The solutions are also called ________________, _________________, and _______________.
Possible Solutions_______________________________________________________________.
Three cases:
1.
2.
3.
Solve by graphing.
1. x 2  6 x  8  0
2
2. 8 x  x  16
3. x 2  4 x  5  0
II. Find the solutions on the graphing calculator.
They are called ______________________________.
Calculator Hint:
1. x 2  5 x  4  0
2
2. x  10 x  23  0
2
3. 2 x  6 x  5  0
2
4. x  4 x  6  0
III. Word Problems
1. The highest bridge in the U.S. is the Royal Gorge Bridge in Colorado. The deck of the bridge is 1053
feet above the river below. Suppose a marble is dropped over the railing from a height of 3 feet above
the bridge deck. How long will it take the marble to reach the surface of the water? Use the formula
h(t )  16t 2  h0 where t is the time in seconds, h0 is the initial height and h(t) is the height when the
marble hits the ground.
2. On March 12, 1999, Adrian Nicholas broke the world record for the longest human flight. He flew 10
miles from his drop point in four minutes and 55 seconds using a specially designed aerodynamic suit.
He jumped from the plane at 35,000 feet and the parachute opened at 500 feet during his free fall. How
long would Mr. Nicholas have been in free-fall had he not used this special suit?
3. Marta throws a baseball with an initial upward velocity of 60 feet per second. Ignoring Marta’s
height, how long after she releases the ball will it hit the ground? Use the formula h(t )  16t 2  v0t ,
where h(t) is the height of the object when it hits the ground, t is the time, and v0 is the velocity in feet
per second.
4. A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. How long
will it take the boulder to hit the ground if it lands at the same elevation from which it was ejected? Use
the same formula as number 2.
Algebra 2 Notes
Section 12-5 Solving by Factoring
Name___________________________
Goal:_________________________________________________________________________________
I. Solving Quadratic Equations
Steps:
1.
2.
3.
*The number of solutions _______________________________________________________________.
Examples:
2
1. x  7 x  12
4
2
2. 10 x  40 x  0
3
2
3. x  2 x  x  2  0
2
4. 3 x  48
2
5. 3 x  24 x  48  0
2
6. 2 x  7 x  15
2
7. x  6 x
II. Write the quadratic equation given the roots.
1. -5 , 6
3.
2 3
,
3 4
2.
1
,2
3
4.
2 1
,
5 2
Math 2 Notes
Section 12-6
Name___________
Goal:_________________________________________________________________________
I. Complete the Square
Steps:
1. Isolate the constant (move the constant to the other side).
2. Make sure the leading coefficient is 1 (if not divide through).
3. Complete the Square (Divide the middle number by 2 and square it).
4. Add that number to both sides of the equation.
5. Factor the left side and combine the right side (Short cut for factoring - Square root the first
term, square root last term, and take the first sign).
6. Square root both sides of the equation. (Remember to put a ± on the right.)
7. Solve the equation.
YOU SHOULD HAVE TWO ANSWERS FOR EACH PROBLEM.
Examples:
1. x 2  4 x  2  0
2. x 2  6 x  16  0
3. 4 x 2  2 x  5  0
5. 4 x 2  8 x  1  0
II. Write the trinomial as the square of a binomial.
1. x 2  2 x  1
2. x 2  10 x  25
3. x 2  12 x  36
4. 3x 2  2 x  8  0
III. Find the value of “C” that makes the trinomial a perfect square.
1. x 2  6 x  " C "
2. x 2  8 x  " C "
3. x 2  4 x  " C "
IV. Square Root Property
1
( x  3) 2  8
4
1. ( x  5)2  7
2.
3. x 2  14 x  49  64
4. 9 x 2  6 x  1  2
5. 4 x 2  12 x  9  4
Algebra 2 Notes
Section 6.5
Name___________________
Goal:__________________________________________________________________________
I. Quadratic Formula
Put the equation in standard form to find a, b, and c.
Examples:
1. 3x 2  5 x  6
2
2. x  6 x  9
2
3. 3x  6 x  4
2
4. 3x  5 x  2
II. Discriminant Formula
What is it used for?_____________________________________________________________________
Discriminant
0
# and Types of Solutions
Negative
Positive Perfect Square
Positive Non a Perfect Square
Find the value of the discriminant and then describe the roots.
2
1. x  10 x  50  0
2
3. 4 x  4 x  17  0
2
2. x  21  4 x
Math 2 Notes
Section Quadratic Word Problems
Name____________________
Goal:______________________________________________________________________
Formulas Needed to Solve:
1. Find two numbers whose sum is 20 and whose product is 96.
2. Helen is making an open top box by cutting a 2 inch square from each corner of a square
piece of cardboard and then folding up the remaining sides. What are the dimensions of the
box if the volume is 392 in2.
3. Find two consecutive odd integers such that twice the square of the second is 1 less than
three times the square of the first.
Consecutive #
Even/IOdd #
N
N
N+1
N+2
N+2
N+4
N+3
N+6
4. A grassy yard 25 feet by 30 feet is surrounded by a walk of uniform width. If the area of the
walk is 200 ft2, how wide is the walk?
5. Find the dimensions of the rectangle if the length is one more than three times the width
and the area is 154 square feet.
6. In a right triangle the hypotenuse is 10 m long, and one leg is 2 m longer than the other leg.
Find the area of the triangle.