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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.