Memorize the following!

1. Memorize the following! 2 2 Three forms of a quadratic equation. Standard: y = ax + bx + c Vertex: y = a(x – h) + k Intercept: y = a(x – p)(x – q) X coordinate of vertex: -b 2a

2. VERTEX FORM THE PARENT FUNCTION of the quadratic family Characteristics: shape is parabolic and symmetric axis of symmetry is vertical has a maximum or a minimum value + k If |a | > 1, parabola narrows If |a | < 1, parabola widens If a < 0, parabola opens down Vertex is (h, k)

3. Graphing from vertex form Plot the vertex ( h, k ) Determine if the parabola opens up or down Draw the axis of symmetry as a dashed vertical line Choose an x value on one side of the vertex, Substitute this value in, solve for y and plot the point. Plot the reflection image of the point. Repeat steps 5 and 6 for an additional point Sketch the parabola.

4. STANDARD FORM + bx + c Effect of a is the same. If a is positive the parabola opens up, if negative, down. Axis of symmetry is the vertical line x = -b 2a Vertex is (,f(x)) The y-intercept is (0, c). Plot this point, then reflect it over the line of symmetry. - or- To complete graph, choose an x value near the vertex, calculate its corresponding y value, plot the point, then reflect it over the line of symmetry.

5. Graphing from standard form • Plot the vertex, the x coordinate is found using Find the y coordinate by plugging the x value into the original function and solving for y. -Determine if the parabola opens up or down -Sketch the axis of symmetry using a dashed line. -Choose a value for x, plug it into the equation and solve for y. Plot this point -Plot the reflection image of this point -Choose another x value and repeat the last 2 steps -Sketch the parabola

6. INTERCEPT FORM Effect of a is the same. The x-intercepts are p and q. The axis of symmetry goes through the x-axis at the midpoint of the segment defined by p and q. The vertex is (x, f(x)), where

7. Graphing from intercept form Plot the x intercepts p and q Draw the axis of symmetry halfway between p and q The x coordinate of the vertex is the x value of the axis of symmetry, plug this into the equation and solve for y. Plot the vertex. Sketch the graph.

8. The path of a baseball after it is hit is modeled by the function h is the height of the baseball in feet and d is the distance in feet the baseball is from home plate. What is the maximum height reached by the baseball? How far is the baseball from home plate when it reaches it’s maximum height?

9. QUADRATIC FUNCTIONS What is the distance between the two towers if the cable supports are 500 feet above the road? The Golden Gate Bridge in San Francisco has two towers that rise 500 feet above the road and are connected by suspension cable as shown. Each cable forms a parabola with the equation + 8 What is the height above the road of a cable at its lowest point?

10. Although a football field appears to be flat, its surface is actually shaped like a parabola so that rain runs off to either side. The cross section of a synthetic field can be modeled by the equation + 1.5 Where x and y are measured in feet. What is the field’s width? What is the maximum height of the field’s surface?

11. A kernel of popcorn contains water that expands when the kernel is heated, causing it to pop. The equations below give the “popping volume”, y (in cubic cm/gram) of popcorn with the moisture content as x (as percent of the popcorn’s weight). Hot-air popping: – 94.8 Hot-oil popping: For hot-air popping, what moisture content maximizes popping volume? What is the maximum volume? For hot-oil popping, what moisture content maximizes popping volume? What is the maximum volume? 5. The moisture content of popcorn typically ranges from 8% to 18%. Examine the graph of both equations for the interval 6. Based on a comparison of both graphs, what general statement can you make about the volume of popcorn produced from hot-air popping versus hot-oil popping for any moisture content in the interval

12. Writing Quadratic Functions in Standard Form Multiply using FOIL combine like terms distribute the -1

13. You have 400 feet of fencing with which you need to enclose a rectangular field along a straight river bank. (There is no need to fence along the river.) Let x represent the width of the field and write an equation to represent the area of the field. Find the dimensions of the field with the maximum area.

14. Warm-up Anne and Linda went out to lunch. When the bill came Anne realized she had only 8 dollars in her wallet. Linda told her not to worry because she had enough money to cover three fifths of the bill. With Anne’s eight dollars and Linda’s money they could pay. How much was the lunch bill? How much did Linda contribute? Factor the following expressions. x2 – x - 6 x2 -3x

15. Factoring • Factor out the GCF • Look for special patterns such as the difference of two squares, perfect square trinomial • FOIL to check your answers!

16. Undo FOIL!

17. Factor out any GCF first!

18. Difference of Squares

19. Perfect Square Trinomials

20. Solving quadratic equations by factoring - Set the equation equal to zero .(You’re making y zero because you’re finding the x intercepts.) -Factor the equation completely. -Set each factored portion equal to zero and solve for the variable.

21. Zero Product Property If A and B are real numbers or algebraic expressions, If AB = 0 then A = 0 or B = 0 In Quadratic equations the x intercepts are also called -solutions -roots -zeros of the function When solving, instructions might say: -Solve the equation. -Find the function’s zeros. -Write the quadratic function in intercept form and find the function’s zeros.

22. You are making a coffee table with a glass top surrounded by a cherry border. The glass is 3 ft. by 3 ft.. You want the cherry borer to be a uniform width. You have 7 square feet of cherry. What should the width of the border be?

23. The dimensions of a rectangular garden were 5 meters by 12 meters. Each dimension was increased by the same amount. The garden then had an area of 120 square meters. Find the dimensions of the new garden.

24. A graphic artist is designing a poster that consists of a rectangular print with a uniform border. The print is to be twice as tall as it is wide. The border is to be 3 inches wide. If the area of the poster is to be 680 square inches find the dimensions of the print.

25. You have made a rectangular stained glass window that is 2 feet by 4 feet. You have 7 square feet of clear glass to create a border of uniform width around the window. What should the width of the border be?

26. Warm-up Solve the following equations by factoring. 16x2 = 8x -1 y = 25x2 -4 15 = 3x2 -12x 5x2 = 30x

27. warm-up Factor the following expressions. X2 + 2x + 1 X2 – 4x + 4 4x2 +12x + 9 Solve the following equations. X2 -10x + 25 = 0 x2 – 16x = -64

28. If x2 = 16 what is the value of x? • Product Property Quotient Property • A square-root expression is simplified if: • No radicand has a perfect square factor other than 1 • There is no radicand in the denominator • Simplify • √12 √45 √6 . 2√6 √98 • Rationalizing the denominator-the process of eliminating a radical from the denominator.

29. Solving quadratic equations by taking square roots. • Rewrite the equation so the squared term is isolated on one side of the equal sign. • Take the square root of each side. • Remember there will be a positive and a negative root. • Ex.

30. You can use square roots to solve some quadratic equations. -you can use this method when there is no linear term Ex. 2x2 + 1 = 17 1/3(x + 5)2 = 7 Remember + and - !!!!!!

31. The height h(in feet) of the object t seconds after it is dropped can be modeled by the function: h = -16t2 + ho (hois the object’s initial height) A stunt man working on the set of a movie is to fall out of a window 100 feet above the ground. For the stunt man’s safety, an air cushion 26 feet wide by 30 feet long by 9 feet high is positioned on the ground below the window. For how many seconds will the stunt man fall before he reaches the cushion? A movie camera operating at a speed of 24 frames per second records the stunt man’s fall. How many frames of film show the stunt man falling?

32. From 1990 to 1993 the number of truck registrations (in millions) in the United States can be approximated by the model R = .29t2 + 45 where t is the number of years since 1990. During which year were approximately 46.16 million trucks registered? In 1992 the average income I(in dollars) for a doctor aged x years could be modeled by: I = -425x2 + 42,500x – 761,000 For what ages did the average income for a doctor exceed $250,000?

33. The aspect ratio of a TV screen is the ratio of the screen’s width to its height. For most TVs the aspect ratio is 4:3. What are the width and height of the screen for a 27 inch TV? Use the Pythagorean Theorem to solve.

34. SOLVE USING = 21 = = =

35. QUADRATICSTHERE’S AN APP FOR THAT! When an object is dropped, its speed continually increases, and therefore its height above the ground decreases at a faster and faster rate. The height, h (in feet) of the object t seconds after it is dropped can be modeled by the function + Where is the object’s initial height. The tallest building in the United States is in Chicago, Illinois. It is 1450 tall. a) How long would it take a penny to drop from the top of this building? b) How fast would the penny be travelling when it hits the ground if the speed is given by s = 32t where t is the number of seconds since the penny was dropped?

36. WHAT IF THE REAL NUMBER SYSTEM ISN’T SUFFICIENT? - WE INVENT SOME MORE NUMBERS! IMAGINARY NUMBERS i =

37. i is the imaginary unit it is equal to the square root of -1 i = = 6i =i A complex number in standard form - a + bi Ex. 2 + 3i A pure imaginary number 2i

38. Adding and Subtracting Complex Numbers • The imaginary number i acts like a variable but when you get substitute -1

39. Multiplying Complex Numbers Substitute -1 for i squared

40. Complex conjugates example (5 + 3i) and (5 – 3i) The product of complex conjugates is always a real number. Dividing Complex Numbers

41. QUIZ 5.4 1. Simplify: • Perform the indicated operation: • 2. (25 + 15i) + (25 – 6i) • 3. (25 + 15i) - (25 – 5i) • 4. (5 + i)(8 + i) • 5. • 6. Write the complex conjugate of 6 – 3i. • 7. Solve: = 120

42. YET ANOTHER WAY TO SOLVEQUADRATIC EQUATIONSCompleting the square -Use inverse operations to move the constant term to the other side of the equal sign -Divide the coefficient of the linear term in half and square it, add it to both sides of the equal sign -Factor the now perfect square trinomial -Solve by taking the square root of both sides Complete the expression to make it a square: + x + ___ - 22x + ___ Rewrite each expression as the square of a binomial

43. COMPLETE THE SQUARETO SOLVE THE EQUATION + 6x – 8 = 0 + 6 x = 8 Move -8 to other side + 6x + 9 = 8 + 9 Add half of the middle term squared to both sides of the quadratic equation + 6x + 9 = 17 Simplify Factor to rewrite trinomial square as square of binomial = 17 both sides x + 3 = Isolate x

44. COMPLETING THE SQUARE (cont’d) The leading coefficient must be one. Divide by the coefficient of if necessary. Example: =

45. WRITING A QUADRATIC EQUATIONIN VERTEX FORM+ k Example: Make room for c Complete the square Rewrite trinomial + 16 Isolate y + 16 -9 Simplify + 7 To name the vertex: + 7 Write in vertex form Vertex is (-3, 7)

46. On dry asphalt the distance (d) in feet needed for a car to stop is given by d = .05s2 + 1.1s Where s is the car’s speed in miles per hour. What speed limit should be posted on a road where drivers round a corner and have 80 feet to stop?

47. You want to plant a rectangular garden along part of a 40 foot side of your house. To keep out animals you will enclose the garden with wire mesh along its three open sides. You will also cover the garden with mulch. If you have 50 feet of mesh and enough mulch to cover 100 square feet, what should the garden’s dimensions be? Solve by writing a quadratic equation and completing the square. x 50-2x garden x 40 ft. house

48. 1. Start with the standard form of quadratic equation: -7x² + 2x + 9 = 0 THE QUADRATIC FORMULA 2. Determine values of a, b and c. a= -7 b = 2 c= 9 3. Replace a, b and c in quadratic formula with values from equation. The “catch-all” method for solving quadratic equations 4. Solve for x. x = -1, x = 9/7

49. EXAMPLESSOLVE USING THE QUADRATIC FORMULA 1. 3x² - 5x = 2 X = 2 or 2. x²+ 2x + 1 = 0 X = -1 3. x² − 6x+ 10 = 0 X = 3

50. THE DISCRIMINANT(“the qualifier” of solutions) b²−4ac STANDARD EQUATION DISCRIMINANT +/0/− #/TYPE OF SOLUTIONS SOLUTION(S) + -7x² + 2x + 9 = 0 2²−4(-7)(9) =256 2/Real -1 and 9/7 x²+ 2x + 1 = 0 2²− 4(1)(1) =0 0 1/Real -1 x² − 6x+ 10 = 0 (-6)²−4(1)(10) =-4 − 2/Imaginary 3 ± ί If the discriminant is positive there are 2 real solutions. If the discriminant is negative there are 2 imaginary solutions. If the discriminant is zero there is one real solution.