6/8/2016Math KM1 Chapter 9: Quadratic Equations and Functions 9.1 The Basics of Solving Quadratic Equations 9.2 The Quadratic Formula 9.3 Applications 7.4 More on Quadratic Equations 9.5 Graphing f(x) = a(x - h) 2 + k 9.6 Graphing f(x) = ax 2 + bx= c 9.7 skip (woo hoo) 9.8 Polynomial and ( Rational Inequalities )

6/8/2016Math KM2 9.1

6/8/2016Math KM3 Square Root Property Solve: x 2 = 9 9.1

6/8/2016Math KM4 What about the graph of: f(x)= x

6/8/2016Math KM5 2x 2 = 14 Square Root Property x 2 = 7 9.1

6/8/2016Math KM6 25x = 0 Imagine this? 9.1

6/8/2016Math KM7 (x + 7) 2 = 4 Think about it? 9.1

6/8/2016Math KM8 (x - 9) 2 = -25 How about this one? 9.1

6/8/2016Math KM9 (x - 3) 2 = 5 One More? 9.1

6/8/2016Math KM10 Quadratic Equations 9.1

6/8/2016Math KM11 x x + ? Complete this Trinomial to make a Perfect Trinomial Square 9.1

6/8/2016Math KM12 A Perfect Trinomial Square has the form: Why Build a Perfect Trinomial Square? Which can be factored into a Binomial squared. Which is the perfect form if we want to use the Square Root Property for solving Quadratic Equations! 9.1

6/8/2016Math KM13 Perfect Trinomial Square +5x x2x xx2x x = (x + 5) 2 x x

6/8/2016Math KM14 Completing the Square 1) Divide to get the leading coefficient = 1 2) Isolate the quadratic and linear terms from the constant. 3) Compute the “complete the square” number: a) Start with the coefficient of x b) Multiply it by ½ c) Square the result 4) Add this number to both sides of equation. 5) Factor & use the Square Root Property 9.1

6/8/2016Math KM15 Solve by Completing the Square 9.1

6/8/2016Math KM16 Solve by Completing the Square: Just the steps! 9.1

6/8/2016Math KM17 Solve by Completing the Square: Just the Steps 9.1

6/8/2016Math KM18 Solve by Completing the Square: Just the Steps 9.1

6/8/2016Math KM19 Solve by Completing the Square: Just the Steps! 9.1

6/8/2016Math KM20 Organize First, then Complete the Square 9.1

6/8/2016Math KM21 9.2

6/8/2016Math KM22 Given a quadratic equation in Standard Form: ax 2 + bx + c = 0 a ≠ 0 The solutions can be found using the coefficients: a, b, and c, and the following formula: Introducing the QUADRATIC FORMULA 9.2

6/8/2016Math KM23 Solve using the Quadratic Formula: Example 1 9.2

6/8/2016Math KM24 Solve using the Quadratic Formula: Example 1 - continued 9.2

6/8/2016Math KM25 Solve using the Quadratic Formula: 9.2

6/8/2016Math KM26 Solve using the Quadratic Formula: Example 2 - continued

6/8/2016Math KM27 Solve using the Quadratic Formula: Example 3 9.2

6/8/2016Math KM28 Solve using the Quadratic Formula: Example 3 - continued 9.2

6/8/2016Math KM29 Solve using the Quadratic Formula: Example 3 - continued 9.2

6/8/2016Math KM30 Solve using the Quadratic Formula: Example 4 9.2

6/8/2016Math KM31 Solve using the Quadratic Formula: Example 4 - continued 9.2

6/8/2016Math KM32 Solve using the Quadratic Formula: Example 4 - continued 9.2

6/8/2016Math KM33 Solve using the Quadratic Formula: Example 5 9.2

6/8/2016Math KM34 Last one? 9.2

6/8/2016Math KM35 Last one? - continued 9.2

6/8/2016Math KM36 Last one? – The End 9.2

6/8/2016Math KM37 Where We Left Off Last Class

6/8/2016Math KM38 9.3

6/8/2016Math KM39 Maria and Ben helped build a community garden. The length of the garden is triple the width. The area of the garden is 507 square feet. What are the dimensions of the garden? 1.88 seconds w 3w Area = length x width 7.3

6/8/2016Math KM seconds h h + 9 Melissa and Roman are putting up a triangular “Coolaroo” shade sail near the community garden. The base of the sail is 9 feet more than the height. The area is 56 square feet. Find the base and height of the sail. 7.3

6/8/2016Math KM seconds h h

6/8/2016Math KM42 Literally Speaking – Solve this! If we know the variables are non- negative, we can write: 7.3

6/8/2016Math KM43 YIKES! Solve for r? 7.3

6/8/2016Math KM44 9.4

6/8/2016Math KM45 is the DISCRIMINANT is the QUADRATIC Formula ax 2 + bx + c = 0, a  0 9.4

6/8/2016Math KM46 Quadratic Formula: The Discriminant Since the Discriminant is under the square root, it will determine the nature of the solutions of the quadratic equation. 9.4

6/8/2016Math KM47 Positive Discriminant ax 2 + bx + c = 0 has two unequal Real solutions If the Discriminant is positive, 9.4

6/8/2016Math KM48 Zero Discriminant ax 2 + bx + c = 0 has one Real solution (a double root) If the Discriminant is zero, 9.4

6/8/2016Math KM49 Negative Discriminant ax 2 + bx + c = 0 has two Complex Conjugate solutions. If the Discriminant is negative, 9.4

6/8/2016Math KM50 Nature of the Solutions: D < 0 Compute the discriminant and describe the nature of the roots. 3x 2 – 4x + 5 = 0 3x 2 – 4x + 5 = 0 has two Complex Conjugate solutions a = 3 b = -4 c = 5 9.4

6/8/2016Math KM51 Compute the discriminant and describe the nature of the roots. 4x 2 – 12x + 9 = 0 a = 4 b = -12 c = 9 Nature of the Solutions: D = 0 4x 2 – 12x + 9 = 0 has one Real solution (a double root) 9.4

6/8/2016Math KM52 Compute the discriminant and describe the nature of the roots. 2x 2 - 3x - 1 = 0 a = 2 b = -3 c = -1 Nature of the Solutions: D > 0 2x 2 – 3x -1 = 0 has two unequal Real solutions. 9.4

6/8/2016Math KM53 Summary: Nature of the Solutions ax 2 + bx + c = 0 DiscriminantSIGNSolutions b 2 - 4ac > 0Positive Two unequal Real Solutions b 2 - 4ac = 0Zero One Real Double Root b 2 - 4ac < 0Negative Two Complex Solutions (Complex Conjugates) 9.4

6/8/2016Math KM54 Working Backwards! 9.4

6/8/2016Math KM55 If the solution set for a quadratic equation is What was the equation? {-2, 7} or 9.4

6/8/2016Math KM56 If the solution set for a quadratic equation is What was the equation? {-2,2} 9.4

6/8/2016Math KM57 If the solution set for a quadratic equation is What was the equation? { 6 } 9.4

6/8/2016Math KM58 If the solution set for a quadratic equation is What was the equation? { 7/2, -1/4 } 9.4

6/8/2016Math KM59 If the solution set for a quadratic equation is What was the equation? { 3i, -3i } 9.4

6/8/2016Math KM60 More Advanced Equations 9.4

6/8/2016Math KM61 a( ? ) 2 + b( ?) + c = 0 au 2 + bu + c = 0 Substitute: Let u = ? Solve for u Un-Substitute Solve for ? Quadratic in Form 9.4

6/8/2016Math KM62 Let u = x 2 9.4

6/8/2016Math KM63...or just solve it! 9.4

6/8/2016Math KM64 Look at the FORM! 9.4

6/8/2016Math KM65... Let’s Lose the Negativity! 9.4

6/8/2016Math KM66 Don’t be Afraid! 9.4

6/8/2016Math KM67 9.5

6/8/2016Math KM68 9.5

6/8/2016Math KM69 9.5

6/8/2016Math KM70 Turn around point highest... or lowest point Minimum Maximum 9.5

6/8/2016Math KM71 mirror line 9.5

6/8/2016Math KM72 Maximum or Minimum? x = 2 Vertex: (2, -3) Opens: Up or Down ? Wide or Narrow ? Axis of symmetry: The minimum value of f(x) is

6/8/2016Math KM73 Vertex x = 2 Axis of symmetry Minimum (2, -3) 7.5

6/8/2016Math KM74 Maximum or Minimum? x = -1 Vertex: (-1, 4) Opens: Up or Down ? Wide or Narrow ? Axis of symmetry: The maximum value of f(x) is

6/8/2016Math KM75 Vertex x = -1 Axis of symmetry Maximum (-1, 4) 9.5

6/8/2016Math KM76 y=1x 2 y=2x 2 y=3x 2 y=(1/2)x 2 y=(1/3)x 2 f(x)= ax 2 9.5

6/8/2016Math KM77 {-4,2} {5/3} { } {3} y=1x 2 y=2x 2 y=3x 2 y=(1/2)x 2 y=(1/3)x 2 (1, 3) (1, 2) (1, 1) (1, 1/2) (1, 1/3) f(x)= ax2 9.5

6/8/2016Math KM f(x) = ax 2 + bx + c f(x) = a(x-h) 2 + k 9.6

6/8/2016Math KM79 1.Which way does it open? i) a > 0 UP (minimum y value) ii) a < 0 DOWN (maximum y value) 2.Locate the VERTEX (h, k) i) f(x) = a(x – h) 2 + k ii) 3.Axis of Symmetry: x = h 4.Locate the y-intercept (0, f(0)) 5.Locate the x-intercept(s) (if there are any) i) Set f(x) = 0 and solve. Parabola Checklist f(x) = ax 2 + bx + c 9.6

6/8/2016Math KM80 {-4,2} {5/3} { } {3} Information Format 9.6

6/8/2016Math KM81 y = 2x 2 + 8x + 5 Opens Upward There is a minimum value Narrow Vertex ( -2, -3) Axis of Symmetry: x = -2 Minimum value of y is -3. EZ intercept: (0, 5) Harder intercepts: 9.6

6/8/2016Math KM82 y = 2x 2 + 8x

6/8/2016Math KM83 y = -6x x

6/8/2016Math KM84 y = -6x x - 2 Opens Downward There is a maximum value Very Narrow Vertex ( 1, 4) Axis of Symmetry: x = 1 Maximum value of y is 4 EZ intercept: (0,-2) Harder intercepts: 9.6

6/8/2016Math KM85 y = -6x x

6/8/2016Math KM Polynomial Inequalities 9.8

6/8/2016Math KM87 Quadratic Inequalities 1 9.8

6/8/2016Math KM88 Quadratic Inequalities 2 9.8

6/8/2016Math KM89 Quadratic Inequalities 3 9.8

6/8/2016Math KM90 Quadratic Inequalities 4 9.8

6/8/2016Math KM91 That’s All for Now!