1. Chapter 6: Quadratic Functions Vogler Algebra II Vogler Algebra II

2. Simplifying Quadratics: Factoring  Simplifying quadratics makes for easier graphing  (x+y) 2 =x 2 +2xy+y 2  (x-y) 2 =x 2 -2xy+y 2  (2x+3) 2 =4x 2 +12x+9  You can apply this property to any expanded quadratic with perfect squares on x and y  Simplifying quadratics makes for easier graphing  (x+y) 2 =x 2 +2xy+y 2  (x-y) 2 =x 2 -2xy+y 2  (2x+3) 2 =4x 2 +12x+9  You can apply this property to any expanded quadratic with perfect squares on x and y

3. Simplifying Quadratics: Factoring  FOIL  First Outer Inner Last  (2x+4)(x-3)  2xx (First)  2x-3 (Outer)  4x (Inner)  4-3 (Last)  2x 2 -6x+4x-12  Simplify: 2x 2 -2x-12  FOIL  First Outer Inner Last  (2x+4)(x-3)  2xx (First)  2x-3 (Outer)  4x (Inner)  4-3 (Last)  2x 2 -6x+4x-12  Simplify: 2x 2 -2x-12

4. Simplifying Quadratics: Factoring  Not all quadratics are easily factorable:  3x 2 +5x+2  Set up a grid  Find a x c (3 x 2)  Find all the factors of 6 that sum to 5  Fill in the grid  Find the GCF of each column and row  Write the expression  (3x+2)(x+1)  Not all quadratics are easily factorable:  3x 2 +5x+2  Set up a grid  Find a x c (3 x 2)  Find all the factors of 6 that sum to 5  Fill in the grid  Find the GCF of each column and row  Write the expression  (3x+2)(x+1) 3x 2 2 3x 2 X1X1 3x 2x

5. Graphing: Translations  In ax 2 +bx+c, c refers to the y- intercept  x 2 +4 is 4 units up from x 2  y=x 2 +k moves a graph up k units  y=x 2 -k also moves a graph down k units  In ax 2 +bx+c, c refers to the y- intercept  x 2 +4 is 4 units up from x 2  y=x 2 +k moves a graph up k units  y=x 2 -k also moves a graph down k units

6. Graphing: Translations  In y=(x-h) 2  h is the distance moved left or right  -h moves right  +h moves left  (x+3) 2 is 3 units left of x 2  x=h is the line of symmetry

7. Graphing: Translations  In ax 2 +bx+c:  +a opens up  -a opens down  To find the line of symmetry:  Find -b/2a  In ax 2 +bx+c:  +a opens up  -a opens down  To find the line of symmetry:  Find -b/2a

8. Solving Quadratics: Factoring  To solve by factoring, make ax 2 +bx+c equal to 0:  0= 3x 2 +5x+2  Then factor:  0=(3x+2)(x+1)  Make each binomial equal to zero and solve:  0=3x+20=x+1  -2=3x -1=x (second solution)  -2/3=x (first solution)  The two solutions are: -2/3 and -1  Solutions to quadratics refer to the x-intercepts  In other words, if the equation is not equal to 0, then we have to make it equal to 0 to solve it  To solve by factoring, make ax 2 +bx+c equal to 0:  0= 3x 2 +5x+2  Then factor:  0=(3x+2)(x+1)  Make each binomial equal to zero and solve:  0=3x+20=x+1  -2=3x -1=x (second solution)  -2/3=x (first solution)  The two solutions are: -2/3 and -1  Solutions to quadratics refer to the x-intercepts  In other words, if the equation is not equal to 0, then we have to make it equal to 0 to solve it

9. Solving Quadratics: Completing the Square  X 2 +10=39  Draw a square with area x 2  Add a rectangle of length 10  Split the rectangle  Find the area  Add the area to 39  Find the square root of the answer (64)  Solve x+5=8 and x+5=-8  X=3 and -13 x x 10 =39 x x 5 5 25 =39+25 =64 + +

10. Graphing: Quadratic modeling  Coordinates:  (0,5), (1, 10), (2, 19)  Use a basic formula: 5=a(0) 2 +b(0)+c 10=a(1) 2 +b(1)+c 19=a(2) 2 +b(2)+c  Coordinates:  (0,5), (1, 10), (2, 19)  Use a basic formula: 5=a(0) 2 +b(0)+c 10=a(1) 2 +b(1)+c 19=a(2) 2 +b(2)+c

11. Graphing: Quadratic modeling  Set up and solve a system for your three equations: 5=a(0) 2 +b(0)+c 10=a(1) 2 +b(1)+c 19=a(2) 2 +b(2)+c  a=2; b=3; c=5  So y=2x 2 +3x+5  Set up and solve a system for your three equations: 5=a(0) 2 +b(0)+c 10=a(1) 2 +b(1)+c 19=a(2) 2 +b(2)+c  a=2; b=3; c=5  So y=2x 2 +3x+5

12. Graphing: Quadratic modeling  A vehicle’s braking distance is found for the following three coordinates:  Find the equation for this vehicle’s braking distance  A vehicle’s braking distance is found for the following three coordinates:  Find the equation for this vehicle’s braking distance Speed (MPH) Distance (ft.) 00 1019 40116

13. Graphing: Quadratic modeling  A ball ’ s trajectory can be found using the equation:  h(t) = -4.9t2 + v o t + h o  The ball has an initial velocity of 14 m/sec and was thrown from a height of 30 meters.  A ball ’ s trajectory can be found using the equation:  h(t) = -4.9t2 + v o t + h o  The ball has an initial velocity of 14 m/sec and was thrown from a height of 30 meters.

14. Solving Quadratics: Quadratic Equation  The quadratic formula only works when y=0 in y=ax 2 +bx+c  So, in 10x 2 -13x-3=0,  The quadratic formula only works when y=0 in y=ax 2 +bx+c  So, in 10x 2 -13x-3=0,

15. Solving Quadratics: Quadratic Equation  Simplify: do what is under the radical sign first:  Both add and subtract from 13  Simplify:  X=3/2 and -1/5  Simplify: do what is under the radical sign first:  Both add and subtract from 13  Simplify:  X=3/2 and -1/5

16. Solving Quadratics: Quadratic Equation  A cat is dropped from a height of 40 feet. Use the formula h=-16t 2 +44t+40. 1.When does it hit the ground? 2.Does it land on it’s feet?  A cat is dropped from a height of 40 feet. Use the formula h=-16t 2 +44t+40. 1.When does it hit the ground? 2.Does it land on it’s feet?

17. Solving Quadratics: Quadratic Equation  Sometimes we want to solve for equations that do not equal 0.  So, make them equal to 0: A baseball is hit from home plate into the outfield. If the ball is hit at a height of 5 ft. and an initial velocity of 147 ft/sec, when will it reach a height of 10 ft?  Write an equation: 10=-16t 2 +147t+5  Get the equation in terms of 0:  0=-16t 2 +147t-5  Solve for t.  T=.03 and 9.15  Which value makes more sense? Why?  Sometimes we want to solve for equations that do not equal 0.  So, make them equal to 0: A baseball is hit from home plate into the outfield. If the ball is hit at a height of 5 ft. and an initial velocity of 147 ft/sec, when will it reach a height of 10 ft?  Write an equation: 10=-16t 2 +147t+5  Get the equation in terms of 0:  0=-16t 2 +147t-5  Solve for t.  T=.03 and 9.15  Which value makes more sense? Why?

18. Discriminant  All quadratics have two solutions, but not all solutions are real:  Discriminant:  b 2 -4ac>0, two real solutions  b 2 -4ac=0, one real solution  b 2 -4ac<0, two complex (imaginary) solutions  All quadratics have two solutions, but not all solutions are real:  Discriminant:  b 2 -4ac>0, two real solutions  b 2 -4ac=0, one real solution  b 2 -4ac<0, two complex (imaginary) solutions

19. Imaginary Numbers  All numbers have square roots, even negative numbers:  √4=2, -2  √-4=2i  Imaginary number: i…for imaginary  i= √-1, so i 2 =-1 X 2 =-100 √x 2= √-100 X= √100 √-1 X=10i and -10i  All numbers have square roots, even negative numbers:  √4=2, -2  √-4=2i  Imaginary number: i…for imaginary  i= √-1, so i 2 =-1 X 2 =-100 √x 2= √-100 X= √100 √-1 X=10i and -10i (√-25)(2i) √25 √-1(2 √-1) 5(2) √-1 2 10(-1)=-10

20. Complex Numbers  Complex numbers: a+bi  4+2i  The conjugate (opposite) is 4-2i  (4+2i)+(3+i)  Only combine like terms: 2i and i are like terms  7+3i  2i(4-7i)  8i-14i 2 =8i-14(-1)=8i+14  Complex numbers: a+bi  4+2i  The conjugate (opposite) is 4-2i  (4+2i)+(3+i)  Only combine like terms: 2i and i are like terms  7+3i  2i(4-7i)  8i-14i 2 =8i-14(-1)=8i+14

21. Complex Numbers  To convert to a+bi form:  Multiply by 1 (the conjugate of the bottom number)  Simplify