1. Special Study of Linear & Quadratic Polynomials Chapter 6

2. What do you already know? Linear Quadratic

3. What do you already know? LinearQuadratic L(x) = ax + b what happens if… b>0 b<0 b=0 a>0 a<0 -1<a<1 (a  0) a = 0 a is undefined Q(x) = ax 2 + bx + c what happens if… c>0 c<0 c=0 a>0 a<0 -1<a<1 (a  0) a = 0 a is undefined

4. Slopes NegativePositive Horizontal Vertical

5. Equation Forms Slope Intercept Standard Horizontal Vertical y = mx + b Ax + By = C y = b x = a This brings us back to the concept: a linear equation has a degree of 1

6. Given any linear equation, one should be able to y = -3x – 7 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope 1.Slope intercept 2.Falling 3.-3 4.-7 5.- -7/(-3) = -7/3 6.-3 7.-7 identify…

7. So what is new regarding Linear polynomials? L(x) = 4x + 7 if you need to “find the zeroes”, x –intercept, solve for 0, etc… Change the format L(x) = 4(x + 7/4) x would have to equal -7/4 for L(x) = 0

8. What do you already know? LinearQuadratic L(x) = ax + b what happens if… b>0 b<0 b=0 a>0 a<0 -1<a<1 (a  0) a = 0 a is undefined Q(x) = ax 2 + bx + c what happens if… c>0 c<0 c=0 a>0 a<0 -1<a<1 (a  0) a = 0 a is undefined

9. this brings us to our quadratics… Q(x) = ax 2 + bx + c Factored form Q(x) = a(x - p)(x - q) where p<q Now what happens? Note: Page 99 – Interval notation

10. Q(x) = a(x - p)(x - q) if a > 0 (-¥, p ) x = p (p,q) x = q (q, ¥ ) Note: Page 99 – Interval notation

11. Q(x) = a(x - p)(x - q) if a > 0 (-¥, p ) x = p (p,q) x = q (q, ¥ ) Q(x) = 0 Note: Page 99 – Interval notation

12. Q(x) = a(x - p)(x - q) if a > 0 (-¥, p ) x = p (p,q) x = q (q, ¥ ) Q(x) > 0 pos Q(x) = 0 Q(x) < 0 neg Q(x) = 0 Q(x) > 0 pos Note: Page 99 – Interval notation

13. (-¥, p ) x = p (p,q) x = q (q, ¥ ) Q(x) > 0 pos Q(x) = 0 Q(x) < 0 neg Q(x) = 0 Q(x) > 0 pos Note: Page 99 – Interval notation Zero at pZero at q Positive Negative

14. Quadratic Equations Make some connections with prior knowledge

15. y=(1/3)x 2 +2x-5

16. What makes an equation Quadratic? The degree is 2. y = x 2 y = ax 2 +bx+c y = a(x-h) 2 + k y = a(x-p)(x-q)

17. What makes an equation Quadratic? y = x 2 y = ax 2 +bx+c y = a(x-h) 2 + k y = a(x-p)(x-q) the basic Standard Form Vertex Form Intercept Form Given any one of the equations, you should be able to convert it into another form. (except for maybe the basic)

18. The basic y = x 2 vertex = (0,0) line of symmetry x = 0 x2x2 y 00 1 11 -24 24 -39 39

19. Standard Form y = ax 2 +bx+c c y-intercept -b/2a x-coordinate of the vertex use substitution to find the y-coordinate x = -b/2a is the line of symmetry. Now you can graph it

20. y = 8x 2 +10x+2 The y- intercept is 2

21. y = 8x 2 +10x+2 The y- intercept is 2 -10/(2*8) = -5/8

22. y = 8x 2 +10x+2 The y-intercept is 2 -10/(2*8) = -5/8 y =8(-5/8)2+10(-5/8)+2 =-1.125 or -9/8 Vertex (-5/8, -9/8)

23. y = 8x 2 +10x+2 The y-intercept is 2 -10/(2*8) = -5/8 y =8(-5/8)2+10(-5/8)+2 =-1.125 or -9/8 Vertex (-5/8, -9/8) so with the line of symmetry we can find the third point (-5/8+-5/8, 2) = (-10/8,2)

24. y = 8x 2 +10x+2 Connect the plotted points

25. y = (8x+2)(x+1) y = 2(4x+1)(x+1) This is the intercept form. Equate the quantities to 0 to find the intercepts 4x+1 = 0 or x+1 = 0 x = -1/4 or x = -1 as shown in the graph. If you factor this quadratic y = 8x 2 +10x+2

26. Intercept Form y = a(x-p)(x-q) p & q x-intercepts a opens up/down wide/skinny can expand to find the vertex and line of symmetry

27. Using either a graphing calculator or an online calculator EXPLORE!! What does “a” denote? What does “c” denote? How do these coefficients effect the graph?

28. Finding the roots (x- intercepts) the quadratic formula will work every time This should be familiar – now match it to the text.

29. re-writing/manipulating the eqn

30. Break this into two parts and rename

31. Break this into two parts u and M

33. to see proof go to section 1.5

34. So our quadratic equation becomes… Q(x) = a ( u 2 – M)

35. Think about what happens if M < 0 No real roots –if a > 0 Minimum –if a < 0 Maximum

36. Think about what happens if M = 0 or a double root same idea regarding max/min –if a > 0 Minimum –if a < 0 Maximum

37. Think about what happens if M > 0 or 2 real roots

38. 6.4 Extremum Values: Maximums and Minimums Find your vertex… -b/2a How do you know if it will be a maximum or a minimum? Identify the intervals…Q(x) negative…Q(x) positive