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Special Study of Linear & Quadratic Polynomials

Special Study of Linear & Quadratic Polynomials. Chapter 6. Linear. Quadratic. What do you already know?. L(x) = ax + b what happens if… b>0 b<0 b=0 a>0 a<0 -1<a<1 (a ¹ 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 ¹ 0) a = 0

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Special Study of Linear & Quadratic Polynomials

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  1. Special Study of Linear & Quadratic Polynomials Chapter 6

  2. Linear Quadratic What do you already know?

  3. L(x) = ax + b what happens if… b>0 b<0 b=0 a>0 a<0 -1<a<1 (a¹0) a = 0 a is undefined Q(x) = ax2 + bx + c what happens if… c>0 c<0 c=0 a>0 a<0 -1<a<1 (a¹0) a = 0 a is undefined What do you already know?Linear Quadratic

  4. Slopes Positive Negative Horizontal Vertical

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

  6. The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Slope intercept Falling -3 -7 - -7/(-3) = -7/3 -3 -7 Given any linear equation, one should be able to y = -3x – 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. L(x) = ax + b what happens if… b>0 b<0 b=0 a>0 a<0 -1<a<1 (a¹0) a = 0 a is undefined Q(x) = ax2 + bx + c what happens if… c>0 c<0 c=0 a>0 a<0 -1<a<1 (a¹0) a = 0 a is undefined What do you already know?Linear Quadratic

  9. this brings us to our quadratics… • Q(x) = ax2 + bx + c • Factored form • Q(x) = a(x - p)(x - q) where p<q • Now what happens? Note: Page 99 – Interval notation

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

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

  12. (-¥, 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 Q(x) = a(x - p)(x - q) if a > 0 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 Positive Negative Positive Zero at p Zero at q Note: Page 99 – Interval notation

  14. Quadratic Equations Make some connections with prior knowledge

  15. y=(1/3)x2+2x-5

  16. What makes an equation Quadratic? • The degree is 2. • y = x2 • y = ax2+bx+c • y = a(x-h)2 + k • y = a(x-p)(x-q)

  17. y = x2 y = ax2+bx+c y = a(x-h)2 + k y = a(x-p)(x-q) the basic Standard Form Vertex Form Intercept Form What makes an equation Quadratic? 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 = x2 • vertex = (0,0) • line of symmetry x = 0

  19. Standard Form y = ax2+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 = 8x2+10x+2 • The y-intercept is 2

  21. y = 8x2+10x+2 • The y-intercept is 2 • -10/(2*8) = -5/8

  22. y = 8x2+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 = 8x2+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 = 8x2+10x+2 • Connect the plotted points

  25. If you factor this quadratic y = 8x2+10x+2 • 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.

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

  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

  32. u and M

  33. u and M to see proof go to section 1.5

  34. So our quadratic equation becomes… • Q(x) = a ( u2 – 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

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