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Quadratic Equations and Functions

Dive into the world of quadratic equations and functions with this comprehensive guide. Learn to identify the quadratic, linear, and constant terms in the standard form ( ax^2 + bx + c ). Discover methods to build quadratic functions by multiplying linear functions and find the axis of symmetry and vertex of parabolas. We also explore how to derive quadratic models from sets of coordinates. Through examples and practice problems, you'll master graphing quadratic functions, understanding parabolas, and solving equations with ease.

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Quadratic Equations and Functions

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  1. Quadratic Equations and Functions The ones with the little two above and to the right of the x

  2. Standard Form ax2 + bx + c Constant term Quadratic term Linear term

  3. Put it in Standard Form • (2x + 3) (x – 4) • x2 + 5x -2x2 + 7 – 3x • (x – 5) ( 3x -1) • x(2x + 4) Now Identify the Quadratic, Linear and Constant Terms

  4. Building Quadratic Functions • You can build a quadratic function by multiplying 2 linear functions g(x) = 3x + 4 f(x) = 2x + 9 f(x) g(x) =(2x+9)(3x+4) f(x) g(x) = (6x2 +35x +36)

  5. Parent Quadratic f(x) = x2 Parabola Axis of Symmetry Vertex

  6. Find the Axis of Symmetry x = -1

  7. Find the Vertex The Vertex always lies on the axis of symmetry x = -1 (-1,-5)

  8. You can find a quadratic model using 3 points Standard form is ax2 + bx + c = y If I know 3 sets of coordinates (x ,y) Then I can substitute to get 3 equations with 3 (a,b,c) unknowns And solve for (a,b,c)

  9. Example a( x)2 + b(x) + c = y a( 2)2 + b(2) + c = 3 a( 3)2 + b(3) + c = 13 a( 4)2 + b(4) + c = 29

  10. Example a(4) + b(2) + c = 3 a(9) + b(3) + c = 13 a(16)+ b(4) + c = 29

  11. Example Let’s get rid of c ! a(4) + b(2) + c = 3 a(9) + b(3) + c = 13 a(16)+ b(4) + c = 29

  12. Let’s move the coefficients out front 4a + 2b + c = 3 a(4) + b(2) + c = 3 9a + 3b+ c = 13 a(9) + b(3) + c = 13 16a + 4b + c = 29 a(16)+ b(4) + c = 29

  13. Pair em’ up -1( ) 4a + 2b + c = 3 -1( ) ( ) 4a + 2b + c = 3 9a + 3b+ c = 13 16a + 4b + c = 29

  14. Add em -4a + -2b + -c = -3 -4a + -2b + -c = -3 9a + 3b + c = 13 16a + 4b + c = 29 12a + 2b = 26 5a + b = 10

  15. Lets get rid of b 12a + 2b = 26 -2( ) 5a + b = 10

  16. Solve for a 12a + 2b = 26 -10a +-2b = -20 2a = 6 a = 3

  17. Solve for b a = 3 -10(3) +(-2b) = -20 -30 + (-2b) = -20 (-2b) = 10 b = -5

  18. Solve for c b = -5 a = 3 4a + 2b + c = 3 4(3) + 2(-5) + c = 3 12 + -10 + c = 3 2 + c = 3 c = 1

  19. Write the quadratic model c = 1 b = -5 a = 3 3( x)2 + -5(x) + 1 = y

  20. Do Now! Page 237 Problems 1 – 15 Problems 16,20,24,26

  21. Parabolas Pair-a-bowl-ahs

  22. Parent Quadratic f(x) = x2 Parabola Axis of Symmetry Vertex

  23. y = x2and y = ½ x2

  24. y = x2and y = - x2

  25. y = x2 , y = x2 +2x, y = x2 +4x and y = x2 +6x

  26. x = 0 x = -1x = -2 X = -4 y = x2y = x2 +2x y = x2 +4x y = x2 +6x

  27. x = 0 x = -1/2x = -1 X = -3/2 y = 2x2y = 2x2 +2x y = 2x2 +4x y = 2x2 +6x

  28. Vertex of a Parabola

  29. Compare Quadratic Absolute value

  30. Graph a Quadratic

  31. Graph a Quadratic • Evaluate the function at another point • Graph that point and it’s reflection across the axis of symmetry • Sketch the curve

  32. Example Graph -1 = x2+2x-y y = x2+2x+1 a = 1 b = 2 c =1 y = (-1)2+2(-1)+1 y = 1+-2+1 = 0 Vertex = (-1 , 0)

  33. Example

  34. Example Graph Vertex = (-1 , 0) y -intercept = c y -intercept =1 Graph (0 , 1) Axis of Symmetry x = -1 Vertex is one unit to the right Reflection is one unit to the left Graph (-2 , 1)

  35. Example

  36. Example Graph evaluate at x = 1 y = (1)2+2(1)+1 y = 1 + 2 + 1 = 4 Graph (1 , 4) Axis of Symmetry x = -1 Vertex is two units to the right Reflection is two units to the left Graph (-3 , 4)

  37. Trace the curve

  38. Example

  39. Do Now! • Page 244 • Problems 1- 4, 7, 10 – 19, 22, 24, 37-39

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