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Math 426

Math 426. FUNCTIONS. QUADRATIC. Any function of the form. y = f ( x ) = ax 2 + bx + c. where a  0 is called a Quadratic Function. Example:. y = 3 x 2 - 2 x + 1. a. b. c. = 3,. = -2,. = 1. Note that if a = 0 we simply have the linear function y = bx + c.

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Math 426

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  1. Math 426 FUNCTIONS QUADRATIC

  2. Any function of the form y = f (x) = ax 2+ bx + c where a 0 is called a Quadratic Function

  3. Example: y = 3x 2 - 2x + 1 a b c = 3, = -2, = 1 Note that if a = 0 we simply have the linear function y = bx + c

  4. Consider the simplest quadratic equation y = x 2 Here a = 1, b = 0, c = 0 Plotting some ordered pairs (x, y) we have: y = f (x ) = x 2 xf (x ) (x, y ) -3 9 (-3, 9) -2 4 (-2, 4) -1 1 (-1, 1) 0 0 (0, 0) 1 1 (1, 1) 2 4 (2, 4) 3 9 (3, 9)

  5. y (-2, 4) 4 3 2 1 (2, 4) (x, y) (-3, 9) (-2, 4) (-1, 1) y = x2 (0, 0) (1, 1) (2, 4) (3, 9) x -3 -2 -1 1 2 3 Vertex (0, 0) A parabola with the y-axis as the axis of symmetry.

  6. Graphs of y = ax 2will have similar form and the value of the coefficient ‘a ’ determines the graph’s shape. y y = 2x 2 4 3 2 1 y = x 2 y = 1/2 x 2 a > 0 opening up x -3 -2 -1 1 2 3

  7. y x a < 0 opening down y = -2x 2 In general the quadratic term ax 2 in the quadratic function f (x ) = ax 2 +bx + c determines the way the graph opens.

  8. Consider f (x ) = ax 2 +bx + c In a general sense the linear term bx acts to shift the plot of f (x ) from side to side and the constant term c (=cx 0) acts to shift the plot up or down. y a > 0 x-intercept c Notice thatc is the y -intercept where x = 0 and f (0) = c a < 0 x c y-intercept Note also that the x -intercepts (if they exist) are obtained by solving: y = ax 2 +bx + c = 0

  9. It turns out that the details of a quadratic function can be found by considering its coefficients a, b and c as follows: (1) Opening up (a > 0), down (a < 0) (2) y –intercept: c (3) x -intercepts from solution of y = ax 2 + bx + c = 0 You solve by factoring or the quadratic formula (4) vertex =

  10. Example:y = f (x ) = x 2 - x - 2 here a = 1, b = -1 and c = -2 (1) opens upwards since a > 0 (2) y –intercept: -2 (3) x -intercepts from x 2 - x - 2 = 0 or (x -2)(x +1) = 0 x = 2 or x = -1 (4) vertex:

  11. y (-1, 0) (2, 0) x -2 -1 0 1 2 y = x 2 - x - 2 -1 -2 -3

  12. Example:y = j (x ) = x 2 - 9 here a = 1, b = 0 and c = -9 (1) opens upwards since a > 0 (2) y –intercept: -9 (3) x -intercepts from x 2 - 9 = 0 or x 2 = 9 x = 3 (4) vertex at (0, -9)

  13. y (-3, 0) (3, 0) x -3 0 3 y = x 2 - 9 -9 (0, -9)

  14. Example:y = g (x ) = x 2 - 6x + 9 here a = 1, b = -6 and c = 9 (1) opens upwards since a > 0 (2) y –intercept: 9 (3) x -intercepts from x 2 - 6x + 9 = 0 or (x - 3)(x - 3) = 0 x = 3 only (4) vertex:

  15. y (0, 9) 9 y = x 2 - 6x + 9 (3, 0) x 3

  16. Example:y = f (x ) = -3x 2 + 6x - 4 here a = -3, b = 6 and c = -4 (1) opens downwards since a < 0 (2) y –intercept: -4 (3) x -intercepts from -3x 2 + 6x - 4 = 0 (there are nox -intercepts here) (4) vertex at (1, -1) Vertex is below x-axis, and parabola opens down!

  17. y x 1 2 (1, -1) -1 -4 y = -3x 2 + 6x - 4 (0, -4)

  18. The Quadratic Formula It is not always easy to find x -intercepts by factoring ax 2 + bx + c when solving ax 2 + bx + c = 0 Quadratic equations of this form can be solved for x using the formula:

  19. Example: Solve x 2 − 6x + 9 = 0 here a = 1, b = -6 and c = 9 Note: the expression inside the radical is called the “discriminant” Note: discriminant = 0 one solution as found previously

  20. Example: Solve x 2 - x - 2 = 0 here a = 1, b = -1 and c = -2 Note: discriminant > 0 two solutions

  21. Example: Find x -intercepts of y = x 2 - 9 Solve x 2 - 9 = 0 a = 1, b = 0, c = -9 Note: discriminant > 0 two solutions x = 3 or x = -3

  22. Example:Find the x -intercepts of y = f (x) = -3x 2 + 6x - 4 a = -3, b = 6 and c = -4 Solve -3x 2 + 6x - 4= 0 Note: discriminant < 0 no Real solutions  there are no x -intercepts as we discovered in an earlier plot of y = -3x 2 + 6x - 4

  23. FUNCTIONS QUADRATIC The end.

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