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Curve Sketching

Curve Sketching. C1 Section 4.6 – 4.7. y = 9 - x 2. y = 2(9 - x 2 ). f( x ). 2f( x ). (2 , 5). (2 , 10). Each point twice as far along y. Stretches in the y-axis. If we have the function y = f( x ) then: y = a f( x ) is a stretch by a factor a in the y-axis:. f( x ). f(2 x ).

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Curve Sketching

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  1. Curve Sketching C1 Section 4.6 – 4.7

  2. y = 9 - x2 y = 2(9 - x2) f(x) 2f(x) (2 , 5) (2 , 10) Each point twice as far along y. Stretches in the y-axis • If we have the function y = f(x) then: • y = af(x) is a stretch by a factor a in the y-axis:

  3. f(x) f(2x) (3 , 0) (1.5 , 0) (2 , 5) (1 , 5) Stretches in the x-axis • If we have the function y = f(x) then: • y = f(ax) is a stretch by a factor1/ain the x-axis: y = 9 - x2 y = 9 – (2x)2 Each point 1/2 as far along x.

  4. (1 , 4) f(x) 0 (3, 1) Transformations of Points • f(x + a) Imagine a function where y = f(x), which has a root at 0, and points (1 , 4) and (3 , 1) lie on the curve: (0 , 4) f(x + 1) -1 (2, 1)

  5. (1 , 4) f(x) 0 (3, 1) Transformations of Points • f(x – a) Imagine a function where y = f(x), which has a root at 0, and points (1 , 4) and (3 , 1) lie on the curve: (2 , 4) f(x - 1) 1 (4, 1)

  6. (1 , 4) f(x) 0 (3, 1) Transformations of Points • f(x) - a Imagine a function where y = f(x), which has a root at 0, and points (1 , 4) and (3 , 1) lie on the curve: f(x) - 4 (1 , 0) -4 (3, -3)

  7. (1 , 4) f(x) 0 (3, 1) Transformations of Points • nf(x) Imagine a function where y = f(x), which has a root at 0, and points (1 , 4) and (3 , 1) lie on the curve: (1 , 8) 2f(x) (3, 2) 0

  8. (1 , 4) f(x) 0 (3, 1) Transformations of Points • f(nx) Imagine a function where y = f(x), which has a root at 0, and points (1 , 4) and (3 , 1) lie on the curve: (0.5 , 4) f(2x) (1.5, 1) 0

  9. (1 , 4) f(x) 0 (3, 1) Transformations of Points • -f(x) Imagine a function where y = f(x), which has a root at 0, and points (1 , 4) and (3 , 1) lie on the curve: -f(x) 0 (3, -1) (1 , -4)

  10. (1 , 4) f(x) 0 (3, 1) Transformations of Points • f(-x) Imagine a function where y = f(x), which has a root at 0, and points (1 , 4) and (3 , 1) lie on the curve: (-1 , 4) f(-x) 0 (-3, 1)

  11. Quadratic Functions • When examining a quadratic for the transformations from y = x2 • Complete the square to get the quadratic into the form y = n(x - a)2 + b • Think about the series of transformations from that…

  12. Quadratics Example… • y = -4x2 + 8x + 3 • = -4(x2 – 2x) + 3 [ Factorise the 4 ] • = -4[ (x – 1)2-1 ] + 3 [Complete the sq] • = -4(x – 1)2 + 7 [ (-4 x -1) + 3 = 7]

  13. Quadratic Example • y = -4x2 + 8x + 3 = -4(x – 1)2 + 7 y = x2 y = (x – 1)2 y = 4(x – 1)2

  14. Quadratic Example • y = -4x2 + 8x + 3 = -4(x – 1)2 + 7 y = 4(x – 1)2 y = - 4(x – 1)2 y = - 4(x – 1)2 + 7 Line of symmetry: x = 1, max at +7, intercept: +3

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