curve sketching
DESCRIPTION
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 ). - PowerPoint PPT PresentationTRANSCRIPT
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Curve Sketching
C1 Section 4.6 – 4.7
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:
y = 9 - x2
y = 2(9 - x2)
Each point twice as far along y.
f(x) 2f(x)
(2 , 5) (2 , 10)
Stretches in the x-axis If we have the function y = f(x) then:
y = f(ax) is a stretch by a factor 1/a in the x-axis:
y = 9 - x2
f(x) f(2x)
y = 9 – (2x)2
Each point 1/2 as far along x.
(3 , 0) (1.5 , 0)(2 , 5) (1 , 5)
Transformations of Points f(x + a)
0
(1 , 4)
(3, 1)
Imagine a function where y = f(x), which hasa root at 0, and points (1 , 4) and (3 , 1) lie on the curve:
f(x) f(x + 1)
-1
(0 , 4)
(2, 1)
Transformations of Points f(x – a)
0
(1 , 4)
(3, 1)
Imagine a function where y = f(x), which hasa root at 0, and points (1 , 4) and (3 , 1) lie on the curve:
f(x) f(x - 1)
1
(2 , 4)
(4, 1)
Transformations of Points f(x) - a
0
(1 , 4)
(3, 1)
Imagine a function where y = f(x), which hasa root at 0, and points (1 , 4) and (3 , 1) lie on the curve:
f(x) f(x) - 4
-4
(1 , 0)
(3, -3)
Transformations of Points nf(x)
0
(1 , 4)
(3, 1)
Imagine a function where y = f(x), which hasa root at 0, and points (1 , 4) and (3 , 1) lie on the curve:
f(x) 2f(x)
0
(1 , 8)
(3, 2)
Transformations of Points f(nx)
0
(1 , 4)
(3, 1)
Imagine a function where y = f(x), which hasa root at 0, and points (1 , 4) and (3 , 1) lie on the curve:
f(x) f(2x)
0
(0.5 , 4)
(1.5, 1)
Transformations of Points -f(x)
0
(1 , 4)
(3, 1)
Imagine a function where y = f(x), which hasa root at 0, and points (1 , 4) and (3 , 1) lie on the curve:
f(x) -f(x)
0
(1 , -4)
(3, -1)
Transformations of Points f(-x)
0
(1 , 4)
(3, 1)
Imagine a function where y = f(x), which hasa root at 0, and points (1 , 4) and (3 , 1) lie on the curve:
f(x) f(-x) 0
(-1 , 4)
(-3, 1)
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 + bThink about the series of transformations from
that…
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]
Quadratic Example y = -4x2 + 8x + 3 = -4(x – 1)2 + 7
y = x2 y = (x – 1)2 y = 4(x – 1)2
Quadratic Example
y = 4(x – 1)2 y = - 4(x – 1)2 y = - 4(x – 1)2 + 7
y = -4x2 + 8x + 3 = -4(x – 1)2 + 7
Line of symmetry: x = 1, max at +7, intercept: +3