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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