VCE Maths Methods: Quadratic Graphs - Transformations, Equations, Coefficients, Lecture notes of Advanced Calculus

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Quadratic graphs

• Transformations of parabolas
• Graphing quadratics
• Finding the equation of quadratics

Transformations of parabolas - dilations

y = ax^2 : a is the dilation factor from the x axis that narrows or widens the parabola. (The curve de fi ned by a quadratic function is a parabola.)

Transformations of parabolas - vertical translations

y = x^2 + k : k is the vertical translation that moves the graph up k units. y = x 2 y = x 2

• 2 y = x 2 − 5

Graphing quadratics from linear factors

x intercept: (-5,0) & (1,0) y = (x − 1 )(x + 5 ) y = x 2

• 4 x − 5 turning point: (-2,-9) y = (x − 1 )(x + 5 ) y intercept: y = ( − 2 − 1 )( − 2 + 5 ) axis of symmetry: x=- x =

Turning point: y = − 5 x = 1 ,x = − 5 x intercepts: y intercept: (0,-5) y = ( − 3 )( − 3 ) = − 9 ( − 2 , − 9 )

Finding the equation - from turning point & another point

1. Decide on turning point or factor form. 2.Find the value of a by substituting another point into the equation. y = a(x − 2 ) 2 − 8 0 = a( 4 − 2 ) 2 − 8 8 = a( 2 2 ) a = 2 y = 2 (x − 2 ) 2 − 8 y = 2 x 2 − 8 x y = 2 (x 2 − 4 x + 4 ) − 8 Another point (4,0) Turning point (2,-8)

y = a(x − h)

2

+ k

Finding the equation - from two x intercepts & another point

1. Decide on turning point or factor form. 2.Find the value of a by substituting another point into the equation. y = a(x − 0 )(x − 4 ) y = ax(x − 4 ) − 8 = a( 2 )( 2 − 4 ) − 8 = a( − 4 ) a = 2 y = 2 x(x − 4 ) y = 2 x 2 − 8 x Another point (2,-8) x intercepts: 0 & 4

y = a(x − m)(x − n)

Finding the equation from three points - simultaneous equations

• Matrices can be used to help solve simultaneous equations of two or more

variables.

• For example, fi nding the equation of a quadratic curve (y = ax

2

+ bx +c)

that passes through three points (-1,6) , (0, 3) & (2, 9).

a

b

c

6 = a( − 1 )

2

+ b( − 1 ) + c

3 = a( 0 )

2

+ b( 0 ) + c

9 = a( 2 )

2

+ b( 2 ) + c

− 1

a

b

c

a

b

c

a

b

c

y = 2 x

2

− x + 3

a b c