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Quadratics 1

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This document discusses exploring properties of quadratic functions through investigating different forms of expression and transformations of quadratic graphs. It examines exploring properties like the vertex and line of symmetry for basic quadratic functions like y=x^2 as well as transformed functions like y=(x-h)^2, looking at shifts along the x-axis and y-axis. Examples are provided of sketching graphs for quadratic functions in different forms and with different transformations, concluding with identifying the vertex and line of symmetry for general quadratic functions of the form y=a(x-h)^2+k.

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1 Sep 26­04:59 p.m. To explore the properties of quadratic functions and their graphs. To investigate the different forms in which quadratic functions can be expressed. To explore the transformations of quadratic functions and their graphs. http://www.youtube.com/watch?v=VSUKNxVXE4E&feature=player_embedded# http://evmaths.jimdo.com/year­11/functions/?logout=1
2 Sep 26­04:59 p.m. f(x) = x2 vertex: line of symmetry:
3 Sep 26­04:59 p.m. What do you expect if                      ? y = x2 y = ­ x2 vertex: line of symmetry:
4 Sep 26­04:59 p.m. Draw                ,               and   y = x2 y = 2 x2 vertex: line of symmetry:
5 Sep 26­04:59 p.m. Conclusions: y = a x 2 The graph of is a parabola with vertex: (0,0) line of symmetry :   x = 0 a > 0 a < 0 |a | > 1 as "a" increases the parabola gets "thinner" 0 < |a| <1 the parabola looks "fatter" "a " produces a stretch along the y-axis
6 Sep 26­04:59 p.m. Sketch the graphs of   and vertex: line of symmetry: vertex: line of symmetry: y = ( x + 3)2 y = ( x ­ 2 )2 y = x2
7 Sep 26­04:59 p.m. Conclusions: parabola moves to the right parabola moves to the left vertex: line of symmetry: vertex: line of symmetry: ( h  , 0  ) x = h ( ­ h  , 0  ) x = ­ h Translation along x­axis ( h > 0 )
8 Sep 26­04:59 p.m. Sketch the graphs of and y=x2 y=x2  ­2 y=x2  +3 vertex: line of symmetry: vertex: line of symmetry:
9 Sep 26­04:59 p.m. Conclusions: parabola moves upwards parabola moves downwards vertex: line of symmetry: ( 0 , k  ) x = 0 Translation  k units along y­axis k > 0 k < 0
10 Sep 26­04:59 p.m. Conclusions: vertex (h , k) (­h , k) In general: represents a parabola • with vertex in (h,k) • axis of symmetry x = h  • a produces a stretch parallel to the y- axis • a > 0 • a < 0
11 Sep 26­04:59 p.m. y = ( x ­ 1 ) 2  + 3 vertex: line of symmetry: y = 2 ( x ­ 3 ) 2    vertex: line of symmetry: (1,3) x=1 (3,0) x=3
12 Sep 26­04:59 p.m. y = ­ 3 x2  + 4    vertex: line of symmetry: y = 3 ( x + 1 ) 2  ­ 2 vertex: line of symmetry: (0,4) x=0 (-1,-2) x=-1 http://members.shaw.ca/ron.blond/QFA.CSF.APPLET/index.html Transformaciones Función Cuadrática.ggb
13 Sep 26­04:59 p.m. For Parabolas of the form What is the y­intercept ? Find the roots of f. Concavity? factorising (if possible) by formula (y = 0)
14 Sep 26­04:59 p.m.   y­intercept  = ­8 roots : ­ 4  and 2  line of symmetry? vertex?
15 Sep 26­04:59 p.m. Line of symmetry is in the middle between the roots : The vertex will be on the line of symmetry: We can also find the line of symmetry by doing : y - intercept: ( 0 , c ) a < 0a >0 Cambios cuadratica.ggb
16 Sep 26­04:59 p.m.  For find: y- intercept: line of symmetry: vertex: roots: Now draw a sketch of the function.
17 Sep 26­04:59 p.m.   y- intercept: line of symmetry: vertex: roots: Now draw a sketch of the function. Express f(x) in the form   
18 Sep 26­04:59 p.m. y = a (x ­ x1) ( x ­ x2)Parabolas of the form : y = ( x ­ 3 ) ( x + 1 ) Roots: Line of symmetry: Vertex: In general: x1  and   x2
19 Sep 26­04:59 p.m. axis of symmetry   vertex root root y­ intercept (0 , c )
20 Sep 26­04:59 p.m. y = (x­2)2 y = ­ x 2  + 1 y = x2  ­ 2 y = x2  + 3  y = (x ­ 3 )2 +5 y= ­ 2 x 2  + 1
Attachments Parabola canonica.ggb Cambios cuadratica.ggb QUADRATIC  FUNCTIONS I  2010.doc Transformaciones Función Cuadrática.ggb

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