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1. 1. 1Sep 26­04:59 p.m.To explore the properties of quadratic functionsand their graphs.To investigate the different forms in whichquadratic functions can be expressed.To explore the transformations of quadraticfunctions and their graphs.http://www.youtube.com/watch?v=VSUKNxVXE4E&feature=player_embedded#http://evmaths.jimdo.com/year­11/functions/?logout=1
2. 2. 2Sep 26­04:59 p.m.f(x) = x2vertex:line of symmetry:
3. 3. 3Sep 26­04:59 p.m.What do you expect if                      ?y = x2y = ­ x2vertex:line of symmetry:
4. 4. 4Sep 26­04:59 p.m.Draw                ,               and  y = x2 y = 2 x2vertex:line of symmetry:
5. 5. 5Sep 26­04:59 p.m.Conclusions:y = a x 2The graph of is a parabola withvertex: (0,0) line of symmetry :   x = 0a > 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. 6. 6Sep 26­04:59 p.m.Sketch the graphs of   andvertex:line of symmetry:vertex:line of symmetry:y = ( x + 3)2y = ( x ­ 2 )2y = x2
7. 7. 7Sep 26­04:59 p.m.Conclusions:parabola moves to the rightparabola moves to the leftvertex:line of symmetry:vertex:line of symmetry:( h  , 0  )x = h( ­ h  , 0  )x = ­ hTranslation along x­axis( h > 0 )
8. 8. 8Sep 26­04:59 p.m.Sketch the graphs of andy=x2y=x2 ­2y=x2 +3vertex:line of symmetry:vertex:line of symmetry:
9. 9. 9Sep 26­04:59 p.m.Conclusions:parabola moves upwardsparabola moves downwardsvertex:line of symmetry:( 0 , k  )x = 0Translation  k units alongy­axisk > 0k < 0
10. 10. 10Sep 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. 11. 11Sep 26­04:59 p.m.y = ( x ­ 1 ) 2 + 3vertex:line of symmetry:y = 2 ( x ­ 3 ) 2  vertex:line of symmetry:(1,3)x=1(3,0)x=3
12. 12. 12Sep 26­04:59 p.m.y = ­ 3 x2 + 4   vertex:line of symmetry:y = 3 ( x + 1 ) 2 ­ 2vertex:line of symmetry:(0,4)x=0(-1,-2)x=-1http://members.shaw.ca/ron.blond/QFA.CSF.APPLET/index.htmlTransformaciones Función Cuadrática.ggb
13. 13. 13Sep 26­04:59 p.m.ForParabolas of the formWhat is the y­intercept ?Find the roots of f.Concavity?factorising (if possible)by formula(y = 0)
14. 14. 14Sep 26­04:59 p.m.  y­intercept  = ­8roots : ­ 4  and 2 line of symmetry?vertex?
15. 15. 15Sep 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 >0Cambios cuadratica.ggb
16. 16. 16Sep 26­04:59 p.m. For find:y- intercept:line of symmetry:vertex:roots:Now draw a sketch of the function.
17. 17. 17Sep 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. 18. 18Sep 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. 19. 19Sep 26­04:59 p.m.axis of symmetry  vertexrootrooty­ intercept(0 , c )
20. 20. 20Sep 26­04:59 p.m.y = (x­2)2y = ­ x 2 + 1y = x2 ­ 2y = x2 + 3 y = (x ­ 3 )2+5y= ­ 2 x 2 + 1