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Warm up

Warm up. Determine whether the graph of each equation is symmetric wrt the x-axis, the y-axis, the line y = x the line y = -x or none. 1. 2. . Lesson 3-2 Families of Graphs. Objective: Identify transformations of simple graphs and sketch graphs of related functions. Family of graphs.

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Warm up

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  1. Warm up Determine whether the graph of each equation is symmetric wrt the x-axis, the y-axis, the line y = x the line y = -x or none. 1. 2.

  2. Lesson 3-2 Families of Graphs Objective: Identify transformations of simple graphs and sketch graphs of related functions.

  3. Family of graphs A family of graphs is a group of graphs that displays 1 or more similar characteristics. Parent graph – the anchor graph from which the other graphs in the family are derived.

  4. Identity Functions • f(x) =x y always = whatever x is

  5. Constant Function • f(x) = c In this graph the domain is all real numbers but the range is c. c

  6. Polynomial Functions f(x) = x2 The graph is a parabola.

  7. Square Root Function • f(x)=

  8. Absolute Value Function • f(x) =|x|

  9. Greatest Integer Function (Step) y=[[x]]

  10. Rational Function y=x-1 or 1/x

  11. Reflections • A reflection is a “flip” of the parent graph. • If y = f(x) is the parent graph: • y = -f(x) is a reflection over the x-axis • y =f(-x) is a reflection over the y-axis

  12. Reflections Parent Graph y =x3 y=f(-x) y=-f(x)

  13. Translations y=f(x)+c moves the parent graph up c units y=f(x) - c moves the parent graph down c units

  14. Translations f(x) +c 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Vertical Translations f(x) +2= x2 + 2 f(x) = x2 0 f(x) - 5 = x2 - 5

  15. Translations y=f(x+c) moves the parent graph to the left c units y=f(x – c) moves the parent graph to the right c units

  16. 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Translations y =f(x+c) Horizontal Translations 5 2 f(x - 5) f(x) f(x + 2) In other words, ‘+’ inside the brackets means move to the LEFT

  17. Translations y=c •f(x); c>1 expands the parent graph vertically (narrows) y=c •f(x); 0<c<1 compresses the parent graph vertically (widens)

  18. y co-ordinates tripled y co-ordinates doubled Points located on the x axis remain fixed. Translations y=cf(x) 30 y = f(x) Stretches in the y direction 3f(x) 20 2f(x) f(x) 10 x 4 -6 2 8 0 6 -4 -2 0 -10 The graph of cf(x) gives a stretch of f(x) by scale factor cin the y direction. -20 -30

  19. y co-ordinates halved y co-ordinates scaled by 1/3 Translations y = cf(x);0<c<1 30 y = f(x) 1/3f(x) 20 ½f(x) f(x) 10 x 4 -6 2 8 0 6 -4 -2 -10 The graph of cf(x) gives a stretch of f(x) by scale factor cin the y direction. -20 -30

  20. Translations y=f(cx); c>1 compresses the parent graph horizontally (narrows) y=f(cx); 0<c<1 expands the parent graph horizontally (widens)

  21. 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 ½ the x co-ordinate 1/3 the x co-ordinate -4 -6 Translations y=f(cx) f(2x) f(3x) f(x) Stretches in x 0 The graph of f(cx) gives a stretch of f(x) by scale factor 1/cin the x direction.

  22. 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Translations y=f(cx) Stretches in x f(1/3x) f(x) f(1/2x) 0 The graph of f(cx) gives a stretch of f(x) by scale factor 1/cin the x direction. All x co-ordinates x 2 All x co-ordinates x 3

  23. Sources http://mrstevensonmaths.wordpress.com/2011/02/07/transformation-of-graphs-2/; August 9,2013

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