1. 3x + 2 = ½ x – 5 2. |3x + 2| > x – 5 < -3x |x + 2| < 15 Algebra II 1

Transformations of parent functions

 Parent function:  the most basic graph in a family of graphs  Transformation  A change in size, shape, position, or orientation of a graph  Translation  A transformation that shifts a graph horizontally or vertically but does not change size or shape 3 Algebra II

 Reflection  A transformation that flips a graph over a line of reflection  Vertical stretch  A transformation that causes the graph of a function to stretch away from the x axis. (multiplied by a factor >1)  Vertical shrink  A transformation that causes the graph of a function to shrink toward the x-axis (multiplied by a factor 0<a<1) 4 Algebra II

 Domain:  The x values of a graph, the distance from left to right  Range :  the y values of a graph, the distance from bottom to top ** Domain and Range must be written in: INTERVAL NOTATION 5 Algebra II

 Domain:  [-4,-1]  Range:  [-4,∞) 6 Algebra II

 Domain:  [-1,5]  Range:  [-4,7] 7 Algebra II

 Domain:  (-∞, ∞)  Range:  [0,∞) 8 Algebra II

9 ConstantLinear f(x) = 1 Domain: (-∞,∞) Range {1} f(x) = x Domain: (-∞,∞) Range: (-∞,∞)

Algebra II 10 Absolute ValueQuadratic f(x) = |x| Domain: (-∞,∞) Range: [0, ∞) f(x) = x 2 Domain: (-∞,∞) Range: [0, ∞)

 R x SR y  Reflect over x-axis (affect the y-values), Shift (horizontal and vertical), Reflect over y-axis (affect the x-values)  y = -(x)  effects y so flips over x axis  y = (x – h) effects x: shift left/right (opposite direction)  y = x + k effects y: shift up/down (same direction)  y =(-x)  effects x so reflect over y 11 Algebra II

12 Linear f(x) = x Vertical Shrink by a factor of ¼ Reflection over the x-axis Vertical shift up 8

Algebra II 13 Constant f(x) = 1 Vertical shift down 4

Algebra II 14 Absolute Value f(x) = |x| Horizontal shrink by a Factor of ⅕ …….So It is also a vertical stretch by a factor of 5 NARROWER

Algebra II 15 Quadratic f(x) = x 2 Horizontal shift right 1 Vertical shift up 4

Algebra II 16 Linear f(x) = x Vertical shift down 7

Algebra II 17 Constant f(x) = 1 Vertical shift down 10

Algebra II 18 Absolute Value f(x) = |x| Vertical shift Up 1

Algebra II 19 Quadratic f(x) = x 2 Reflection over the x-axis

Algebra II 20 Quadratic f(x) = x 2 Vertical shrink by a factor of ⅛ WIDER

Algebra II 21 Absolute Value f(x) = |x| Vertical stretch by a factor of 6 NARROWER

22 Algebra II 11. Identify the function family of f(x) = ⅓|-x| + 4 and describe the domain and range. Use a graphing calculator to verify your answers.

23 Algebra II 11b. Identify the function family of f(x) = -2(x + 3) 2 – 8 and describe the domain and range. Use a graphing calculator to verify your answers.

24 Algebra II

25 Algebra II 13.

26 Algebra II 14.

27 Algebra II 15.

28 Algebra II 16. Graph g(x) = x – 4 and its parent function. Then describe the transformation.

29 Algebra II 17.

30 Algebra II 18. Graph p(x) = -x 2 and its parent function. Then describe the transformation.

31 Algebra II 19. Graph k(x) = -x and its parent function. Then describe the transformation.

32 Algebra II 21. g(x) = x h(x) = (x – 2) m(x) = -|x| 23. g(x) = 2|x| 24. h(x) = ½x g(x) = 3x 26. h(x) = 3/2x c(x) = 0.2 |x – 2| 28. g(x) = - |x + 5| h(x) = -0.25x 2 + 4

33 Algebra II 31. The table shows the height y of a dirt bike x seconds after jumping off a ramp. What type of function can you use to model this data? Estimate the height after 1.75 seconds.

34 Algebra II Use a graphing calculator to graph the function and its parent function. Then describe the transformation. 32. h(x) = -¼x d(x) = 3(x – 1)

35 Algebra II 35.

36 Algebra II Identify the function family to which g belongs. Compare the graph of g to its parent function and describe the transformation. 1. g(x) = -x g(x) = x g(x) = 2 – 0.2x 4. g(x) = 2 I x I g(x) = 2.2(x + 2) 2

37 Algebra II 6.