1. Algebra 2 Families of Functions Lesson 2-6 Part 2

2. Goals Goal To analyze transformations of functions. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Reflection Vertical Stretch Vertical Compression

4. Essential Question Big Idea: Function What other transformations can you perform on a parent function?

5. Definition Reflection - is a transformation that flips a figure across a line called the line of reflection. Each reflected point is the same distance from the line of reflection, but on the opposite side of the line.

6. Reflections You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure. l You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image. Example:The figure is reflected across line l.

7. Reflections – continued… reflects across the y axis to line n (2, 1)  (-2, 1) & (5, 4)  (-5, 4) Reflection across the x-axis: the x values stay the same and the y values change sign. (x, y)  (x, -y) Reflection across the y-axis: the y values stay the same and the x values change sign. (x, y)  (-x, y) Example:In this figure, line l : reflects across the x axis to line m. (2, 1)  (2, -1) & (5, 4)  (5, -4) ln m

8. Reflections Reflection Across y-axisReflection Across x-axis Reflections

9. Reflecting Functions Start with flips graph vertically (across the x-axis) flips graph horizontally (across the y-axis)

10. Reflections about the x-axis Given the graph you get the graph by reflecting the first graph across the x-axis. Changing y to  y reflects the graph across the x-axis.

11. Reflections about the x-axis

12. Reflections about the y-axis Given the graph you get the graph by reflecting the first graph across the y-axis. Changing x to  x reflects the graph across the y-axis.

13. Reflections about the y-axis

14. y = f (–x) y = f (x) y = –f (x) Reflections The graph of the function y = f ( – x) is the graph of y = f (x) reflected in the y-axis. The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x-axis. x y

15. Your Turn: f(x) = x 2 - 3 f(x) = -(x 2 - 3) Reflection in the x-axis x y Describe the transformation

16. Your Turn: f(x) = (x – 3) 2 f(x) = (-x-3) 2 Reflect in the y-axis x y Describe the transformation

17. You can transform a function by transforming its ordered pairs. When a function is reflected, the original graph and the graph of the transformation are congruent because the size and shape of the graphs are the same. Reflecting Graphs

18. Reflecting Across an Axis Reflecting the Graph of a Function Across an Axis For a function (a) the graph of is a reflection of the graph of f across the x-axis. (b) the graph of is a reflection of the graph of f across the y-axis.

19. reflection across x-axis Identify important points from the graph and make a table. xy–y–y –5–3–1(–3) = 3 –20– 1(0) = 0 0–2– 1(–2) = 2 20 – 1(0) = 0 5–3 – 1(–3) = 3 Multiply each y-coordinate by – 1. The entire graph flips across the x-axis. Example: Use a table to perform the transformation of y = f(x). Use the same coordinate plane as the original function.

20. reflection across x-axis xy–y –24–4 –100 02–2 22 f Multiply each y-coordinate by –1. The entire graph flips across the x-axis. Use a table to perform the transformation of y = f(x). Use the same coordinate plane as the original function. Your Turn:

21. Use a table to perform the transformation of y = f(x). Graph the function and the transformation on the same coordinate plane. Reflection across y-axis Your Turn:

22. Given the graph of sketch the graph of (a) (b) Solution (a) (b)

23. Nonrigid Transformations Horizontal shifts, vertical shifts, and reflections are called rigid transformations because the basic shape of the graph is unchanged. These transformations change only the position of the graph in the coordinate plane. Nonrigid transformations are those that cause a distortion – a change in the shape of the original graph. These transformations include vertical stretch, vertical compression, horizontal stretch, and horizontal compression.

24. Imagine grasping two points on the graph of a function that lie on opposite sides of the y-axis. If you pull the points away from the y-axis, you would create a horizontal stretch of the graph. If you push the points towards the y-axis, you would create a horizontal compression. Vertical Stretches and Compressions

25. Vertical Stretching Given the graph you get the graph by stretching the first graph vertically by a factor of k. Multiplying f(x), the output of the function, by k>1 stretches the graph vertically by a factor of k.

26. Vertical Stretching of the Graph of a Function If c > 1, the graph of is obtained by vertically stretching the graph of by a factor of c. In general, the larger the value of c, the greater the stretch. Vertical Stretching

28. Vertical Compression Given the graph you get the graph by compressing the first graph vertically by a factor of k. Multiplying f(x), the output of the function, by 0<k<1 compresses the graph vertically by a factor of k.

29. Vertical Compressing of the Graph of a Function If the graph of is obtained by vertically compressing the graph of by a factor of c. In general, the smaller the value of c, the greater the compression. Vertical Compression

31. Vertical Stretching or Compression Rules The graph of y = af(x) can be obtained from the graph of y = f(x) by stretching vertically for |a| > 1, or compressing vertically for 0 < |a| < 1. (The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

32. Vertical Stretching and Compressing If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c. If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) compressed vertically by c. Example: y = 2x 2 is the graph of y = x 2 stretched vertically by a factor of 2. – 4– 4 x y 4 4 y = x 2 is the graph of y = x 2 compressed vertically by a factor of. y = 2x 2

33. Examples Compression along the y-axis by a factor of 1/10 Stretch along the y-axis by a factor of 2

34. Your Turn: f(x) = (x – 3) 2 -1 f(x) = 3((x-3) 2 -1) Stretch by a factor 3 in the y-axis x y Describe the transformation

35. Identify important points from the graph and make a table. Use a table to perform a vertical stretch of y = f(x) by a factor of 2. Graph the transformed function on the same coordinate plane as the original figure. xy2y2y –132(3) = 6 002(0) = 0 222(2) = 4 42 Multiply each y-coordinate by 2. Your Turn:

36. Use a table to perform the transformation of y = f(x). Graph the function and the transformation on the same coordinate plane. Vertical compression by a factor of. f Your Turn:

37. Transformations

38. Combining Transformations of Graphs Example Describe how the graph of can be obtained by transforming the graph of Sketch its graph. Solution Since the basic graph is the vertex of the parabola is shifted right 4 units. Since the coefficient of is –3, the graph is stretched vertically by a factor of 3 and then reflected across the x- axis. The constant +5 indicates the vertex shifts up 5 units. shift 4 units right shift 5 units up vertical stretch by a factor of 3 reflect across the x-axis

39. Graphs:

40. Caution in Translations of Graphs The order in which transformations are made is important. If they are made in a different order, a different equation can result. –For example, the graph of is obtained by first stretching the graph of by a factor of 2, and then translating 3 units upward. –The graph of is obtained by first translating horizontally 3 units to the left, and then stretching by a factor of 2.

41. Essential Question Big Idea: Function What other transformations can you perform on a parent function? A reflection flips a function across a line, usually an axis. The equation of the reflected function changes the sign of either the inputs or outputs. Vertical stretch and compression multiply the outputs of a function by the same factor.

42. Assignment Section 2-6 Part 2, Pg 117 – 120;#1 – 6 all, 8 – 36 even.