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C ollege A lgebra

C ollege A lgebra. Functions and Graphs (Chapter1). Objectives. Cover the topics in Section ( 1-5):Graphs and Transformations. Library of Elementary Functions. Vertical and horizontal Shifts. Reflection, Expansions, and Contractions. Chapter1. Absolute Value Function.

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C ollege A lgebra

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  1. College Algebra Functions and Graphs (Chapter1)

  2. Objectives Cover the topics in Section ( 1-5):Graphs and Transformations • Library of Elementary Functions. • Vertical and horizontal Shifts. • Reflection, Expansions, and Contractions. Chapter1

  3. Absolute Value Function Identity Function g(x) f(x) 5 5 x –5 5 x –5 5 g(x) = |x| f(x) = x –5 1. 2. Graphs and Transformations Six Basic Functions 3 Chapter1

  4. Square Function Cube Function h(x) m(x) 5 5 x –5 5 x –5 5 2 3 h(x) = x m(x) = x –5 3. 4. Graphs and Transformations Six Basic Functions 4 Chapter1

  5. Cube-Root Function Square-Root Function p(x) n(x) 5 5 x –5 5 x 5 3 n(x) = x p(x) = x –5 5. 6. Graphs and Transformations Six Basic Functions 5 Chapter1

  6. Graphs and Transformations Vertical Translation: Y-k = f(x) Horizontal Translation: y = f(x-h) Reflection: y = – f(x) Reflect the graph of y = f(x) in the x axis Vertical Expansion and Contraction: y = A f(x) k > 0 Shift graph of y = f(x) up k units k < 0 Shift graph of y = f(x) down k units h > 0 Shift graph of y = f(x) right h units h < 0 Shift graph of y = f(x) left h units A > 1 Vertically expand graph of y = f(x) by multiplying each ordinate value by A 0 < A < 1 Vertically contract graph of y = f(x) by multiplying each ordinate value by A 6 Chapter1

  7. Graphs and Transformations 7 Chapter1

  8. Graphs and Transformations Example 1: Sketch the graph of the function f(x)=|x|+1. Do not plot points, but instead apply transformations to the graph of a standard function. Solution 8 Chapter1

  9. Graphs and Transformations Example 2: Sketch the graph of the function f(x)=|x-1|. Do not plot points, but instead apply transformations to the graph of a standard function. Solution 9 Chapter1

  10. y g(x) = |x| + 3 8 f(x) = |x| 4 h(x) = |x| –4 x 4 -4 -4 Graphs and Transformations Example 3:Use the graph of f (x) = |x| to graph thefunctions g(x) = |x| + 3 and h(x) = |x| – 4. Example 3:Use the graph of f (x) = |x| to graph thefunctions g(x) = |x| + 3 and h(x) = |x| – 4. Solution 10 Chapter1

  11. Graphs and Transformations Example 4: Solution 11 Chapter1

  12. Graphs and Transformations Example 5: Solution 12 Chapter1

  13. Graphs and Transformations Example 6: 13 Chapter1

  14. Graphs and Transformations 14 Chapter1

  15. y y 4 4 x x -4 -4 (–1, –2) Graphs and Transformations Example(7) Graph the function using the graph of Then a horizontal shift 5 units left. First make a vertical shift 4 units downward. (4, 2) (0, 0) (4, –2) (–5, –4) (0, – 4) 15 Chapter1

  16. Graphs and Transformations 16 Chapter1

  17. Graphs and Transformations 17 Chapter1

  18. Graphs and Transformations Example(8) Sketch the graph of the function -|x| . Do not plot points, but instead apply transformations to the graph of a standard function. 18 Chapter1

  19. Graphs and Transformations Example(9) Sketch the graph of the function 2|x| . Do not plot points, but instead apply transformations to the graph of a standard function. 19 Chapter1

  20. Graphs and Transformations Example(10) 20 Chapter1

  21. Graphs and Transformations 21 Chapter1

  22. y y 4 4 x x 4 4 – 4 -4 Graphs and Transformations Example(11) Graph y = –(x + 3)2 using the graph of y = x2. Then shiftthe graphthree units to the left. First reflectthe graphin the x-axis. y = x2 (–3, 0) y =–(x + 3)2 y =–x2 22 Chapter1

  23. y 4 is the graphof y = x2shrunk vertically by. x –4 4 Graphs and Transformations Example(12) Vertical Stretching and Shrinking If c > 1 then the graph of y= cf(x) is the graph of y = f(x) stretched vertically by c. If 0 < c < 1 then the graph of y = cf(x) is the graph of y = f(x) shrunk vertically by c. y = x2 y =2x2 Example: y =2x2 is the graph of y = x2stretched verticallyby 2. Chapter1

  24. y 4 is the graph of y = |x| stretched horizontallyby . x -4 4 Graphs and Transformations Example(13) Horizontal Stretching and Shrinking If c > 1, the graph of y = f(cx) is the graph of y = f(x) shrunk horizontally by c. If 0 < c < 1, the graph of y = f(cx) is the graph of y = f(x) stretchedhorizontally by c. y = |2x| Example:y = |2x|is the graph of y = |x| shrunk horizontally by 2. y = |x| Chapter1

  25. 5 –5 5 –5 Graphs and Transformations Example(14) Sketch the graphs given by Chapter1

  26. y y 8 8 4 4 x x -4 4 4 Step 3: -4 Step 4: Graphs and Transformations Example(15) Graph using the graph of y = x3. Step 1: y = x3 Step 2: y =(x + 1)3 Chapter1

  27. Graphs and Transformations Example(16) Chapter1

  28. Graphs and Transformations Chapter1

  29. Graphs and Transformations Chapter1

  30. Graphs and Transformations Example(17) Chapter1

  31. Graphs and Transformations Chapter1

  32. Graphs and Transformations Chapter1

  33. End of the Lecture Let Learning Continue

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