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Chapter 2: Analysis of Graphs of Functions

Chapter 2: Analysis of Graphs of Functions. 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs 2.4 Absolute Value Functions: Graphs, Equations, Inequalities, and Applications

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Chapter 2: Analysis of Graphs of Functions

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  1. Chapter 2: Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs 2.4 Absolute Value Functions: Graphs, Equations, Inequalities, and Applications 2.5 Piecewise-Defined Functions 2.6 Operations and Composition

  2. 2.3 Stretching, Shrinking, and Reflecting Graphs 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.

  3. 2.3 Vertical Shrinking Vertical Shrinking of the Graph of a Function If the graph of is obtained by vertically shrinking the graph of by a factor of c. In general, the smaller the value of c, the greater the shrink.

  4. 2.3 Horizontal Stretching Horizontal stretching of a graph of a function If 0 < c < 1, then the graph of y = f (cx ) is a horizontal stretching of the graph of y = f ( x ). In general, smaller the value of c, greater the stretch. Horizontal stretching changes x - intercepts but not the y - intercept

  5. 2.3 Horizontal Shrinking Horizontal shrinking of a graph of a function If c > 1, then the graph of y = f (cx ) is a horizontal shrinking of the graph of y = f ( x ). In general, larger the value of c, greater the shrink Horizontal shrinking changes x - intercepts but not the y - intercept

  6. 2.3 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.

  7. 2.3 Example of Reflection Given the graph of sketch the graph of (a)(b) Solution (a) (b)

  8. 2.3 Reflection with the Graphing Calculator

  9. reflect across the x-axis shift 5 units up shift 4 units right vertical stretch by a factor of 3 2.3 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.

  10. Graphs:

  11. 2.3 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.

  12. 2.3 Transformations on a Calculator- Generated Graph Example The figures show two views of the graph and another graph illustrating a combination of transformations. Find the equation of the transformed graph. Solution The first view indicates the lowest point is (3,–2), a shift 3 units to the right and 2 units down. The second view shows the point (4,1) on the graph of the transformation. Thus, the slope of the ray is Thus, the equation of the transformed graph is First View Second View

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