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Analysis of Graphs: Symmetry, Shifts, Stretching, Shrinking, and Reflection

In this chapter, we explore the various transformations that can be applied to graphs of functions. This includes horizontal and vertical shifts, stretching and shrinking, reflecting graphs across axes, and combining multiple transformations. We also discuss the importance of the order in which transformations are applied and caution in translations. Examples and calculations are provided to illustrate these concepts.

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Analysis of Graphs: Symmetry, Shifts, Stretching, Shrinking, and Reflection

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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 2.5 Piecewise-Defined Functions 2.6 Operations and Composition

  2. Vertical Stretching of the Graph of a Function If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical stretching of the graph of by applying a factor of c. 2.3 Vertical Stretching

  3. 2.3 Vertical Shrinking Vertical Shrinking of the Graph of a Function If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical shrinking of the graph of by applying a factor of c.

  4. 2.3 Horizontal Stretching and Shrinking • Horizontal Stretching and Shrinking of the Graph of a Function • If a point lies on the graph of then the point lies on the graph of • If then the graph of is a horizontal stretching of the graph of • (b) If then the graph of is a horizontal shrinking of the graph of

  5. 2.3 Reflecting Across an Axis Reflecting the Graph of a Function Across an Axis For a function defined by the following are true. (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.

  6. 2.3 Example of Reflection Given the graph of sketch the graph of (a)(b) Solution (a) (b)

  7. 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.

  8. Graphs:

  9. 2.3 Caution in Translations of Graphs • The order in which transformations are made can be important. Changing the order of a stretch and shift can result in a different equation and graph. • 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.

  10. 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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