Learning Objectives for Section 2.2 MAT 103 SPRING 2009 Learning Objectives for Section 2.2 Elementary Functions; Graphs and Transformations You will become familiar with some elementary functions. You will be able to transform functions using vertical and horizontal shifts. You will be able to transform functions using stretches and shrinks. You will be able to graph piecewise-defined functions.

Six Graphs of Common Functions ___________ Function ___________ Function Domain: _________ Domain: _________ Range: _________ Range: _________

Six Graphs of Common Functions (continued) Domain: _________ Range: _________

Six Graphs of Common Functions (continued) Domain: _________ Range: _________

Vertical and Horizontal Translations MAT 103 SPRING 2009 Vertical and Horizontal Translations TRANSLATIONS, also called shifts, are simple transformations of the graph of a function whereby each point of the graph is translated (shifted) a certain number of units vertically and/or horizontally. The shape of the graph remains the same. Vertical translations are shifts upward or downward. Horizontal translations are shifts to the right or to the left.

Vertical and Horizontal Translations Example Graph the function on your calculator and describe the transformation that the graph of must undergo to obtain the graph of h(x).

Vertical and Horizontal Translations Vertical and Horizontal Translations of the Graph of y = f(x) For h, k positive real numbers: Vertical shift k units UPWARD: Vertical shift k units DOWNWARD: Horizontal shift h units to the RIGHT: Horizontal shift h units to the LEFT:

Transformations of f(x) = x2 MAT 103 SPRING 2009 Transformations of f(x) = x2 Given Describe each transformation of the graph of y1.

Vertical and Horizontal Translations (continued) Take a look* Name the transformations that f(x) = x2 must undergo to obtain the graph of g(x) = (x + 3)2 – 5 Solution: You can obtain the graph of g by translating f 3 units ___________ and 5 units ____________________. f(x) g(x)

Example Write the resulting function if the graph of y = x3 is shifted 5 units to the right and 8 units upward.

Reflecting Graphs A REFLECTION is a mirror image of the graph in a certain line. Reflection in the x-axis: h(x) =  f(x)

Reflecting Graphs (continued) Take a look* Sketch the graph of and . Solution: You can obtain the graph of g by reflecting f in the x-axis. Parent Function g(x) f(x) Reflection in x-axis

Example First name the parent function (one of the six common graphs). Then, describe the transformations of each of the following graphs as compared to the graph of its parent function. Then sketch the graph of the transformed function. a)

MAT 103 SPRING 2009 b) c)

Nonrigid Transformations Translations and reflections are called rigid transformations because the basic shape of the graph is NOT changed. Nonrigid transformations cause the original shape of the graph to change or become distorted. We will look briefly at two nonrigid transformations: VERTICAL STRETCHES AND SHRINKS.

Vertical Stretch and Shrink A VERTICAL STRETCH causes the graph to become more elongated (skinnier) A VERTICAL SHRINK causes the graph to become squattier (wider). For y = f(x), y = A f(x) is a vertical stretch if A > 1. y = A f(x) is a vertical shrink if 0 < A < 1. Example: Notice that the vertex of the parabola does not change; the graph just becomes narrower or wider depending upon the value of A.

Examples First name the parent function (one of the six common graphs). Then describe the transformations of each of the following graphs as compared to the graph of its parent function. Then sketch the graph of the transformed function. a)

MAT 103 SPRING 2009 b) c)

Summary of Graph Transformations MAT 103 SPRING 2009 Summary of Graph Transformations See Handout: Transformations of the graph of y = f(x)

Extra Practice: Domain, Range, and Transformations of Common Function More Practice See Handout: Extra Practice: Domain, Range, and Transformations of Common Function

Piecewise-Defined Functions MAT 103 SPRING 2009 Piecewise-Defined Functions The absolute value of a real number x can be defined as Notice that this function is defined by different rules for different parts of its domain. Functions whose definitions involve more than one rule are called piecewise-defined functions. Graphing one of these functions involves graphing each rule over the appropriate portion of the domain.

Example of a Piecewise-Defined Function MAT 103 SPRING 2009 Example of a Piecewise-Defined Function Graph the function x y