Presentation on theme: "Learning Objectives for Section 2.2"— Presentation transcript:
1Learning Objectives for Section 2.2 MAT SPRING 2009Learning Objectives for Section 2.2Elementary Functions; Graphs and TransformationsYou 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.
2Six Graphs of Common Functions ___________ Function ___________ FunctionDomain: _________ Domain: _________Range: _________ Range: _________
3Six Graphs of Common Functions (continued) Domain: _________Range: _________
4Six Graphs of Common Functions (continued) Domain: _________Range: _________
5Vertical and Horizontal Translations MAT SPRING 2009Vertical and Horizontal TranslationsTRANSLATIONS, 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.
6Vertical 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).
7Vertical 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:
8Transformations of f(x) = x2 MAT SPRING 2009Transformations of f(x) = x2GivenDescribe each transformation of the graph of y1.
9Vertical 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 – 5Solution:You can obtain the graph of g by translating f 3 units ___________and 5 units ____________________.f(x)g(x)
10ExampleWrite the resulting function if the graph of y = x3 is shifted 5 units to the right and 8 units upward.
11Reflecting GraphsA REFLECTION is a mirror image of the graph in a certain line.Reflection in the x-axis: h(x) = f(x)
12Reflecting Graphs (continued) Take a look*Sketch the graph of andSolution:You can obtain the graph of g by reflecting f in the x-axis.Parent Functiong(x)f(x)Reflection in x-axis
13ExampleFirst 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)
15Nonrigid 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.
16Vertical 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.
17ExamplesFirst 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)
19Summary of Graph Transformations MAT SPRING 2009Summary of Graph TransformationsSee Handout:Transformations of the graph of y = f(x)
20Extra Practice: Domain, Range, and Transformations of Common Function More PracticeSee Handout:Extra Practice: Domain, Range, and Transformations of Common Function
21Piecewise-Defined Functions MAT SPRING 2009Piecewise-Defined FunctionsThe absolute value of a real number x can be defined asNotice 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.
22Example of a Piecewise-Defined Function MAT SPRING 2009Example of a Piecewise-Defined FunctionGraph the functionxy