Presentation on theme: "3.4 Graphs and Transformations"— Presentation transcript:
1 3.4 Graphs and Transformations Define parent functions.Transform graphs of parent functions.
2 Parent FunctionsParent functions are used to illustrate the basic shape and characteristics of various functions.The rules of transforming these functions can be applied to ANY function.
3 Parent Functions constant function identity (linear) function absolute-value function
4 greatest integer function Parent Functionsgreatest integer functionquadratic functioncubic function
5 Parent Functions reciprocal function square root function cube root function
6 Vertical Shifts Vertical shift upward c units. Vertical shift downward c units.
7 Example #1 Shifting a Graph Vertically Vertical shift 3 units up.Vertical shift 5 units down.
8 Horizontal Shifts Horizontal shift left c units. Horizontal shift right c units.
9 Example #2 Shifting a Graph Horizontally Horizontal shift 2 units right.Horizontal shift 4 units left.
10 ReflectionsReflection over the x-axis.Reflection over the y-axis.
11 Example #3 Reflecting a Graph Reflection over the x-axis.Reflection over the y-axis.
12 Vertical Stretches & Compressions Given a function with the transformation:Every point of the function is changed byIf c > 1, the graph of f is stretched vertically, away from the x-axis, by a factor of c.If c < 1, the graph of f is compressed vertically, toward the x-axis, by a factor of c.
13 Example #4 Vertical Stretches & Compressions Vertical compression by a factor ofVertical stretch by a factor of 2.
14 Horizontal Stretches & Compressions Given a function with the transformation:Every point of the function is changed byIf c > 1, the graph of f is compressed horizontally, toward the y-axis, by a factor ofIf c < 1, the graph of f is stretched horizontally, away from the y-axis, by a factor of
15 Example #5 Horizontal Stretches & Compressions Horizontal compression by a factor ofHorizontal stretch by a factor of 5 .
16 Combining Transformations If a < 0, reflect over the y-axis.Stretch or compress horizontally by a factor ofShift the graph horizontally b units left or right.If c < 0, reflect over the x-axis.Stretch or compress vertically by a factor of .Shift the graph vertically d units up or down.
17 Example #6 Combining Transformations Describe the transformations on the following functions, then graph.A.)Horizontal compression by a factor of 1/3.Shift 4 units right.Reflect over x-axis.Shift 2 units up.Apply transformations usingthe order of operations.
18 Example #6 Combining Transformations Describe the transformations on the following functions, then graph.B.)Reflection over the y-axis.Horizontal stretch by a factor of 4.Shift 4 units left.Vertical stretch by a factor of 2.Shift 4 units down.Apply transformations usingthe order of operations.