Unit 2: “Graph-itti!” • Lesson 1: 2.1 Symmetry (3-1) • Lesson 2: 2.2 Graph Families (3-2, 3-3) • Lesson 3: 2.3 Inverses (3-4) • Lesson 4: 2.4 Continuity (3-5) • Lesson 5: 2.5 Extrema (3-6) • Lesson 6: 2.6 Rational Functions (3-7)
Unit Two: “Graph-itti” In this unit we will learn… • STANDARD 2.1: use algebraic tests to determine symmetry in graphs, including even-odd tests (3-1) • STANDARD 2.2: graph parent functions and perform transformations to them (3-2, 3-3) • STANDARD 2.3: determine and graph inverses of functions (3-4) • STANDARD 2.4: determine the continuity and end behavior of functions (3-5) • STANDARD 2.5: use appropriate mathematical terminology to describe the behavior of graphs (3-6) • STANDARD 2.6: graph rational functions (3-7)
STANDARD 2.1: use algebraic tests to determine symmetry in graphs, including even-odd tests (3-1) • In this lesson we will… • Discuss what symmetry is and the different types that exist. • Learn to determine symmetry in graphs. • Classify functions as even or odd.
What is Symmetry? • Point Symmetry: Symmetry about one point • Figure will spin about the point and land on itself in less than 360º.
Formal Definition: P’ M P
Symmetry to Origin: • This is the main point we look at for symmetry. • Let’s build some symmetry!
Lines We Are Interested In… • x-axis • y-axis • y = x • y = -x
Homework: • HW 2.1: P 134 #15 – 35 odd
Warm-up: • Get a piece of graph paper and a calculator. • Graph the following on separate axii:
STANDARD 2.2: graph parent functions and perform transformations to them (3-2) In this section we will… • Identify the graphs of some simple functions. • Recognize and perform transformations of simple graphs. • Sketch graphs of related functions.
Families of Graphs: • Any function based on a simple function will have the basic “look” of that family. • Multiplying, dividing, adding or subtracting from the function may move it, shrink it or stretch it but won’t change its basic shape.
Let’s do some… Send One person from your group to get a white board with a graph on it, a pen and an eraser.
STANDARD 2.2: graph parent functions and perform transformations to them (3-3) In this section we will… • Use function families to graph inequalities.
Homework: • HW1 2.2: P 143 #13-29 odd, 33 • HW2 2.2: P 150 #21-31 odd
STANDARD 2.3: determine and graph inverses of functions (3-4) In this section we will… • Determine inverses of relations and functions. • Graph functions and their inverses.
Inverse Relations: • An inverse of function will take the answers (range) from the function and give back the original domain.
Finding the inverse of a relation: • Easy!!! Just switch the domain and range! • Are they both functions?
Property of Inverse Functions: • If f(x) and f –1(x) are inverse functions, then • In other words… • Two relations are inverse relations iff one relation contains the element (b,a) whenever the other relation contains (a,b). • Does this remind you of something?
A function and its inverse… • Are reflections of each other over the line y = x.
Quick and dirty test: • If the original function passes the HORIZONTAL line test then the inverse will be a function. • Let’s check our parent graphs.
Proving Inverses: • If two functions are actually inverses then both the composites of the functions will equal x. • You must prove BOTH true.
The Handy, Dandy Build Your Own Inverse Kit: • Replace f(x) with y (it is just easier to look at this way). • Switch the x and y in the equation. • Resolve the equation for y. • The result is the inverse. • Now check!
Word Problem: • The fixed costs for manufacturing a particular stereo system are $96,000, and the variable costs are $80 per unit. • A. Write an equation that expresses the total cost C(x) as a function of x given that x units are manufactured.