Precalc0103to0104

# Precalc0103to0104 - (Section 1.3: Basic Graphs and...

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(Section 1.3: Basic Graphs and Symmetry) 1.3.1 SECTION 1.3: BASIC GRAPHS and SYMMETRY LEARNING OBJECTIVES • Know how to graph basic functions. • Organize categories of basic graphs and recognize common properties, such as symmetry. • Identify which basic functions are even / odd / neither and relate this to symmetry in their graphs. PART A: DISCUSSION • We will need to know the basic functions and graphs in this section without resorting to point-plotting. • To help us remember them, we will organize them into categories. What are the similarities and differences within and between categories, particularly with respect to shape and symmetry in graphs? (We will revisit symmetry in Section 1.4 and especially in Section 1.7.) • A power function f has a rule of the form fx () = x n , where the exponent or power n is a real number. • We will consider graphs of all power functions with integer powers, and some power functions with non-integer powers. • In the next few sections, we will manipulate and combine these building blocks to form a wide variety of functions and graphs.

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(Section 1.3: Basic Graphs and Symmetry) 1.3.2 PART B: CONSTANT FUNCTIONS If fx () = c , where c is a real number, then f is a constant function . • Any real input yields the same output, c . If = 3 , for example, we have the input-output model and the flat graph of y = 3 , a horizontal line, below. PART C: IDENTITY FUNCTIONS If = x , then f is an identity function . • Its output is identical to its input. 6 ± f ± 6 ± 10 ² f ²± 10 • There are technically different identity functions on different domains. The graph of y = x is the line below.
(Section 1.3: Basic Graphs and Symmetry) 1.3.3 PART D: LINEAR FUNCTIONS If fx () = mx + b , where m and b are real numbers, and m ± 0 , then f is a linear function . In Section 0.14, we graphed y = mx + b as a line with slope m and y -intercept b . If = 2 x ± 1 , for example, we graph the line with slope 2 and y -intercept ± 1 . PART E: SQUARING FUNCTION and EVEN FUNCTIONS Let = x 2 . We will construct a table and graph f . x Point x Point 0 0 0, 0 0 0 0, 0 1 1 1, 1 ± 1 1 ± 2 4 2, 4 ± 2 4 ± 2, 4 3 9 3, 9 ± 3 9 ±

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(Section 1.3: Basic Graphs and Symmetry) 1.3.4 TIP 1 : The graph never falls below the x -axis, because squares of real numbers are never negative . Look at the table. Each pair of opposite x values yields a common function value fx () , or y . Graphically , this means that every point x , y on the graph has a “mirror image partner” ± x , y that is also on the graph. These “mirror image pairs” are symmetric about the y -axis . • We say that f is an even function . (Why?) A function f is even ± f ² x = , ³ x ´ Dom f ± The graph of y = is symmetric about the y- axis . Example 1 (Even Function: Proof) Let = x 2 . Prove that f is an even function. § Solution Dom f = ± . ± x ² ± , f ± x = ± x 2 = x 2 = Q.E.D. (Latin: Quod Erat Demonstrandum) • This signifies the end of a proof. It means “that which was to have been proven, shown, or demonstrated.” TIP 2 : Think: If we replace x with ± x as the input, we obtain equivalent outputs . §
(Section 1.3: Basic Graphs and Symmetry) 1.3.5

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## This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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Precalc0103to0104 - (Section 1.3: Basic Graphs and...

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