1.3 Graphs of Functions 2015 Digital Lesson

Warm-up/ Quiz Practice Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

Slide Copyright © 2010 Pearson Education, Inc. Closed and Open Intervals When a set includes the endpoints, the interval is a closed interval and brackets are used. When a set does not include the endpoints, the interval is an open interval and parentheses are used.

Slide Copyright © 2010 Pearson Education, Inc. Union Symbol An inequality in the form x 3 indicates the set of real numbers that are either less than 1 or greater than 3. The union symbol U can be used to write this inequality in interval notation as

Precalculus 1.3 Graphs of Functions Objectives Determine intervals on which functions are increasing, decreasing, or constant. Determine relative maximum and relative minimum values of functions. Identify and graph piecewise-defined functions. Identify even and odd functions.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 The graph of a function f is the collection of ordered pairs (x, f(x)) where x is in the domain of f. Definition of Graph x y 4 -4 (2, –2) is on the graph of f(x) = (x – 1) 2 – 3. (2, –2) f(2) = (2 – 1) 2 – 3 = 1 2 – 3 = – 2

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 x y 4 -4 The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists. The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain. Domain Range Domain & Range

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–8 Illustration of Domain and Range.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 x y – 1 1 Example: Find the domain and range of the function f (x) = from its graph. The domain is [–3,∞). The range is [0,∞). Range Domain Example: Domain & Range (–3, 0)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 x y 4 -4 Vertical Line Test A relation is a function if no vertical line intersects its graph in more than one point. Vertical Line Test This graph does not pass the vertical line test. It is not a function. This graph passes the vertical line test. It is a function. y = x – 1 x = | y – 2| x y 4 -4

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 decreasing on an interval if, as x values increase, y values decrease. constant on an interval if, as x values increase, y values remain constant. The graph of y = f (x): increases on (– ∞, –2), decreases on (–2, 1.5), increases on (1.5, ∞). Increasing, Decreasing, and Constant Functions A function f is: increasing on an interval if, as x values increase, y values increase. (1.5, – 2) x y ( – 2, 3) –2–2 2

Slide Copyright © 2010 Pearson Education, Inc. Increasing, Decreasing, and Endpoints The concepts of increasing and decreasing apply only to intervals of the real number line and NOT to individual points. Do NOT say that the function f both increases and decreases at the point (0, 0). Decreasing: (–∞, 0) Increasing: (0, ∞)

Slide Copyright © 2010 Pearson Education, Inc. Ex 2: Determining intervals of increase or decrease Ex 2: Determining intervals of increase or decrease Use the graph of Decreasing: and interval notation to identify where f is increasing or decreasing. Solution Increasing:

You try: find the intervals on which the function increases, decreases, or is constant. Remember that intervals are x-values. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 increases on (– 5, –3), (-1,1), (4, 5) decreases on (– ∞, -5), (1,4) Constant on (–3, -1), (5, ∞)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Minimum and Maximum Values A function value f(a) is called a relative minimum of f if it is less than all other function values (y values) nearby. x y A function value f(a) is called a relative maximum of f if it is greater than all other function values (y values) nearby. Relative minimum Relative maximum

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–16 Section 1.3, Figure 1.24, Illustration of Definition of Relative Minimum and Relative Maximum, pg. 33

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Graphing Utility: Approximating a Relative Minimum Graphing Utility: Approximate the relative minimum of the function f(x) = 3x 2 – 2x – 1. – 6 6 6

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 x y 4 -4 A piecewise-defined function is composed of two or more functions. Piecewise-Defined Functions f(x) = 3 + x, x < 0 x 2 + 1, x 0 Use when the value of x is less than 0. Use when the value of x is greater or equal to 0. (0 is not included.) open circle (0 is included.) closed circle

Let’s try one together Sketch the graph of

You try: Sketch the graph of

Even and Odd Functions Some functions are even, some are odd, and some are neither. You can determine if a function is even, odd, or neither with an algebraic test, or by looking at its graph. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–22 Symmetry of Graphs

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 A function f is even if for each x in the domain of f, f (– x) = f (x). Even Functions x y f (x) = x 2 f (– x) = (– x) 2 = x 2 = f (x) f (x) = x 2 is an even function. For polynomials, every term in an even function will have an even power. Constants are ok, they are zero power. Symmetric with respect to the y-axis.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 A function f is odd if for each x in the domain of f, f (– x) = – f (x). Odd Functions x y f (x) = x 3 f (– x) = (– x) 3 = –x 3 = – f (x) f (x) = x 3 is an odd function. For polynomials, every term in an odd function will have an odd power, Constants are not ok. Symmetric with respect to the origin.

Example Are the following functions even, odd, or neither? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25

Homework Pg. 38: odd, 29, 45, 47, odd, 109 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26

HWQ State intervals of increase and decrease on the graph of. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27