1. Graphs of Functions Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 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

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 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)

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 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

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 decreasing on an interval if, for any x 1 and x 2 in the interval, x 1 f (x 2 ), constant on an interval if, for any x 1 and x 2 in the interval, f (x 1 ) = f (x 2 ). The graph of y = f (x): increases on ( – ∞, – 3), decreases on ( – 3, 3), increases on (3, ∞). Increasing, Decreasing, and Constant Functions A function f is: increasing on an interval if, for any x 1 and x 2 in the interval, x 1 < x 2 implies f (x 1 ) < f (x 2 ), (3, – 4) x y ( – 3, 6) –2–2 2

6. Increasing/Decreasing Intervals

7. Calculator Mode Screen Keep your calculator in degree mode until we get to trigonometry.

8. Standard Window Screen This is where you change the size of your graph. Xmin is the leftmost point on your graph. Xmax is the rightmost point on your graph. Xscl is the scale you are using on your x-axis. Ymin is the lowest point on your graph. Ymax is the highest point on your graph. Yscl is the scale that you are using on your y-axis.

9. The Math Menu

10. Full, Horizontal, G-T Options This alteration is made in the MODE options.

11. Grids FORMAT is located in the text above the ZOOM button located at the top center of your calculator.

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Minimum and Maximum Values A function value f(a) is called a relative minimum of f if there is an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). x y A function value f(a) is called a relative maximum of f if there is an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). Relative minimum Relative maximum

13. Relative Max and Min

14. Relative Max/Min

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Graphing Utility: Approximating a Relative Minimum Graphing Utility: Approximate the relative minimum of the function f(x) = 3x 2 – 2x – 1. – 6 6 6

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 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

17. Piecewise Functions TEST is located in the text right above the MATH button on the left side of the calculator.

18. Piecewise Functions

20. Greatest Integer Function

21. Greatest Integer Functions

22. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 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. Symmetric with respect to the y-axis.

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 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. Symmetric with respect to the origin.

24. Even/Odd Functions Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24

25. Even/Odd Functions Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25

26. Even/Odd Function Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26