1 § Elementary Functions: Graphs and Transformations The student will learn about: functions, domain, range, transformations. a “Library of Elementary Functions”, and

22 Definition of a Function Example – Each student in this class has a York College identification number. Def. – a function f is a rule that assigns to each number x in a set a number f (x). That is, for each element in the first set there corresponds one and only one element in the second set. The first set is called the domain, and the set of corresponding elements in the second set is called the range. That’s one – not two, not three, but one, 1, uno, un, no more no less but exactly one.

33 Functions Defined by Equations 2x – y = 1 x 2 + y 2 = 25 (Not a function – graph it.) Note if x = 3 then y = both 4 and – 4. Finding the domain: Eliminate square roots of negatives Eliminate division by zero

4 A Beginning Library of Elementary Functions The identity function;f (x) = x

5 A Beginning Library of Elementary Functions The second degree function; h (x) = x 2

6 A Beginning Library of Elementary Functions The cube function;m (x) = x 3

7 A Beginning Library of Elementary Functions The square root function; n (x) = √x h (x) = x 2

8 A Beginning Library of Elementary Functions The cube root function;p (x) = 3 √x m (x) = x 3

9 A Beginning Library of Elementary Functions The absolute value function; g (x) = |x|

10 A Beginning Library of Elementary Functions The sine function; g (x) = sin x

11 A Beginning Library of Elementary Functions The cosine function; g (x) = cos x

12 A Beginning Library of Elementary Functions The tangent function; g (x) = tan x Etc!

13 Vertical Shifts In the following discussions it is assumed that f (x) is from the previous library although the discussion is true for all functions. The graph of y = f (x) + k is the same as that of y = f (x) except it is shifted vertically k units. If k is positive the shift is up and if k is negative the shift is down.

14 Vertical Shifts y = x 2 y = x 2 + 3

15 Horizontal Shifts In the following discussions it is assumed that f (x) is from the previous library although the discussion is true for all functions. The graph of y = f (x + h) is the same as that of y = f (x) except it is shifted horizontally – h units. I.e. if h is positive the shift is to the left and if h is negative the shift is to the right.

16 Horizontal Shifts y = x 2 y = (x + 3) 2

17 Reflections, Expansions, and Contractions In the following discussions it is assumed that f (x) is from the previous library although the discussion is true for all functions. Consider the graph of y = A f (x) in comparison to y = f (x): there are three cases.

18 Reflections y = x 2 y = - x 2 Case 1: If A is negative than y = A f (x) is an x axis reflection of y = f (x).

19 Contractions y = x 2 y = ⅓ x 2 Case 2: If | A | is less than 1 then y = A f (x) looks like y = f (x) but has a vertical contraction (squish) by multiplying each y value by A. 1 1

20 Expansions y = x 2 y = 3x 2 Case 3: If | A | is greater than 1 then y = A f (x) looks like y = f (x) but has a vertical expansion (stretch) by multiplying each y value by A. 1 3

21 Summary Vertical Translation: y = f (x) + kk > 0 shifts f (x) up k units. k < 0 shifts f (x) down k units. Horizontal Translation: y = f (x + h)h > 0 shifts f (x) left h units. h < 0 shifts f (x) right h units. Reflection: y = - f (x) Reflect the graph of f (x) in the x axis. Vertical Expansion/Contraction: y = A f (x)A > 1 Expands f (x) vertically by a multiple of A. 0 < A < 1 Contracts f (x) vertically by a multiple of A.

22 Combined Transformations The above transformations may be combined. y = A f(x + h) + k y = 2 (x – 3) y = x 2 }

23 Summary. We were shown some basic “library” functions that will be used in the text. We saw the transformations that can be applied to these functions to create more interesting functions.

24 ASSIGNMENT § 1-1.5; Handout