1. A Library of Parent Functions MATH 109 - Precalculus S. Rook

2. Overview Section 1.6 in the textbook: – Identifying Parent Functions – Graphing Piecewise functions 2

3. Identifying Parent Functions

4. The textbook defines the simplest form of a function as a parent function Being able to extract the parent from a complicated function is an important skill – Facilitates sketching (Section 1.7) – Allows us to predict the behavior of the complicated function What follows is a summary of the important features of important parent functions – Become very comfortable with these functions 4

5. Linear Equations versus Linear Functions In Section 1.3, we discussed linear equations before we discussed functions in 1.4 No different with function notation: – E.g. Instead of finding a linear equation that passes through (-4, 8) and (2, 1), we say find a linear function where f(-4) = 8 and f(2) = 1 Know how to do this from Section 1.3 5

6. Constant Function (Horizontal Line) f(x) = c – x-intercept: None if c ≠ 0 Infinite if c = 0 – y-intercept: (0, c) – Domain: (-oo, +oo) – Range: f(c) – Stays constant on the interval (-oo, +oo) – Even function Symmetric to the y-axis 6

7. Identity Function f(x) = x – x-intercept: (0, 0) – y-intercept: (0, 0) – Domain: (-oo, +oo) – Range: (-oo, +oo) – Increases on the interval (-oo, +oo) – Odd function Symmetric to the origin 7

8. Quadratic Function f(x) = x 2 – x-intercept: (0, 0) – y-intercept: (0, 0) – Domain: (-oo, +oo) – Range: [0, +oo) – Decreases on the interval (-oo, 0) and increases on the interval (0, +oo) – Even function Symmetric to the y-axis 8

9. Cubic Function f(x) = x 3 – x-intercept: (0, 0) – y-intercept: (0, 0) – Domain: (-oo, +oo) – Range: (-oo, +oo) – Increases on the interval (-oo, +oo) – Odd function Symmetric to the origin 9

10. Square Root Function – x-intercept: (0, 0) – y-intercept: (0, 0) – Domain: [0, +oo) – Range: [0, +oo) – Increases on the interval (0, +oo) – Neither odd nor even No symmetry 10

11. Reciprocal Function – x-intercept: None – y-intercept: None – Domain: (-oo, 0) U (0, +oo) – Range: (-oo, 0) U (0, +oo) – Decreases on the interval: (-oo, 0) U (0, +oo) – Odd function Symmetric to the origin 11

12. Absolute Value Function f(x) = |x| – x-intercept: (0, 0) – y-intercept: (0, 0) – Domain: (-oo, +oo) – Range: [0, +oo) – Decreases on the interval (-oo, 0) and increases on the interval (0, +oo) – Even function Symmetric to the y-axis 12

13. Identifying Parent Functions (Example) Ex 1: Identify the parent function: a)b) c)d) e)f) 13

14. Graphing Piecewise Functions

15. Recall that a piecewise function is comprised of multiple functions over different intervals To graph a piecewise function: – Use a table of values and/or knowledge of the shape of the function for each interval ALWAYS include the end values e.g. Use t = 2 when sketching h(t) = t 2 + 3 Use t = 2 and t = 4 when sketching h(t) = 7t Use t = 4 when sketching h(t) = 2t – 1 15

16. Graphing Piecewise Functions (Example) Ex 2: Graph the piecewise function: a) b) 16

17. Summary After studying these slides, you should be able to: – Identify the shape of the parent functions – Understand the key characteristics of the parent functions – Graph a piecewise function Additional Practice – See the list of suggested problems for 1.6 Next lesson – Transformations of Functions (Section 1.7) 17