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

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

Identifying Parent Functions

Identifying Parent Functions • • • 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

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

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

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

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

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

Square Root Function –

f



x

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

Reciprocal Function –

f

 1

x

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

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

Identifying Parent Functions (Example)

Ex 1:

Identify the parent function: a)

f x

 5  3 c)

h

    8   2  5 e)

q

 2

x

 5 b)

g

 9 2

x

 2 d)

p

   15

x

 7 f)

r

 6 2

x

 3  1 13

Graphing Piecewise Functions

Graphing Piecewise Functions • • 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 sketching h(t) = 7t

h

    

t

2  7 2

t t

,   3 1 , ,

t t

2    2 4

t

 4 15 h(t) = 2t – 1

Graphing Piecewise Functions (Example)

Ex 2:

Graph the piecewise function: a)

f

   

x

2

x

2

x

   3 2 , 1 ,  ,

x

1

x

   

x

3 1  3 b)

g

  3  

x x

  3 ,

x

1 ,

x

 1  1 16

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