Transcript Angles, Degrees, and Special Triangles
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