Objectives
 To find the difference quotient.
 Understand and use piecewise functions
 Identify intervals on which a function increases, decreases, or is constant.
 Use graphs to locate relative maxima or minima.
 Identify even or odd functions & recognize the symmetries.
 Graph step functions.

REVIEW of Lesson 1.2
Reminder: Domain Restrictions
For FRACTIONS:
 No zero in denominator!
ex .
7
0
 undefined
For EVEN ROOTS:
 No negative under even root!
 2 , 4 x  2

Find the domain of each (algebraically)and write in interval notation.
a ) ( ) | 3 x

2 | b ) ( )

1 x

4
) ( )
 x

7 d ) ( )

7
 x

Functions & Difference Quotients
 Useful in discussing the rate of change of function over a period of time
 EXTREMELY important in calculus
(h represents the difference in two x values)
 DIFFERENCE QUOTIENT FORMULA: f ( x
 h )
 f ( x ) h

Difference Quotient
The average rate of change (the slope of the secant line)

If f(x) = -2x 2 + x + 5, find and simplify each expression.
 A) f(x+h)

If f(x) = -2x 2 + x + 5, find and
 simplify each expression.
B) f ( x
 h )
 f ( x ) h

Your turn: Find the difference quotient: f(x) = 2x 2 – 2x + 1 f x
 h )

2( x
 h )
2

2( x h ) 1 f x
 h )

2( x
2 
2 xh
 h
2 
2 x

2 h

1 f x
 h )

2 x
2 
4 xh

2 h
2 
2 x

2 h

1 f x h h
) f x

2 x
2 
4 xh

2 h
2 
2 x
 h
2 h
  x
2 
2 x

1)

PIECEWISE FUNCTIONS
 Piecewise function – A function that is defined differently for different parts of the domain; a function composed of different “pieces”
Note: Each piece is like a separate function with its own domain values.
 Examples: You are paid $10/hr for work up to
40 hrs/wk and then time and a half for overtime.

10 ,

40
15 ,

40



 Check Point 2:
Use the function


20 if 0
 t

60
60) if t

60 to find and interpret each of the folllowing:
 a) C(40) b) C(80)

 Draw the first graph on the coordinate plane.
 Be sure to note where the inequality starts and stops. (the interval)

Erase any part of the graph that isn’t within that interval.

See p.1015
for more problems.

10

9

8

7

6

5

4

3

2


1

6

7

8

9

10

2

3

4

5
  y
3 x x

10
9
8
7
6
5
3
4
3
2
1
1 if x if x


1
1
2 3 4 5 6 7 8 9 10 x

 A function is described by intervals, using its domain, in terms of x-values.
 Remember:

refers to "positive infinity"

refers to "negative infinity"

Increasing and Decreasing
Functions
 Increasing : Graph goes “up” as you move from left to right.
x
1
 x
2
, f ( x
1
)
 f ( x
2
)
 Decreasing : Graph goes “down” as you move from left to right.
x
1
 x
2
, f ( x
1
)
 f ( x
2
)
 Constant : Graph remains horizontal as you move from left to right.
x
1
 x
2
, f ( x
1
)
 f ( x
2
)

 Check Point 1 – See middle of page 166.

Find the Intervals on the Domain in which the
Function is Increasing, Decreasing, and/or
Constant

Relative Maxima and Minima
 based on “y” values

 maximum – “peak” or highest value minimum – “valley” or lowest value

Relative Maxima and Relative
Minima

Even & Odd Functions & Symmetry
 Even functions are those that are mirrored through the y-axis. (If –x replaces x, the y value remains the same.) (i.e. 1 st quadrant reflects into the
2 nd quadrant)
 Odd functions are those that are mirrored through the origin. (If –x replaces x, the y value becomes –y.) (i.e. 1 st quadrant reflects into the 3 rd quadrant or over the origin )

 Determine whether each function is even, odd, or neither.
a) f(x) = x 2 + 6 b) g(x) = 7x 3 - x

Determine whether each function is even, odd, or neither.
c) h(x) = x 5 + 1

c) a) b)
Your turn: Determine if the function is even, odd, or neither.
f ( x )

2 ( x

4 )
2 
2 x
2
Even
Odd
Neither