1.2 Basic Classes of Functions
MD Calculus - Chapter 1: Functions and Graphs
1.
2.
3.
4.
5.
6.
Calculate and interpret slope
linearfunctions
Identify the degree of a polynomial
fHe axts
Find roots of a quadratic
Describe odd and even polynomials
Identify a rational function
Where a, b are constants
Describe the graphs of power and root
functions
When ! > 0, $(&) is increasing
7. Explain the difference between algebraic and
When ! < 0, $(&) is decreasing
transcendental functions
When ! = 0, $(&) is constant
8. Graph a piecewise function
TAI
Slope M
9. Sketch a transformed graph
f
slope interceptform
y mxtb
0 b yintercept
Point slopeform
xz x
standardform
y y mix x I
AxtBy
C
allowsforvertical
lines
x y apointon tu ein
Example 12: Consider the line passing through the points (11, -40 and (-4, 5)
a. Find the slope of the line
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Is
b. Find an equation for this linear function in
point-slope form
y s zcx 4
c. Find an equation in slope-intercept form
y 5 34 41
yS
I
s
y 3 x Eg Es
1,0
5 3
78minutes
Examples 13: Jessica leaves her house at 5:50 am and goes for a 9-mile run. She returns to her house at 7:08
am. Assume Jessica runs at a constant pace.
a. Describe the distance D (in miles) Jessica runs
as a linear function of her run time t (in
minutes).
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b. Sketch a graph of D.
d
D
c. Interpret the meaning of the slope.
q
Shemoves 3 miles ever 26 minutes
I
78
t
For integers * ≥ 0 and constants !! , !!"# , … !$ , !# , !% , a polynomial is of the form:
Degree
fx
anxhtan.ph
azx2t ai tao
n 1 linear
n z quadratic
leading
n 3 Cubic
Coefficent
Power Functions
even
odd
end
behavior
2eros roots
fly axb
fl
x
fl
X
_fix
fix
symmetricalacross
y axis
symmetrical aboutorigin
How $(&) behaves as & → ∞ and & → −∞
Where the graph intersects the x-axis
Example 14: For the following functions,
i.
describe the behavior of $(&) as & → ±∞
ii.
Find all zeros of f.
iii.
Sketch a graph of f.
a. $(&) = −2& $ + 4& − 1
b. $(&) = & & − 3& $ − 4&
neg even
xta
x
x
positive odd
3 3 2
1
454744
X
41
1
2 3
4X
4
XCX4 xxl
4
02929,0707
X
o
a
Example 15 (modified with easier numbers): A company is interested in predicting the amount of revenue it
will receive depending on the price it charges for a particular item. The company arrives at the following
quadratic function to model revenue R (in thousands of dollars) as a function of price per item p:
:(;) = ; · (−; + 26) = −;$ + 26; for 0 ≤ ; ≤ 26.
a. Predict the revenue if the company sells the
item at a price of p = $5.
Pyles
45
zs
t 2615
oo
iosg
b. Find the zeros of this function and interpret
the meaning of the zeros.
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c. Sketch a graph of R.
d. Use the graph to determine the value of p
that maximizes revenue. Find the maximum
revenue.
R
The Marx is tee vertex
p
B
13ft 2643
R 13
169 338
o
26
price
fG
Rational Function
PCIe
gets
169thosanddollais
Where ;(&), ?(&) are polynomials and ?(&) ≠ 0
it
CG
Root Function
&'"#
Example 16: Find the domain and range of $(&) = (')$
5
2
Fo
Rs X
D x txt 215
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f
x
R ugly1315
(skipped example b: $(&) = √4 − & $ )
Example 17: For each function, determine the domain.
2
&
a. $(&) = ' ! "#
110
$')(
1 ex D4
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Il
d. $(&) = √2& − 1
4 3 20
372 4
14 3
2413
2
2
41 0
1 41
"
c. $(&) = √4 − 3&
11
b. $(&) = &' ! )*
11
Transcendental functions
Functions that can’t be described by basic algebraic operations
not
peondag
Extrigexponentials logs
Example 18: Classify each function as algebraic or transcendental.
a) $(&) =
b) $(&) = 2'
√' " )#
*')$
algebra
!
c) $(&) = sin(2&)
exponential strascendenta
trig
transcendental
rhorizontalshift
Transformations
a
fCb X h
n
r
Vertical Horizontal
Stretch Stretch
compress
leavertical
compress
Example 21: Sketch each graph using a sequence of transformations of a well-known function.
a) $(&) = −|& + 2| − 3
a
1
h k Cz 3
b
b) $(&) = 3√−& + 1
a 3
bi l
h k Co 1
s
shift
Example 19: Sketch a graph of the piecewise-defined
function:
$(&) = E
& + 3,
(& − 2)$ ,
&<1
&≥1
Example 20: In a big city, drivers are charged variable rates for parking in a parking garage. They are charged
$10 for the first hour or any part of the first hour and an additional $2 for each hour or part thereof up to a
maximum of $30 for the day. The parking garage is open from 6 a.m. to 12 midnight.
a. Write a piecewise-defined function that describes the cost C to park in the parking garage as a function
of hours parked x.
b. Sketch a graph of this function C(x).
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iz
Cx
µ
30
oaxel
taxes
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102 118
n
30
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zu
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18
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