Transcript Thinking Mathematically by Robert Blitzer

Unit 1 Functions and Graphs
1.1: Modeling and Equation Solving
Focus:
Factor completely.
2
25r  16
= ( 5r )2 – ( 4 )2
2
What goes here?
25x  60x  36
= ( 5x )2 + 2(5x)( -6 ) + (-6 )2
What goes here? What goes here?
What goes here?
2
= ( 5r + 4 )( 5r – 4 )
7x  28x
2
 7x(x  4)
3
 7x(x  2)(x  2)
= ( 5x +-6 )
= ( 5x – 6 )2
2x  11x  5
2
 (2x  1)(x  5)
A Numerical Model is the most basic kind of model
in which numbers (or data) are analyzed to gain
insights into phenomena.
U.S. Prison Population (Thousands)
%
Male
%
Female
Year
Total
Male Female
1980
1985
316
480
304
459
12
21
96.2
95.6
3.8
4.4
1990
1995
2000
740
1085
1382
699
1021
1290
41
64
92
94.5
94.1
93.3
5.5
5.9
6.7
Is the proportion
of female prisoners
over the years
increasing?
Yes
An Algebraic Model uses formulas to relate variable
quantities associated with the phenomena being studied.
A pizzeria sells a rectangular 18” by 24” pizza for the
same price as its large round pizza (24” diameter). If both
pizzas have the same thickness, which option gives the
most pizza for the money?
Solution:
We need to compare the areas of the pizzas.
Rectangular pizza:
Round pizza:
A = lw
A = r2
= (18in)(24in)
= (12in)2
= 432 in2
= 144in2
= 452.4 in2
The round pizza is larger and therefore gives more for the money.
You are choosing between two long-distance telephone
plans. Plan A has a monthly fee of $20 with a charge of
$0.05 per minute for all long-distance calls. Plan B has
a monthly fee of $5 with a charge of $0.10 per minute
for all long-distance calls.
Express the monthly cost for plan A, f, as a function of the
number of minutes of long-distance calls in a month, x.
f(x) = 20 + 0.05x
Express the monthly cost for plan B, g, as a function of the
number of minutes of long-distance calls in a month, x.
g(x) = 5 + 0.10x
For how many minutes of long-distance calls will the
costs for the two plans be the same?
We are interested in how many minutes of long-distance
calls, x, result in the same monthly costs, f and g, for the
two plans. Thus, we must set the equations for f and g
equal to each other. We then solve the resulting linear
equation for x.
0.05x + 20 = 0.10x + 5
-0.05x - 5 - 0.05x - 5
15 = 0.05x
0.05 0.05
300 minutes = x
A Graphical Model is a visual representation of a
numerical model or an algebraic model that gives insight
into the relationships between variable quantities.
From the data table of prison populations, let t be the number of
years after 1980 and let F be the percentage of females in the
prison population from year 0 to year 20. Create a scatter plot
of the data.
F
t
F
0 3.8
5 4.4
t
10
5.5
15
5.9
20
6.7
Understanding the Viewing Rectangle
20
15
10
5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
-5
-10
[-2, 3] by [-10, 20]
x min
x max
y min
y max
2.5
3
Complete Student Checkpoint
Choose the correct viewing rectangle and label the tick
marks.
[-8,10] by [-8,16]
[-8,12] by [-8,16]
16
12
8
4
-8 -6 -4 -2
-4
-8
2
4
6
8 10 12
Day 2
From the data table of prison populations, let t be the number of
years after 1980 and let F be the percentage of females in the
prison population from year 0 to year 20. Create a scatter plot
of the data.
F
t
F
0 3.8
5 4.4
Using two coordinates we
can write the equation.
t
10
5.5
15
5.9
20
6.7
This pattern looks linear. Use a line of best fit to to find an
algebraic model by finding the equation of the line.
Use the point-slope formula and calculate the slope
from the two coordinates (0,3.8) and (20,6.7)
y  y1  m(x  x1 )
6.7  3.8
y  (3.8) 
(x  (0))
20  (0)
y  3.8  0.145x
3.8
3.8
y  0.145x  3.8
This does a very nice job of modeling the data.
Solving an equation algebraically.
Find all real numbers x for which
6x  11x  10x
2
2
11x  10x 11x  10x
3
6x  11x  10x  0
2
x 6x  11x  10  0
3

2
3x
2

x 2x  5 3x  2   0
-60
-15
4
-11
x  0 or 2x  5  0 or 3x  2  0
2
5
x  0 or x  or x  
3
2
2
2x 6x2 4x
-5 -15x -10
Solve the equation algebraically and graphically.
x  10  4x
10  4x 10  4x
2
x 2  4x  10  0
(4)  (4)2  4(1)(10)
x
2(1)
4  56
x
2
x  1.74 or x  5.74
and graphically
x 2  4x  10  y
x  1.74 or x  5.74
Solve the equation algebraically and graphically.
x  x 1
x x
(3)  (3)2  4(1)(1)
x
2(1)
x  1 x
 
x
2
3 5
x
2
 1  x 
2
x  1  2x  x 2
x
x
2
0  x  3x  1
and graphically
make right side =0
x  2.62 or x  0.38
x  x 1
x  x 1 0
x  x 1 y
Looking at the
graph, this
is the only
x-intercept,
zero or root
Grapher Failure
Graph the equation
y  3 / (2x  5)
The graph never intercepts the x-axis. Why?
3
y cannot
0
equal zero
2x  5
0 2x  5   3
03
Where is the graph undefined?
x  2.5
3
y
when denominator equals 0
2x  5
2x  5  0
Modeling and
Equation Solving