College Algebra
J 0.4 Applications of Linear Systems
Use the Inverse to solve a Matrix Equation.
Name:_----'-_.:.......:::::...--t-
_
=
If the matrix system: AX
B has a solution,
then the solution is X A-1B
=
% of People in U.S. living
below poverty level
You will use your graphing calculator to solve each problem.
1. Finding a model for quadratic data.
The bar graph shows the percentage of people in the U.S.
living below the poverty level for selected years from 1990
through 2003. The data can be modeled with a quadratic
function. y ax2 + bx + c.
13.5
14
=
13
12
a)
We can write the data as ordered pairs (x, y) where x is
number of years after 1990 and y is percentage of people
living below poverty level. Therefore the points
corresponding to each year are:
1990:
(0,13.5)
20,00:·
(,U ~ iL3)
2002:
(12)
11
10
*---~---~--~--~
2000
1990
1-;;).
I)
(13
2003:
J
2002
l.:),G
=
b) Using 3 data points, we ~ill solve for the coefficients (a, b, c) in the quadratic model y
ax2
system of linear equations using the points corresponding to the years 1990,2002, and 2003.
) ~
I
5 -::;
~{ 0)2-+ b (OJ
c)
13. 5
.
C\'2..) 2- ~ b ( rz) 't" (
I d... i >
, ,S·=
tc.
~(\3Jl-tb('3)
The linear system can be written as
I~/£
AX = B where X = (a, b, c)
:X[·l~
~~1 = [~] = [:::~.
'''tt
,-is
X
13 (
+ bx + c.
2 b +L
-;::. I erG- "t"l~b + c..
-, Lf '1~
,A is coefficient matrix and B is the constant
Enter A on the graphing calculator and find A-1
B
Write a
::c.
Il.. I
TC-
2003
c
=[
1
d) Use the Inverse Matrix Method X = A -1B to determine the coefficients.
(a,b,c)
e)
f)
=
(.0?f17) -,
5q'5{,;) I~,!;)
r:/!/l'7
=
g) According to this model, what would the poverty level be in the year 2013 (t
2.
2.
5
6q3"
Substitute your values for (a, b, c) into the model y =
+ bx + c. Y = •
>( -,
y rl3 .•
Graph the function y
ax2 + bx + c on the calculator. Use Trace or Table to verify that the points from part a) lie
on the graph. Does it agree?
ax2
= 23)
Suppose 3 foods are determined to have the following vitamin content per ounce:
Food 1 contains: 20 units Vitamin A, 20 units Iron, and 10 units Calcium.
Food 2 contains: 30 units Vitamin A, 10 units Iron, and 10 units Calcium.
Food 3 contains: 10 units Vitamin A, 10 units Iron, and 30 units Calcium.
? ~
10, 85
ezb
F;- ~ I
~
A
:r.;:l
0
30
c. ~ ~
~~
..J~
10
1"0
30
Suppose a diet must consist precisely of 220 units of vitamin A, 180 units of Iron, and 340 units of Calcium. How
many ounces of each food type should be served?
Define the variables:
Let x = # ounces of Food 1, y = # ounces Food 2,
a) State the system of equations representing this situation.
6)0)( + 30'1 t-(~ l - 22-0
c)
x.
Solve the system using Inverse Matrix method X
(f.. I lj )'t;)
= # ounces
2.2.0
180
3'1b
Food 3
=,
+ 10 'j + io "l:
8o
10 'I.. + (0 '11" 3>01:
"" 3 «fa
b) The system can be written as AX = B, where:
.?b
z
7tr+
e-
.;)
A=[;g~~~l
=
Cf, to )
L.3
X=[;]
B=~j~J
l~fO
10 1\0 3~
z
A -1B. What is the quantity of each food to be served?
(A.(\,tJ
J O.4 Applications
College Algebra
of Linear Systems
Name:_-+--->o".""='.........,.;-t-
_
You are creating an application for Angry Birds, the popular online gam
download to your SmartPhone.
The application uses a lot of parabolic motion
(quadratics) and your task is to determine the function describing the desired motion
'\jj
ofthe projectile being launched!
Problem 1: Determine a quadratic function y = ax2 + bx + c that fits each scenario below. Pick 3 points on each
parabolic curve and substitute into y = ax2 + bx + c to form the system of3 equations. Determine the coefficients
(a, b, c) and write the function that models the curve.
a) Choose 3 points on the curve below.
_-,--L_:2._, "-I-.J,)'---_C_IO_J_?-_) __ C_q_J_H_, _3_)
CO
_
b) Substitute each point into y = ax2
the system of3 equations below:
+ bx + c and write
c
4 =-ct,(2.) 2.-t6l~) t
S I 5~
A - "Z.~
_'--------l--..........::.~
-
I
I
c) se the I erse Matrix Method and your graphing
calculator to solve for X = (a, b, c).
(a,b,c)
= L-/~3q ) ':),~r6) -" 3q 3 ')
I
=
+ bx + c. The
d) Substitute (a, b, c) into y ax2
that models the shown curve is:
equation
2
y
= -, 'l~qx + d.ls1-£~ - ..~q 3
e) Graph the curve from d) on your calculator. Determine the maximum height of the curve.
The maximum is at (x, y)
=
L 5 • " ) 7. \)Does your maximum
V ~ S.
agree with the image shown above?
Problem 2: You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually
strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground,
y, after x secons. Consider the data in the table:
=
a) Find the quadratic function y
ax2 + bx + c whose graph passes
through the given points. Write the system of3 equations below. Use
the Inverse Matrix method and your graphing calculator to solve for the
coefficients a, b, c.
2 ~ i.f ~
'2. -t b ( \) ~ (.
o;( ()
{L'7
System of equations:
J
lc ~
0 tt
-e,
a.( 3) 2 t
b( 3) -\
ct ( Lf) '2- t b( 4') t C.
'2.
y=_-~l..........::~~X
~_~_O_~_+ __ ~_O~O
b) Use the function in part a) to find the value for y when x = 5.
j
-==-
b
c.
hI'\-":> ~
~o~i
x, seconds after the
ball is thrown
y, ball's height, in
feet above the ground
1
224
176
104
3
4
Pr==-
-
)
\
,~ tt
CJ
L
3
'"
\
b~
I
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College Algebra
J 0.4 Applications of Linear Systems
Name:_-I-~",,=~+-
_
You are creating an application for Angry Birds, the popular online gam
download to your SmartPhone.
The application
uses a lot of parabolic motion
(quadratics) and your task is to determine the function describing the desired motion
~
ofthe
projectile
being launched!
Problem 1: Determine a quadratic function y = ax2 + bx + c that fits each scenario below. Pick 3 points on each
parabolic curve and substitute into y
ax2 + bx + c to form the system of3 equations. Determine the coefficients
(a, b, c) and write the function that models the curve.
=
a) Choose
oose Lnoi
3 pomts on the curve below.
(q )H, 31
,
L?.,~"
J--'- (GJ n.)
T
_-c-- __
CO
_
b) Substitute each point into y = ax2
the system of3 equations below:
•••••••••
4 :.C(,(2.)
J
System of equations:
2
_'----+---=.-
5
A -- 7..~
8I ~
{1
+ bx + c and write
-t6l ,,) l'
(s) '1 t
t: CL
C
2.
b( G) --t C.
q,~ "'0..('1)2 -tb ('1)
-t'
c.
I
I
c)~ se the I erse Matrix Method and your graphing
calculator to solve for X = (a, b, c).
(a,b,c)
= L-/~'3q ) ,=,,~-=r6) -" 3q 3 ')
I
=
+ bx + c. The
d) Substitute (a, b, c) into Y ax2
that models the shown curve is:
y = -,
(:r~qx2 T d.fsj-b'( -.,
equation
~q 3
e) Graph the curve from d) on your calculator. Determine the maximum height of the curve.
The maximum is at (x, y)
=
[5.~) 7. \)Does
V ~ 3.
your maximum agree with the image shown above?
Problem 2: You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually
strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground,
y, after x secons. Consider the data in the table:
=
a) Find the quadratic function y
ax2 + bx + c whose graph passes
through the given points. Write the system of3 equations below. Use
the Inverse Matrix method and your graphing calculator to solve for the
2 ~ '-f ~
coefficients a, b, c.
System of equations:
{l
J
7- lo ~
0'1
-e,
0....( l ) '2. -t
b( \) ~ L
a.( 3)
b( 3)
~
(
2t
c.
00\;
Lf) 2 t k>( £.(') t
C
'2.
y=_-~l~~~X
~_~_O_~_+
__~_O~O
b) Use the function in part a) to fmd the value for y when x = 5.
J -;: 0
hl't-~ ~
1Ao~l
x, seconds after the
ball is thrown
I
3
4
A:;,
y, ball's height, in
feet above the ground
224
176
104
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9'
3
,~ tt
L
,
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\
b~
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