Math 111I Sec 001 Name
Group Quiz 5 Solutions
Directions: Read each question carefully and answer in the space provided. The use of calculators is allowed, but not necessary. There are 10 points possi
Name
Chapter 2: Linear Functions
MATH 111 Sec. 009
Goals of Chapter 2
Calculate and interpret average rates of change
Understand how representations of data can be biased
Recognize that a constant rat
EXAM PLE 5
Solution
Figure 1 0
Table 2
Vertex
Focus
SECTION 11.2 The Parabola 731
Finding the Equation of a Parabola
Find the equation of a parabola with vertex at (0, 0) if its axis of symmetry is
790 CHAPTER AnalyticGeometry
An equation of the form of equation (2), with a2 .7 312, is the equation of an
ellipse with center at the origin, foci on the x-axis at (c, 0) and (11,0), where
c2 = a2 b2
780 CHAPTERII AnalyticGeometry
Recall that we obtained equation (2) after placing the focus on the positive
xaxis. If the focus is placed on the negative x-axis. positive yaxis, or negative y-axis,
a
792 CHAPTERII Analyticceemetry
Table 3 Equations M an ElliP5E= CEI'ItEf at (h. it): Major Axis Parallel to 3 Coordinate Axis
C t M ' AX. F ' V t' E 1:.
NOTE It is not recommended that on or a, '5 on o
Figure 24
Figure 25
SECTION 11.3 The Ellipse 791
_ Analyzing the Equation of an Ellipse
Solution
EXAMPLE 4
Solution
Analyze the equation: 9.1:2 + y2 = 9
To put the equation in proper form, divide
778 CHAPTERII AnalyticGeometry
11.2 The Parabola
PREPARING FOR THIS SECTION Before getting started, review the following:
- Distance Formula (Section 2.1, p. 151) I Completing the Square
794 CHAPTERII hnalvticGeometrv
Solution Set up a rectangular coordinate system so that the center of the ellipse is at the
origin and the major axis is along the x-axis. The equation of the ellipse is
SECTION 11.4 The Hyperbola 797
84. Show that the graph of an equation of the form _ _ D2 E2
(b)Isapo1nt1f + F = 0.
Ax2+Cy2+Dx+Ey+F=U, A0,Cv5 4A 46132 E2
where A and C are .Df the Same Sign:l (C) Ctls
782 CHAPTERII AnalyticGeometrv
Figure 11 Axis gf
symmetry
X = h
5 F=(h,k+a)
(a) o ME = 43(X a (b) (y ME = 4a(x h) (c) (x m2 = 43o a
EXAM PLE 6 Finding the Equation of a Parabola, Vertex Not at the
Figure 33
x2 y2
E _ E =
V1=(3=0)
Transverse
axis
F1:
Figure 34
THEOREM
EXAM PLE 1
Solution
SECTION 11.4 The Hyperbola 799
Equation of a Hyperbola: Center at (0,. 0);Transverse Axis along the xAxis
786 CHAPTERII AnalyticGeemetrv
and its depth is 1 inch. Hew far frem the vertex sheuld the
light bulb be placed se that the rays will be reected parallel
te the axis?I
66. Censtructing a Headlight A s
7 Symbolic Math Toolbox
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A =[12: 4 5];
b = [u: V];
z=A\b
result in
sol =
[ 5/3*u+2f3*v]
[ 4f3*U-1/3*v]
Z =
[ -5/3*U+2l3*v]
[ 4/3*u-1/3*v]
Thus 5 and z produce the same solution, although the results
788 CHAPTERII AnalyticGeometry
convenient to let 2o denote the constant distance referred to in the definition. Then,
if P = (x, y) is any point on the ellipse, we have
d(F1, P) + d072, P) = 2111 Burn
SECTION 11.2 The Parabola 779
THEOREM Equation of a Parabola:Vertex at (0, 0), Focus at (a,0),a > 0
The equation of a parabola with vertex at (0,0), focus at (a, U), and directrix
x = a, a > U, is
y2
784 CHAPTERII AnalyticGeometry
and its focus is at (D, a). Since (4, 3) is a point on the graph, we have
42 =4a(3) x2 =4ex=4,y=5
a E E3 | f
3 II US or El.
The receiver should be located 1% feet (1 foo
798 CHM-Tenn AnalyticGeometrv
Figure 32 position the hyperbola so that its transverse axis coincides with a coordinate axis.
d(F1,P) 0'02: P) = 3:23 Suppose that the transverse axis coincides with the
SECTION 11.3 The Ellipse 793
EXAMPLE 6 Analyzing the Equation of an Ellipse
Analyze the equation: 4.3.:2 + y2 8.1: + 4y + 4 = 0
Solution Proceed to complete the squares in .1: and in y.
4x2+y28x+4y+4=
SECTION 11.2 The Parabela 783
3 Solve Applied Problems Involving Parabolas
_ J - Parabelas nd their way inte many applicatiens. Fer example. as discussed in
Sectien 4.4, suspensien bridges have cables
SECTION 11.3 The Ellipse 795
In Prahlerns 1726. final the vertices and faci af each ellipse. Graph each eqaatian.
2 yz 2 2 2 2 2
\17.I+= _I|y= _x_|_y_= - 2+y_=
.25 4 1 189 41 199 251 20x 161
\21. 4x2+
796 CHAPTERII AnalyticGeemetry
Applications and Extensions
69. Semielliptical Arch Bridge A11 arch in the shape ef the
9 upper half ef an ellipse is used te suppert a bridge that is te
span a river 20
11.3 The Ellipse
Figure 17
Miner axis
PREPARING FOR THIS SECTION Before getting started, review the following:
Distance Formula (Section 2.1, p. 151) *- Symmetry (Section 2.2, pp. 160162
SECTION 11.2 The Parabela 785
In Prebtems 1936, find the eqneticn cf the perebcfe described. Find the twcr paints that dene the fetus rectum, end graph the eqneticn.
\ 19. Fecus at (4, U); vertex at (
SECTION 11.3 The Ellipse 789
Notice in Figure 19 the right triangle formed by the points (0, 0), (c, 0), and
(0, b). Because b2 = e2 c2 (or b2 + c2 = e2), the distance from the focus at (c, 0)
to the
A MATLAB Quick Reference
S
Sparse Matrix Functions
These functions allow you to operate on a special
type of matrix, sparse. Using these functions you
can convert full to sparse, visualize, and operat
Solving Equations
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0]
Pi]
1/6*pi]
5f6*pi]
I_II_|_|I_EU'I
Several Algebraic Equations
Now let's look at systems of equations. Suppose we have the system
x2y2=0
_Z=
x2 or.
and we want to solve for xand
7 Symbolic Math Toolbox
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:Ep ars to have redundant components. This is due to the first equation
y? = U. which has two solutions in xandy x = it]. y = i0. Changing the
equations to
eqsi = 'x*2*y2=1,
Data Analysis and Fourier Transform Functions
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Low Level Functions
qrdelete Delete column from QR
factorization
qrinsert Insert column in QR
factorization
Data Analysis and Fourier Transform
Function
Specialized Math Functions
%
m Specialized Math Functions
an" atanh Inverse tangent and inverse This set of functions includes Bessel, elliptic,
hyperbolic tangent .
_ gamma, factorial, and others.
at