Clicker(Ques,ons(and(Addenda( - Phillip Compeau | CMU...

Clicker(Ques,ons(and(Addenda(

Phillip(Compeau(MATH(3C(

Winter(2013(

Q.(2.1:(Pythagorean(Theorem(Ques,on(

What(is(the(length(of(the(missing(side?((A)  5(

B)  8(

C)  10(D)  14(E)  25(( b(=(12(

Q.(2.2:(Proving(the(PT(

What(is(the(area(of(the(big(square?((A)  a2(+(b2(

B)  (a(+(b)2(

C)  c2(+(2ab(D)  Both(A(and(B(E)  Both(B(and(C((

Q.(2.3:(Television(Length(

What(is(the(height(of(a(40Sinch(4:3(television(screen?((A)(20(inches(B)(24(inches(C)(28(inches(D)(32(inches(E)(36(inches(

Q.(3.1:(Dividing(Strings(

Do(you(think(that(this(process(can(be(done(with(any(pair(of(segments?((A)  Yes,(and(I(feel(very(confident.(B)  Yes,(but(I(don’t(feel(very(confident.(C)  No,(but(I(don’t(feel(very(confident.(D)  No,(and(I(feel(very(confident.((ANS:(NO…not(all(numbers(are(ra,onal((see(notes)(

Q.(3.2:(Greek(Ladder(

What(is(the(next(row(of(the(ladder?((A)  m(=(82((((((n(=(58(B)  m(=(89((((((n(=(63(C)  m(=(92((((((n(=(65(D) m(=(99((((((n(=(70(E)  m(=(106((((n(=(75(

((

m(((((((((((((((((((((n#3((((((((((((((((((((((2(7((((((((((((((((((((((5(17(((((((((((((((((((12(41  (((((((((((((((((29(?(((((((((((((((((((((((?(

Q.(3.3:(Decimals(to(Frac,ons(

Convert(0.93939393…to(a(frac,on.((A)  94/99(B)  31/33(C)  930/990(D)  933/999(E)  330/338(

((

Q.(4.1:(Intervals(

Find(the(union(and(intersec,on(of((S∞,(1](and((S1,(2).((A)  Union(=((S∞,(1](;(Intersec,on(=((S1,(1)(B)  Union(=((S1,(1](;(Intersec,on(=((S∞,(2)(C)  Union(=(((S∞,(1](;(Intersec,on(=([1,(2)(D)  Union(=((S∞,(S1](;((Intersec,on(=((S1,(2]((E)  Union(=((S∞,(2)(;((Intersec,on(=((S1,(1](

((

Q.(4.2:(Distance(Formula(

Based(on(the(distances(between(each(pair(of(points,(the(points((S5,(6),((0,(8)(and((S3,(1)(form(the(ver,ces(of(what(kind(of(triangle?((A)  Equilateral((three(equal(sides)(B)  Isosceles((two(equal(sides)(C)  Scalene((no(equal(sides)(D)  Right(E)  Both(B(and(D(

Q.(4.3:(Compu,ng(Slope(

Find(the(slope(of(the(segment(connec,ng(the(points((1,(1)(and((2,(4).((A)  3(B)  S3(C)  1/3(D)  5/3(E)  3/5(

Q.(5.1:(Similar(Triangles(

The(two(triangles(in(the(figure(are(similar.((What(is(b/c?((A)  1/4(B)  1/2(C)  2/5(D)  4/5(E)  We(can’t(conclude(

anything.(

a(

b(

c(2(

4(5(

Q.(5.2:(PointSslope(Form(

Find(the(pointSslope(form(of(the(line(that(passes(through(the(points((1,(1)(and((2,(4).((A)  y(–(1(=(3(x(–(1)(B)  y(–(4(=(3(x(–(2)(C)  y(–(1(=(3x(–(1(D)  y(–(4(=(3x(–(2(E)  More(than(one(of(the(above.(

Q.(5.3:(Changing(Currency(

Jay(wants(to(change($(to(Euros.(His(hotel(charges(a($5(fee(in(addi,on(to(2%(of(money(changed,(then(offers(him(0.75(Euros(for(every($1.((Give(a(formula(for(the(Euros((E)(he(receives(in(terms(of(dollars((D).(a)  E(=(0.75((1.02D(–(5)(b)  E(=(0.75((0.98D(–(5)(c)  E(=(0.75((0.98D)(–(5(d)  E(=(0.75*0.98(D(–(5)(e)  E(=(0.98((0.75D(–(5)(

Q.(6.1:(Zipcar(Zipcar(offers(two(car(rental(op,ons:(1.(Pay($10(for(every(hour(you(drive(in(a(month.(2.(Pay($50(for(the(en,re(month,(plus($8(per(hour.(Arer(how(much(monthly(driving(is(plan(2(cheaper?((A)  25(hours(B)  50(hours(C)  100(hours(D)  Plan(2(is(always(cheaper(than(plan(1.(E)  Plan(2(is(never(cheaper(than(plan(1.(

Q.(6.2:(Perpendicular(Lines(

What(are(the(slopes(of(the(two(lines?((A)(L1:(d/a(((((L2:((e/a((((B)(L1:(d/a(((((L2:(Se/a((((C)(L1:(b/a(((((L2:((c/a(((D)(L1:(b/a(((((L2:(Sc/a(((E)(L1:(b/d(((((L2:(Sc/e((

a(

b(

c(

d(

e(

L1(

L2(

Q.(6.3:(Perpendicular(Lines(

What(is(the(slope(of(the(line(perpendicular(to(the(line(2x(+(3y(=(6?((A)  2/3(B)  3/2(C)  S2/3(D)  S3/2(E)  0(

Q.(6.4:(Midpoint(and(Distance(Formula((

What(is(the(distance(between(points((S1,(3),((2,(S5)(and(the(midpoint(of(the(segment(connec,ng(them?((A)  Distance(=(√(65);(((Midpoint(=((3/2,(S4)(B)  Distance(=(√(65);(((Midpoint(=((1/2,(S1)(C)  Distance(=(√(5);(((((Midpoint(=((1/2,(S1)(D)  Distance(=(√(73);(((Midpoint(=((1/2,(S1)(E)  Distance(=(√(73);(((Midpoint(=((3/2,(S4)(

Q.(7.1:(Equa,on(of(Circle(

A(diameter(of(a(circle(has(endpoints((4,(S3)(and((S2,(5).((Give(the(equa,on(of(the(circle.((A)  (x(–(3)2(+((y(+(1)2(=(10(B)  (x(–(1)2(+((y(–(1)2(=(25(C)  (x(+(1)2(+((y(+(1)2(=(25(D)  (x(S(1)2(+((y(S(1)2(=(100(E)  (x(+(1)2(+((y(+(1)2(=(100(

Q.(7.2:(Comple,ng(the(Square(

Find(the(center(and(radius(of(the(circle(given(by(x2(+(y2(–(10x(+(6y(=(3.((A)  Center(=((S5,(3);((Radius(=(37(B)  Center(=((5,(S3);((Radius(=(37(C)  Center(=((S5,(3);((Radius(=(√(37)(D)  Center(=((5,(S3);((Radius(=(√(37)(E)  This(equa,on(doesn’t(define(a(circle.(

Q.(7.3:(Perimeter(of(Inscribed(Square(

Find(the(perimeter(of(the(square.((A)  2(B)  2√(2)(C)  4(D)  4√(2)(E)  6(

r(=(1(

Arhcimedes’s(96Sgon(

1/15/13 10:59 PMNOVA | Infinite Secrets | Approximating Pi (non-Flash) | PBS

Page 2 of 2http://www.pbs.org/wgbh/nova/archimedes/pi-nf.html

24-Sided Polygoninscribed perimeter = 3.1326circumscribed perimeter = 3.1597



actual value of [pi] = 3.1416

Note: All figures rounded off to four decimal places.

ContemplatingInfinityPhilosophically, theconcept remains amind-bender.

Working with InfinityMathematicians havebecome increasinglycomfortable with theconcept.

Great SurvivingManuscriptsAncient documentsoffer a tantalizingglimpse of lostcultures.

The ArchimedesPalimpsestFollow the 1,000-year-long journey ofthe Archimedesmanuscript.

Approximating PiSee Archimedes'geometrical approachto estimating pi.

Infinite Secrets homepage | NOVA homepage

© | Created September 2003

Q.(8.1:(Mystery(Graph(What(will(the(graph(look(like?(

A( B( C(

D( E(

Ellipse(Anima,on(

Here’s(an(anima,on(of(how(to(draw(an(ellipse(with(2(toothpicks(and(some(string:((htp://www.youtube.com/watch?v=7UD8hOsSvaI((

Q.(8.2:(Intercepts(of(Ellipse(

Find(the(xS(and(ySintercepts(of(the(ellipse(given(by(the(equa,on(9x2(+(4y2(=(16.((A)  (2,(0);((S2,(0);((0,(4/3);((0,(S4/3)(B)  (2,(0);((S2,(0);((0,(3/4);((0,(S3/4)(C)  (0,(2);((0,(S2);((4/3,(0);((S4/3,(0)(D)  (0,(2);((0,(S2);((3/4,(0);((S3/4,(0)(

Ellip,cal(Whispering(Galleries(

(((

Ellip,cal(Orbits((Halley’s(Comet)(

Lithotripsy((Kidney(Stone(Therapy)(

Q.(9.1:(Mystery(Graph(What(will(the(graph(look(like?(

A(

B(

D(

C(

E(

Q.(9.2:(Minimum(Value(of(Parabola(

Give(the(minimum(ySvalue(of(y(=(2(x(+(3)2(+(7.((A)  2(B)  3(C)  S3(D)  7(E)(There(is(no(minimum(ySvalue.(

Q.(9.3:(Standard(Form(of(Parabola(

Write(the(standard(form(of(the(parabola(that(has(vertex((3,(S5)(and(passes(through((1,(S3).((A)  y(=((1/2)(x(–(3)2(–(5(B)  y(=((1/2)(x(+(3)2(–(5(C)  y(=((S3/4)(x(–(1)2(–(3(D)  y(=(2(x(+(3)2(+(5(E)  y(=(2(x(–(3)2(–(5(

Parabolic(Oven(

Ligh,ng(the(Olympic(Flame(

Parabolic(Microphone(

Headlights(

Parabolas(Represent(Projec,les(

Q.(10.1:(Projec,le(Mo,on(

The(height(of(a(baseball(at(,me(t(is(given(by(y(=(S16t2(+(32t(+(5.((Find(its(maximum(height.((A)  5(feet(B)  21(feet(C)  5001(feet(D)  16(feet(E)  11(feet(

Q.(10.2:(Projec,le(Mo,on(

Height((y)(of(baseball:(y(=(S16t2(+(32t(+(5.((Find(any(,mes(t(at(which(the(height(=(17.((A)  t(=(1(B)  t(=(3/2(C)  t=(1/2,(t(=(3/2(D)  t(=(1/3,(t(=(5/3(E)  t(=(0,(t(=(2(

Archimedes’(Claw(

Did(Archimedes(Make(a(Death(Ray?(

A(Parabolic(Death(Ray(Model(

Hotel(Vdara((Las(Vegas)(

Q.(11.1:(Fermat’s(Last(Theorem(

There(are(plenty(of(integers(sa,sfying(the(equa,on(a2(+(b2(=(c2((e.g.,(a(=(3,(b(=(4,(c(=(5).((Do(you(think(there(are(integers(a,(b,(c(such(that(an(+(bn(=(cn(for(some(n(bigger(than(2?((A)  Yes(B)  No(

Fermat(and(Wiles(

Fermat((1637):(an(+(bn(=(cn(?(

Wiles((1995):(NOPE(

Cardano(and(the(Solu,on(of(the(Cubic(

We(have(solved(the(equa,on(ax2(+(bx(+(c(=(0.((Next(ques,on:(How(do(we(solve(the(“cubic(equa,on”:((ax3(+(bx2(+(cx(+(d(=(0((Published(by(Cardano((1545)(

Three solutions to the equation ax

3 + bx

2 + cx+ d = 0:

x1 = b

3a

1

3a3

s1

2

2b3 9abc+ 27a2d+

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

1

3a3

s1

2

2b3 9abc+ 27a2d

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

x2 = b

3a

+1 +

p3

6a3

s1

2

2b3 9abc+ 27a2d+

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

+1

p3

6a3

s1

2

2b3 9abc+ 27a2d

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

x3 = b

3a

+1

p3

6a3

s1

2

2b3 9abc+ 27a2d+

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

+1 +

p3

6a3

s1

2

2b3 9abc+ 27a2d

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

?(

?(

The(“Golden(Ra,o”(

Golden(Ra,o(in(Art(

Golden(Ra,o(in(Architecture(

(c)x

2

5+

y2

9= 1

(d)x

2

9+

y2

5= 1

5. What is the equation of the ellipse graphed below?

−1 1 2 3 4 5 6 7

−1

1

2

3

x

y

(a)(x + 3)2

16+

(y + 1)2

4= 1

(b)(x + 3)2

64+

(y + 1)2

16= 1

(c)(x − 3)2

16+

(y − 1)2

4= 1

(d)(x − 3)2

64+

(y − 1)2

16= 1

6. Find equations for the asymptotes of the hyperbola given by the equation y2−

x2

2= 4.

(a) y = ±1

2x

(b) y = ±1√

2x

(c) y = ±√

2x

(d) y = ±2x

2

Q.(11.2:(Equa,on(of(Ellipse(Give(the(equa,on(of(this(ellipse.(

(c)x

2

5+

y2

9= 1

(d)x

2

9+

y2

5= 1


−1 1 2 3 4 5 6 7

−1

1

2

3

x

y

(a)(x + 3)2

16+

(y + 1)2

4= 1

(b)(x + 3)2

64+

(y + 1)2

16= 1

(c)(x − 3)2

16+

(y − 1)2

4= 1

(d)(x − 3)2

64+

(y − 1)2

16= 1


x2

2= 4.

(a) y = ±1

2x

(b) y = ±1√

2x

(c) y = ±√

2x

(d) y = ±2x

2

(c)x

2

5+

y2

9= 1

(d)x

2

9+

y2

5= 1


−1 1 2 3 4 5 6 7

−1

1

2

3

x

y

(a)(x + 3)2

16+

(y + 1)2

4= 1

(b)(x + 3)2

64+

(y + 1)2

16= 1

(c)(x − 3)2

16+

(y − 1)2

4= 1

(d)(x − 3)2

64+

(y − 1)2

16= 1


x2

2= 4.

(a) y = ±1

2x

(b) y = ±1√

2x

(c) y = ±√

2x

(d) y = ±2x

2

Q.(12.1:(Vertex(of(Polynomial(

What(is(the(“vertex”(of(y(=(3(x(–(1)4(+(2?((A)  (1,(2)(B)  (1,(S2)(C)  (S1,(2)(D)  (S1,(S2)(

Graphing(y(=(3x3(–(4x2(–(5x(+(2(

Q.(12.2:(Ra,onal(Zero(Theorem(

Determine(the(possible(ra,onal(zeroes(of(y(=(7x3(+(4x2(–(45x(+(18.((A)  ±1,(±1/2,(±1/3,(±1/6,(±1/9,(±1/18,(±7,(±7/2,(

±7/3,(±7/6,(±7/9,(±7/18(B)  ±1,(±2,(±3,(±6,(±9,(±18,(±1/7,(±2/7,(±3/7,(±6/7,(

±9/7,(±18/7(C)  ±1,(±7(D)  ±1,(±2,(±3,(±6,(±9,(±18(E)  All(ra,onal(numbers(

Graphing(y(=(x3(+(3x2(–(2x(–(6((

Q.(13.1:(Factor(Theorem(

Recall(that(x(=(2(is(a(zero(of(y(=(3x3(–(4x2(–(5x(+(2.((What(remains(arer(factoring(out((x(–(2)?((A)  3x2(–(10x(–(25(B)  3x2(–(10x(+(15(C)  3x2(+(2x(–(9(

D)  3x2(+(2x(–(1(E)  S5x2(+(4x(S7(

Q.(13.2:(Mul,plicity(of(Zeroes(

Find(the(zeroes(of(y(=((x2(–(3)4(x5(+(x4(–(12x3),(with(mul,plici,es.((A)  x(=(√(3)((mult.(8)(B)  x(=(√(3)((mult.(4);(x(=(S√(3)((mult.(4)(C)  x(=(0((mult.(3);(x(=(S4((mult.(1);(x(=(3((mult.(1)(D)  x(=(0((mult.(3);(x(=(S4((mult.(1);(x(=(3((mult.(1);(

x(=(√(3)((mult.(4);(x(=(S√(3)((mult.(4)(E)  There(are(no(zeroes.(

Three solutions to the equation ax

3 + bx

2 + cx+ d = 0:

x1 = b

3a

1

3a3

s1

2

2b3 9abc+ 27a2d+

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

1

3a3

s1

2

2b3 9abc+ 27a2d

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

x2 = b

3a

+1 +

p3

6a3

s1

2

2b3 9abc+ 27a2d+

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

+1

p3

6a3

s1

2

2b3 9abc+ 27a2d

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

x3 = b

3a

+1

p3

6a3

s1

2

2b3 9abc+ 27a2d+

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

+1 +

p3

6a3

s1

2

2b3 9abc+ 27a2d

q(2b3 9abc+ 27a2d)2 4 (b2 3ac)3

?(

?(

Higher(Degree(Polynomials?(

1.  There(is(a(formula(for(finding(zeroes(of(degree(4(polynomials((very(complicated).((((

2.  There(are(NO(formulas(for(finding(zeroes(of(polynomials(of(degree(5,(6,(7,(etc.(

Graphing(y(=(3x3(–(4x2(–(5x(+(2(

Graphing(y(=(x5(–(20x3(

Q.(14.1:(Polynomial(EndSBehavior(Which(descrip,on(matches(the(polynomial?((((((A)  Odd(degree;(lead(coefficient(<(0(B)  Odd(degree;(lead(coefficient(>(0(C)  Even(degree;(lead(coefficient(<(0(D)  Even(degree;(lead(coefficient(>(0(

(c) x = 0, mult. 3; x = −4, mult. 1; x = 3, mult. 1

(d) x = 0, mult. 3; x = −4, mult. 1; x = 3, mult. 1; x =√

3, mult. 4; x = −√

3,mult. 4

(e) f(x) has no zeros.

5. Which description matches the graph of the polynomial below?

(a) odd degree, lead coefficient negative

(b) even degree, lead coefficient negative

(c) odd degree, lead coefficient positive

(d) even degree, lead coefficient positive


(a) odd degree, lead coefficient negative

(b) even degree, lead coefficient negative

(c) odd degree, lead coefficient positive

(d) even degree, lead coefficient positive


2

Q.(14.2:(Degree(of(Polynomial(

What(is(the(degree(of(y(=(S4(x(–(1)3(x(+(4)4(x(+(7)2?((A)  3(B)  4(C)  9(D)  12(E)  24(

Q.(14.3:(Real(and(Complex(Numbers(

True(or(False:(Every(real(number(is(a(complex(number.((A)  True,(and(I(feel(confident.(B)  True,(but(I(don’t(feel(confident.(C)  False,(but(I(don’t(feel(confident.(D)  False,(and(I(feel(confident.(E)  I(don’t(understand(the(ques,on.(

Q.(15.1:(Dividing(Complex(Numbers(Divide((8(+(2i)/(2(–(4i).((A)  2/5(+((9/5)i(B)  S1/6(+((3/4)i(C)  0(D)  S2(+((3/4)i(E)  24/5(+((28/5)i(

Graphing(y(=(2(x2(–(4x(+(5)(x+√(3))(x(–(√(3))(((

Q.(15.2:(Func,ons(Does(the(table(represent(a(func,on(f(x)?((((A)  Yes,(and(I’m(confident(B)  Yes,(but(I’m(not(confident(C)  No,(but(I’m(not(confident(D)  No,(and(I(am(very(confident(E)  I(am(confident(of(both(“Yes”(and(“No”(

(b) Yes, but I am not very confident

(c) No, but I am not very confident

(d) No, and I am very confident

5. The set of points (x, y) which satisfy the equation (x − 1)2 + (y + 3)2 = 52 can berepresented via a mathematical function relating the x and y variables.

(a) True, and I am very confident.

(b) True, but I am not very confident.

(c) False, but I am not very confident.

(d) False, and I am very confident.

6. Does the table represent a function, y = f(x)?

x 1 2 3 4f(x) 2 3 2 4

(a) Yes, and I am very confident





x 1 2 2 4f(x) 2 3 1 3





8. Does this sentence describe a function? Wanda is two years older than I am.





2

Q.(16.1:(Func,on(as(Table(Does(the(table(represent(a(func,on(f(x)?((((A)  Yes,(and(I’m(confident(B)  No,(but(I’m(not(confident(C)  No,(but(I’m(not(confident(D)  No,(and(I(am(very(confident(E)  I(am(confident(of(both(“Yes”(and(“No”(




5. The set of points (x, y) which satisfy the equation (x − 1)2 + (y + 3)2 = 52 can berepresented via a mathematical function relating the x and y variables.






x 1 2 3 4f(x) 2 3 2 4






x 1 2 2 4f(x) 2 3 1 3





8. Does this sentence describe a function? Wanda is two years older than I am.





2

(b) y = −3x + 6

(c) y = −3x + 2

(d) y = −x + 6

(e) y = 6x − 2

(f) y = x − 2

22. Which of the following functions is not increasing?

(a) The elevation of a river as a function of distance from its mouth

(b) The length of a single strand of hair as a function of time

(c) The height of a person from age 0 to age 80

(d) The height of a redwood tree as a function of time

23. Which of these graphs does not represent y as a function of x?

24. Calculate the average rate of change of the function f(x) = x2 between x = 1 andx = 3.

(a) 8

6

(b) y = −3x + 6

(c) y = −3x + 2

(d) y = −x + 6

(e) y = 6x − 2

(f) y = x − 2








(a) 8

6

Q.(16.2:(Which(is(not(a(func,on?(

A(

(b) y = −3x + 6

(c) y = −3x + 2

(d) y = −x + 6

(e) y = 6x − 2

(f) y = x − 2








(a) 8

6

(b) y = −3x + 6

(c) y = −3x + 2

(d) y = −x + 6

(e) y = 6x − 2

(f) y = x − 2








(a) 8

6

B(

C(

D(

Q.(16.3:(Domain(of(Func,on(Find(the(domain(of(the(func,on((A)  t(>(7(B)  t(≤(7(C)  t(=(7(D)  t(<(7(E)  S7(<(t(<(7(





14. Could this table represent a linear function?

x 1 2 4 8f(x) 10 9 7 3





15. True or False? All linear functions are examples of direct proportionality.

(a) True, and I am very confident

(b) True, but I am not very confident

(c) False, but I am not very confident

(d) False, and I am very confident

16. Find the domain of the function f(x) = 1x−2 .

(a) x = 2

(b) x = 2

(c) x < 2

(d) all real numbers

17. Find the domain of the function g(t) = 2+t√

t−7.

(a) t > 7

(b) t ≥ 7

(c) t = 7

(d) all real numbers

4

Graph(of(Ra,onal(Func,on(

2/6/13 y = (x+3)/(-2(x-2)(x+1) - Google Search

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Graph for (x+3)/((‑2)*(x-2)*(x+1))

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16 18

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-5

-4

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1

2

3

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5

6

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8

9

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12

x: -2.43616112 y: -0.044250129

(y-(x^2)^(1/3)) - Wolfram Alphawww.wolframalpha.com/input/?i=x%5E2%2B(y.../3))%5E2...1A description for this result is not available because of this site's robots.txt – learnmore.

Wolfram|Alpha Pro: Step-by-step Solutionswww.wolframalpha.com/pro/step-by-step-math-solver.htmllimit of (x^2+5x+6) / (x+2) as x approaches -2;; take the derivative: cos(x+1)/x;;derivative of cos(x+1)/(x^2+3);; integrate e^x/(e^(2x)+2e^x+1);; y'(x)=y(x)+x+x y(x)+1 ...

Finding Intercepts From an Equationwww.mathsisfun.com/algebra/finding-intercepts-equation.htmlExample: Find the intercepts of y = x2 - 4. x intercept: ... Question 1 Question 2Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 ...

SparkNotes: Quadratics: Graphing Parabolaswww.sparknotes.com › SparkNotes › Math Study Guides › QuadraticsIn fact, as x increases by 1 , starting with x = 0 , y increases by 1, 3, 5, 7,… ... Thegraph is shifted up 3 units from the graph of y = x 2 , and the vertex is (0, 3) .

[PDF] Section 12.4 Supplementary Questions S1 Suppose that three ...www2.cscamm.umd.edu/~cdavis/ch12suppQandA.pdfFile Format: PDF/Adobe Acrobat - Quick ViewThus, E(Xi)=0.5 and E(X2 i )=0.5 for i = 1,2,3. Determine the value of. E[(X1 - 2X2 +X3)2]. S3 Suppose that X and Y are independent random variables for which ...

(x+3)^2+(y+1)^2=4 - Algebrawww.algebra.com › Algebra › Quadratic-relations-and-conic-sectionsYou can put this solution on YOUR website! Hi, (x+3)^2+(y+1)^2=4 ||Circle with C(-3,-1)and radius of 2 (y+2)^2=16px-1) ||The standard form is ...

SOLUTION: Find the corresponding y values for x= -2,-1,0,1,2,3 if y ...www.algebra.com › Algebra › Systems-of-equationsSOLUTION: Find the corresponding y values for x= -2,-1,0,1,2,3 if y=x^2-x-2. I havetried for 2 days to make this problem work but I keep getting stuck along the ...

Equations - QuickMath.com - Automatic Math Solutionswww.quickmath.com/pages/modules/graphs/equations/index.phpIt will plot functions given in the form y = f(x), such as y = x2 or y = 3x + 1, as well ...Plot y = 3x^2 + 1 and y = 2x^3 - 4 from x = -3 to x = 3, y = -17 to y = 17 with an ...

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y = (x+3)/(-2(x-2)(x+1) Sign in

+You Search Images Maps Play YouTube News Gmail Drive Calendar More

y = x + 3−2 x − 2( ) x +1( )

Q.(17.1:(Hole(of(Ra,onal(Func,on(Which(of(the(following(func,ons(has(a(hole(at((1,4)?((

7. Which of the following functions has a hole at (1,4)?

a. f(x) =x − 1

(x − 1)(x − 5)

b. f(x) =x − 1

(x + 1)2

c. f(x) =4

x − 1

d. f(x) =(x − 1)(11x + 1)

(x − 1)(x + 2)

8. Which of the following functions has a zero, a vertical asymptote and a horizontal asymp-tote?

a. f(x) =x − 4

(x − 4)(x − 5)

b. f(x) =(x + 2)(x2 + 1)

(x − 4)(x2 + 7)

c. f(x) =x2 + 5

(x − 4)(x − 5)

d. f(x) =(x − 5)(x2 + 8)

(x − 4)

9. Which of these functions has no vertical asymptotes?

a. f(x) =x − 7

(x − 7)(x − 5)

b. f(x) =x

x2− x − 1

c. f(x) =1

x − 2

d. f(x) =x2

− 9x + 20

(x − 4)(x − 5)

10. Which of the following functions has a hole, one zero, an oblique asymptote and novertical asymptote?

a. f(x) =(x − 7)(x2 + 1)

(x − 7)(x − 5)

b. f(x) =(x − 7)(x2

− 1)

(x − 7)(x − 2)

c. f(x) =(x − 7)(x3

− 4)

(x − 7)(x2 + 5)

d. f(x) =x − 7

(x − 7)(x − 5)

www.analyzemath.com Page 4 of 5

Q.(17.2:(Horizontal(Asymptotes(Which(of(the(following(func,ons(has(no(horizontal(asymptote?(

5. Find the function f whose graph is given below.

a. f(x) =x − 2

x2 + 3x − 4

b. f(x) =2 − x

x2 + 3x − 4

c. f(x) =x − 2

x2− 3x − 4

d. f(x) =2 − x

x2− 3x − 4

1

2

3

4

5

6

−1

−2

−3

−4

−5

−6

1 2 3 4 5 6−1−2−3−4−5−6x

y = f(x)

0

6. Which of the following functions has no horizontal asymptote?

a. f(x) =x2

− 2

x3− 9x − 4

b. f(x) =5x4

− 23− x + 7

−x3 + 3x2− 4

c. f(x) =9x2

− 2x − 3

x2 + 8x − 2

d. f(x) =1

x2− x

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Graph for (x+3)/((‑2)*(x-2)*(x+1))

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16 18

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

1

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x: -2.43616112 y: -0.044250129

(y-(x^2)^(1/3)) - Wolfram Alphawww.wolframalpha.com/input/?i=x%5E2%2B(y.../3))%5E2...1A description for this result is not available because of this site's robots.txt – learnmore.

Wolfram|Alpha Pro: Step-by-step Solutionswww.wolframalpha.com/pro/step-by-step-math-solver.htmllimit of (x^2+5x+6) / (x+2) as x approaches -2;; take the derivative: cos(x+1)/x;;derivative of cos(x+1)/(x^2+3);; integrate e^x/(e^(2x)+2e^x+1);; y'(x)=y(x)+x+x y(x)+1 ...

Finding Intercepts From an Equationwww.mathsisfun.com/algebra/finding-intercepts-equation.htmlExample: Find the intercepts of y = x2 - 4. x intercept: ... Question 1 Question 2Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 ...

SparkNotes: Quadratics: Graphing Parabolaswww.sparknotes.com › SparkNotes › Math Study Guides › QuadraticsIn fact, as x increases by 1 , starting with x = 0 , y increases by 1, 3, 5, 7,… ... Thegraph is shifted up 3 units from the graph of y = x 2 , and the vertex is (0, 3) .

[PDF] Section 12.4 Supplementary Questions S1 Suppose that three ...www2.cscamm.umd.edu/~cdavis/ch12suppQandA.pdfFile Format: PDF/Adobe Acrobat - Quick ViewThus, E(Xi)=0.5 and E(X2 i )=0.5 for i = 1,2,3. Determine the value of. E[(X1 - 2X2 +X3)2]. S3 Suppose that X and Y are independent random variables for which ...

(x+3)^2+(y+1)^2=4 - Algebrawww.algebra.com › Algebra › Quadratic-relations-and-conic-sectionsYou can put this solution on YOUR website! Hi, (x+3)^2+(y+1)^2=4 ||Circle with C(-3,-1)and radius of 2 (y+2)^2=16px-1) ||The standard form is ...

SOLUTION: Find the corresponding y values for x= -2,-1,0,1,2,3 if y ...www.algebra.com › Algebra › Systems-of-equationsSOLUTION: Find the corresponding y values for x= -2,-1,0,1,2,3 if y=x^2-x-2. I havetried for 2 days to make this problem work but I keep getting stuck along the ...

Equations - QuickMath.com - Automatic Math Solutionswww.quickmath.com/pages/modules/graphs/equations/index.phpIt will plot functions given in the form y = f(x), such as y = x2 or y = 3x + 1, as well ...Plot y = 3x^2 + 1 and y = 2x^3 - 4 from x = -3 to x = 3, y = -17 to y = 17 with an ...


y = (x+3)/(-2(x-2)(x+1) Sign in


y = x + 3−2 x − 2( ) x +1( )

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Graph for (100*x^2-2*x+6)/(2*x^2-5*x-1)

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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x: 0.65899784 y: -14.0407885

Trinomial Factoring - MathScore.comwww.mathscore.com/math/skills/TrinFac.htmlFactors of -12: 1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, and -12. Since c is .... 2. 6x 3 - 100x 2y- 34xy 2 ... Test: (x - 2)(x - 3) => - 3x + - 2x = - 5x This combination matches.

[DOC] Alg2A Factoring and Equations Packetwww.sfponline.org/uploads/.../alg2a1stpacketfactoringandequations....File Format: Microsoft Word - Quick View5) (3x - 5)(2x + 8) 6) (11x - 7)(5x + 3) 7) (4x - 9)(9x + 4) 8)(x - 2)(x + 2). 9) (x - 2)(x - 2) ...1) 14x9 - 7x7 + 21x5 2) 26x4y - 39x3y2 + 52x2y3 - 13xy4. 3) 32x6 - 12x5 ...

Algerbra 2 help meeeeeeeeeeeeee? - Yahoo! Answersanswers.yahoo.com › ... › Science & Mathematics › Mathematics2 answers - Jul 22, 2011Classify the expression 7x3 + 2x2 – 9x as a monomi… ... My Y! Yahoo! ... Simplify4x2(2x2 – 8x + 6) ... 4x2(x – 3)(x – 1) ... Factor completely 100x2 + 60x + 9 ... followingis a factor of 15x2 – 14x – 8? Answer 5x – 2. None of the above 5x + 16 ... 3x – 2 3 2x– 3. Which of the following is a factor of 512x3 – 125?

Difference of two squares - A complete course in algebrawww.themathpage.com/alg/difference-two-squares.htmExample 1. Multiply (x3 + 2)(x3 − 2). Solution. Recognize the form: (a + b)(a − b). Theproduct will be the difference of two squares: (x3 + 2)(x3 − 2) = x6 − 4. x6 is ...

2x^2+5x+8=6 - solutionwww.geteasysolution.com/2x%5E2+5x+8=6Simple and best practice solution for 2x^2+5x+8=6 equation. ... Set the factor '(1 + 2x)'equal to zero and attempt to solve: Simplifying 1 + 2x = 0 Solving 1 + 2x ...

Solve By Completing the Square -- Algebra.Helpwww.algebrahelp.com/worksheets/view/.../completingthesquare.quiz1. x2 + 5x + 6 = 0. x = step-by-step. 2. x2 + 2x + 1 = 0. x = step-by-step ... 7. 4y2 + 8y+ -21 = 0. y = step-by-step ... 100x2 + -100x + 9 = 0. x = step-by-step ...

Dividing Polynomials - Math - Tutorvista.commath.tutorvista.com › ... › Polynomial › Operations with PolynomialsAnswer: The final answer is Quotient = 6x2 + 2x + 5 and remainder value is 10. ... Step1: In given polynomial, 2x4 + x2 - 5x + 2, the leading coefficient is 2.

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y = 100x2 − 2x + 6

2x2 − 5x −1

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Graph for ((‑(3*x^3))-4*x^2+π)/(x^2-4)

-30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45

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x: -0.96703339 y: -0.689741754

Wolfram|Alpha Blog : The Top 100+ Sines of Wolfram|Alphablog.wolframalpha.com/2012/.../24/the-top-100-sines-of-wolframalp...Aug 24, 2012 – Let's start with the simple input sin(x) for the function itself: ... Plot3dsin(sin(x + i y)) from x =-pi to pi and y =-pi to pi · Polar plot min(sin(x), sin(sqrt(2) x),sin(sqrt(3) x), ... Periodicity sin(4x + pi/3) ... Simplify sin(2 arctan(3x) + pi) ...

Calculus&Mathematica: Parametrization Helpsocrates.math.osu.edu/students/help/param.phpA: Because it isn't _usually_ too big a help until you run into 3-dimensions or higher. Forexample, when you write y = 3x, the coordinates to graph this one are (x,y) ... howabout trying x[t] = 2*Cos[t] and y[t] = 2*Sin[t] (and let t go from 0 to 2pi)?

y=2sin(x-pi/2)find the amplitude, phas shift, and period for ... - Algebrawww.algebra.com › Algebra › Trigonometry-basicsSOLUTION: y=2sin(x-pi/2)find the amplitude, phas shift, and period for the graph ofeach function. y=cos(2x-pi/4 y=-4sin(2x/3+pi/6) y= 5/4cos(3x-2pi) y= 2 ...

[PDF] Finding areas by integration - Math Centrewww.mathcentre.ac.uk/resources/uploaded/mc-ty-areas-2009-1.pdfFile Format: PDF/Adobe Acrobat - Quick Viewis given by the integral from x = 0 to x = 1 of the curve y = x(x − 1)(x − 2) = x3 − 3x2 +... x = 1 to x = 2: B = ∫. 2. 1 y dx. = ∫. 2. 1. (x3. − 3x. 2 + 2x)dx. = [ x4. 4 −. 3x3 .... (x2+ x + 4)dx. = [ x3. 3. + x2. 2+ 4x]. 3. 1. = [27. 3 + 9. 2 + 12] − [1. 3 + 1. 2 + 4] ...

Finding areas by integration - Maths Tutorwww.mathtutor.ac.uk/integration/findingareasbyintegration/exercise1/3. y = x2 + 3x from x = 1 to x = 3. y = x2 - 4 from x = -2 to x = 2. y = x - x2 from x =0 to x = 2. Find the area enclosed by ... area: 1 (i). Attempt the following questionsNotation: enter (e.g.) 15/4, pi, sqrt(2) ... evaluate (4x - x2)dx. 6. 0. . . . area: ...

Definite Integrals | [email protected] › Calculus › IntegrationFor example, the answer of $\int x dx$ is $\fracx^22$ $+ C$, where C is an arbitraryconstant. Now, let us say that .... Solution: $\int_1^3 [3x^2 - 2x + 1] dx =3\int_1^3 x_2 dx - 2 \int_1^3 x dx + \int_1^3 1 dx$ .... When $y = 1$, $t =tan^-11$ = $\frac\Pi 4$ ... Question: Solve $\int 5\sqrt4x - 2dx$ Solution: ...

Jiskha Homework Help - Search: Basic Calculus-Relative Extremawww.jiskha.com/search/index.cgi?query=Basic+Calculus-Relative...Use the first derivative test for the function f(x)=x^3-3x+2 ... y' = cosx - sinx = 0 = cosx= sinx cosx/cosx = sinx/cosx 1 = tanx x = pi/4, 5pi/4 Make a sign chart... y&#. .... Findthe relative extrema of f(x) = -2x^2 + 4x + 3 I don't know how to use the ...


y = (-3x^3 - 4x^2 + pi)/(x^2 - 4) Sign in



y = −3x3 − 4x2 +πx2 − 4

Q.(18.1:(Interest(Why(do(banks(pay(you(interest(on(your(savings?((A)  The(law(requires(them(to(pay(you(interest.(B)  They(are(paying(you(a(fee(to(invest(your(

money.(C)  If(you(have(your(money(in(that(bank,(you(are(

likely(to(get(a(loan(from(them.(D)  The(government(pays(the(bank(for(every(

customer(they(have(saving(money.(

Q.(19.1:(Reducing(an(Expression(Reduce((S8)2/3(+((16)S3/4.((A)  S12(B)  S4(C)  S31/8(D)  31/8(E)  33/8(

“20%”(at(Different(Compoundings(

Q.(20.1:(Con,nuous(Interest(Which(is(beter(at(the(end(of(one(year:(An(account(that(pays(10.5%(interest(annually(or(one(that(pays(10%(con,nuously?((A)  10.5%(annually.(B)  10%(con,nuously.(C)  They(are(the(same.(D)  There(is(no(way(to(tell.(

Q.(20.2:(Exponen,al(Func,ons(Which(of(the(following(is(an(exponen,al(func,on(that(goes(through((2,(3)(and((3,(1)?((A)  f(x)(=((3/4)*2x(B)  f(x)(=(12*(1/2)x(C)  f(x)(=(12*(1/4)x(D)  f(x)(=(27*(1/3)x(

Q.(20.3:(Exponen,al(Func,ons(An(exponen,al(func,on(has(an(xSintercept.((A)  True,(and(I(am(confident.(B)  True,(but(I(am(not(confident.(C)  False,(but(I(am(not(confident.(D)  False,(and(I(am(confident.(E)  Some(exponen,al(func,ons(do(and(some(

don’t.(

Q.(20.4:(Exponen,al(Decay(Caffeine(leaves(the(body(at(a(con,nuous(rate(of(17%(per(hour.((How(much(caffeine(is(ler(in(the(body(8(hours(arer(drinking(100(mg(of(caffeine?(((A)  389.62(mg(B)  22.52(mg(C)  25.67(mg(D)  There(is(no(way(to(tell.(

Napier’s(Logarithms(Table(

Q.(21.1:(Logarithms(

(a) logd k = p

(b) logk d = p

(c) logp d = p

(d) logk p = d

13. What is the value of log11 86? (Calculators are allowed.)

(a) .4049

(b) .5383

(c) 1.8576

(d) −2.0564

14. What is 3 = log2 8 in exponential form?

(a) 28 = 3

(b) 32 = 8

(c) 83 = 2

(d) 23 = 8

15. What is k = logm q in exponential form?

(a) mk = q

(b) kq = m

(c) mq = k

(d) qm = k

16. What is 42 = 16 in logarithmic form?

(a) log2 4 = 16

(b) log4 16 = 2

(c) log4 2 = 16

(d) log16 4 = 2

17. What is 3−1 =1

3in logarithmic form?

5


What(is(logb(S1)?((A)  S1(B)  1/b(C)  We(need(to(know(what(b(is.(D)  Undefined(for(any(b.(


(c) (d)

8. Which of the following functions have vertical asymptotes of x = 3?

(a) y = ln(x/3)

(b) y = ln(x − 3)

(c) y = ln(x + 3)

(d) y = 3 lnx

9. log!

M−NM+N

"

=

(a) 2 log M

(b) 2 log N

(c) −2 log N

(d) log(M − N) − log(M + N)

10. If log10(x − a) = n, then x =

(a) 10a+n

(b) a + 10n

(c) n + 10a

(d) n + a10

11. What is the exponential form of logr m = j?

(a) rj = m

(b) jr = m

(c) mj = r

(d) rm = j

12. What is the logarithmic form of kp = d?

4

Reduce(the(expression(as(much(as(possible.(


(c) (d)

8. Which of the following functions have vertical asymptotes of x = 3?

(a) y = ln(x/3)

(b) y = ln(x − 3)

(c) y = ln(x + 3)

(d) y = 3 lnx

9. log!

M−NM+N

"

=

(a) 2 log M

(b) 2 log N

(c) −2 log N

(d) log(M − N) − log(M + N)

10. If log10(x − a) = n, then x =

(a) 10a+n

(b) a + 10n

(c) n + 10a

(d) n + a10

11. What is the exponential form of logr m = j?

(a) rj = m

(b) jr = m

(c) mj = r

(d) rm = j

12. What is the logarithmic form of kp = d?

4

Q.(22.2:(Logarithm(Equa,ons(

(a) log3(−1) =1

3

(b) log−1

1

3= 3

(c) log 1

3

3 = −1

(d) log3

1

3= −1

18. What is the inverse of the following function:

P = f(t) = 16 ln(14t)

(a) f−1(P ) = 114e

16P

(b) f−1(P ) = 114e

P/16

(c) f−1(P ) = 114 ln(P/16)

(d) f−1(P ) = ln 1614 P

19. Solve for x if 8y = 3ex.

(a) x = ln 8 + ln 3 + ln y

(b) x = ln 3 − ln 8 + ln y

(c) x = ln 8 + ln y − ln 3

(d) x = ln 3 − ln 8 − ln y

20. Solve for x if y = e + 2x

(a) x = ln y−1ln 2

(b) x = ln(y−1)ln 2

(c) x = ln yln 2 − 1

(d) x = ln(y−e)ln 2

21. Write the following expression using a single logarithmic function:

ln(2x3 + 1) + 5 ln(3 − x) − ln(6x5 + 2x + 1).

(a) ln(−6x5 + 2x3− 7x + 15)

(b) ln[(2x3 + 1)(15 − 5x)(−6x5− 2x − 1)]

6

Q.(22.3:(Func,on(Shirs(

(a) I

(b) II

(c) III

(d) IV

13. How is the graph of y = 2x−1 + 3 obtained from the graph of y = 2x?

(a) Move 1 down and 3 right

(b) Move 1 left and 3 up

(c) Move 1 up and 3 right

(d) Move 1 right and 3 up

6

How(is(the(graph(of(y(=([1/(x(–(1)](+(3(obtained(from(the(graph(of(y(=(1/x(?(

Q.(23.1:(Shirs(and(Reflec,ons(

Take(f(x)(and(1.(shir(it(right(h(units.((2.(Reflect(the(result(of(1(across(the(ySaxis.((3.(Reflect(the(result(of(2(across(the(xSaxis.((Shir(the(result(of(4(up(k(units.((The(end(result(is:((A)  f(x(+(h)(+(k(B)  f(x(–(h)(+(k(C)  Sf(Sx(–(h)(+(k(D)  Sf(Sx(+(h)(+(k(E)  None(of(the(above(

Q.(23.2:(Composi,on(with(Equa,ons(

If(f(x)(=(x2(+(6(and(g(x)(=(x(–(3,(what(is(f(g(x))?((A)  x2(+(3(B)  x2(–(6x(+(15(C)  x2(–(3(D)  x3(–(3x2(+(6x(–(18(

Q.(23.3:(Composi,on(with(Graphs(What(is(g(f(4))?((A)  S1(B)  0(C)  1(D)  2(E)  3(

(a) -1

(b) 0

(c) 1

(d) 2

(e) 3

(f) 5

4. The graphs of f and g are shown in the figure below. Estimate the value of f(g(2)).

(a) -1

(b) 0

2

Q.(24.1:(Func,on(Inverse(

(b) x2 − 6x + 15

(c) x2 − 3

(d) x3 − 3x2 + 6x − 18

23. Which of the following functions IS invertible?

(a) f(x) = −x4 + 7

(b) g(x) = e3x/2

(c) h(x) = cos(x)

(d) k(x) = |x|

24. Let f(x) = x − 2 and g(x) = 3 − x2. Find g(f(2)).

(a) -3

(b) 0

(c) 3

(d) 2

25. If P = f(t) = 3 + 4t, find f−1(7).

(a) 31

(b) 1

7

(c) 0

(d) 1

26. Let f(x) = x2 and g(x) = x + 2. True or false? The domain of the functionf

gis R, all

real numbers.





27. Let f(x) = x2 − 4 and g(x) =√

x. Find (g f)(x) and the domain of g f .

9

Q.(24.2:(Increasing/Decreasing(Func,ons(

Any(func,on(f(x)(that(has(an(inverse(must(be(either(increasing(or(decreasing.((A)  True,(and(I(feel(confident.(B)  True,(but(I(don’t(feel(confident.(C)  False,(but(I(don’t(feel(confident,(D)  False,(and(I(feel(confident.(

Q.(24.3:(Finding(an(Inverse(Find(the(inverse(of(f(x)(=(1/x.((A)  f(S1(y)(=(y(B)  f(S1(y)(=(y2(C)  f(S1(y)(=(1/y(D)  f(S1(y)(=(xy(E)  No(inverse(of(f(exists.(

Eratosthenes(on(Cosmos(Click(here(for(a(video(explaining(why(Eratosthenes((240(BC)(thought(the(world(was(round,(and(how(he(was(inspired(to(find(the(circumference(of(the(earth.(

Q.(26.1:(Arc(Length(

Find(the(length(of(the(arc(spanned(by(an(angle(of(240°(on(a(circle(of(radius(3(feet.((A)  2(feet(B)  4(feet(C)  2π(feet(D)  4π(feet(E)  6π(feet(

Eratosthenes’s(Observa,on(

Q.(26.2:(Eratosthenes(on(Mars(A(Mar,an(decides(to(do(Eratosthenes’s(experiment(and(finds(that(the(angle(of(the(sun(in(the(sky(changes(18°(between(two(towns(650(miles(apart.((What(is(the(circumference(of(Mars?((A)  13,000(miles(B)  6,500(miles(C)  6,500π(miles(D)  1,300π(miles(E)  We(don’t(know(enough(about(the(problem.(

Q.(26.3:(Angles(in(the(Plane(What(is(a(possible(value(for(this(angle?(((A)  S45°(B)  320°(C)  280°(D)  S405°(E)  685°(

Q.(27.1:(Similar(Triangles(

The(two(triangles(in(the(figure(are(similar.((What(is(b/c?((A)  1/4(B)  1/2(C)  2/5(D)  4/5(E)  We(can’t(conclude(

anything.(

a(

b(

c(2(

4(5(

Q.(27.2:(Finding(Cosine(

(What(is(cos(60°)?((A)  ½(B)  2(C)  √(3)/2(D)  2/√(3)(E)  √(3)(

Q.(28.1:(Calcula,ng(Sine(

What(is(sin(A)?((A)  b/c(B)  c/b(C)  a/c(D)  a/b(E)  We(can’t(say(from(the(

informa,on(given.(

a(

b(

c(

Q.(28.2:(Calcula,ng(Cosine(

If(sin(θ)(=(5/7(,(find(cos(θ).((A)  24/49(B)  √(24)/7(C)  5/7(D)  2/7(E)  We(can’t(say(from(the(informa,on(given.(

Q.(28.3:(Calcula,ng(Sine(

Find(sin(135°).((A)  ½(B)  √(3)/2(C)  1/√(2)(D)  S1/√(2)(E)  We(can’t(say(from(the(informa,on(given.(

Recall:(Archimedes’(96Sgon(

1/15/13 10:59 PMNOVA | Infinite Secrets | Approximating Pi (non-Flash) | PBS

Page 2 of 2http://www.pbs.org/wgbh/nova/archimedes/pi-nf.html




actual value of [pi] = 3.1416

Note: All figures rounded off to four decimal places.

ContemplatingInfinityPhilosophically, theconcept remains amind-bender.

Working with InfinityMathematicians havebecome increasinglycomfortable with theconcept.

Great SurvivingManuscriptsAncient documentsoffer a tantalizingglimpse of lostcultures.

The ArchimedesPalimpsestFollow the 1,000-year-long journey ofthe Archimedesmanuscript.

Approximating PiSee Archimedes'geometrical approachto estimating pi.

Infinite Secrets homepage | NOVA homepage

© | Created September 2003

Q.(29.1:(Area(Formula(

Give(a(formula(for(the(area(of(this(triangle.((A)  (½)ab*sin(C)(B)  (½)bc*sin(A)(C)  (½)ac*sin(B)(D)  All(of(the(above(E)  None(of(the(above(

Q.(29.2:(Law(of(Sines(Classroom Voting Questions: Precalculus

The Law of Sines and the Law of Cosines

1. Given a triangle with angles A, B, and C and opposite sides a, b, and c, find themeasurements of the remaining angle and sides assuming that B = 30, C = 100 andb = 20 feet.

(a) A = 50, a ≈ 39.39 feet, c ≈ 30.64 feet

(b) A = 50, a ≈ 10.50 feet, c ≈ 20.26 feet

(c) A = 50, a ≈ 30.64 feet, c ≈ 39.39 feet

(d) A = 230, a ≈ 30.64 feet, c ≈ 39.39 feet

2. A big pine tree has grown so that it is tilted 3 from vertical toward the sun. Whenits shadow is 20 feet long, the angle of elevation from the tip of its shadow to the topof the tree is 60. Approximately how tall is the tree (i.e. what is its length)?

(a) 32 feet

(b) 38 feet

(c) 44 feet

(d) 331 feet

3. True or False: Two angles and a side determine a unique triangle.





4. True or False: Given the measurements of any two angles and one side of an obliquetriangle, the triangle can be completely solved using the law of sines.





1

Classroom Voting Questions: Precalculus

Trigonometric Functions: Amplitudes, Periods, and

Graphs

1. Which of the following is the approximate value for the sine and cosine of angles Aand B in the figure below.

(a) sin A ≈ 0.5, cos A ≈ 0.85, sin B ≈ −0.7, cos B ≈ 0.7

(b) sin A ≈ 0.85, cos A ≈ 0.5, sin B ≈ −0.7. cos B ≈ 0.7

(c) sin A ≈ 0.5, cos A ≈ 0.85, sin B ≈ 0.7, cos B ≈ 0.7

(d) sin A ≈ 0.85, cos A ≈ 0.5, sin B ≈ 0.7, cos B ≈ 0.7

2. The amplitude and period of the function below are

1

Q.(30.1:(Unit(Circle(Approximate(sine(and(cosine(for(A(and(B.((A)  sin(A(≈(0.5,(cos(A(≈(0.85,(

sin(B(≈(S0.7,(cos(B(≈(0.7(B)  sin(A(≈(0.85,(cos(A(≈(0.5,(

sin(B(≈(S0.7,(cos(B(≈(0.7(C)  sin(A(≈(0.5,(cos(A(≈(0.85,(

sin(B(≈(0.7,(cos(B(≈(0.7(D)  sin(A(≈(0.85,(cos(A(≈(0.5,(

sin(B(≈(0.7,(cos(B(≈(0.7(

Q.(30.2:(Unit(Circle(Use(the(unit(circle(to(find(cos(300°).((A)  0(B)  1/2(C)  S1/2(D)  √(3)/2(E)  S√(3)/2(

Unit(Circle(

Q.(30.3:(Unit(Circle(Use(the(unit(circle(to(find(tan(210°).((A)  √(3)(B)  S√(3)(C)  1/√(3)(D)  S1/√(3)(E)  Undefined(

Q.(30.4:(Bounds(for(Tangent(

We(have(the(inequali,es(S1(≤(sin(θ)(≤(1(and(S1(≤(cos(θ)(≤(1.((Do(you(think(that(we(have(similar(inequali,es(for(tan(θ)?((A)  Yes,(and(I(feel(confident.(B)  Yes,(but(I(don’t(feel(confident.(C)  No,(but(I(don’t(feel(confident.(D)  No,(and(I(feel(confident.(E)  I(don’t(understand(the(ques,on.(

Q.(31.1:(Radians(What(is(the(radian(measure(of(a(216°(angle?((A)  108π(B)  5π/6(C)  6π/5(D)  8π/9(E)  Impossible(to(tell.(

Unit(Circle(with(Radians(

Q.(31.2:(Unit(Circle(Use(the(unit(circle(to(find(sin(2π/3).((A)  0(B)  1/2(C)  1/√(2)(D)  √(3)/2(E)  1(

Q.(32.1:(Amplitude(and(Period(Give(the(amplitude(and(period(of(this(graph.((A)  A(=(2,(P(=(2(B)  A(=(2,(P(=(3(C)  A(=(2,(P(=(½(D)  A(=(3,(P(=(2(E)  A(=(3,(P(=(1/2(

(a) Amplitude = 2, Period = 2

(b) Amplitude = 2, Period = 3

(c) Amplitude = 2, Period = 1/2

(d) Amplitude = 3, Period = 2

(e) Amplitude = 3, Period = 1/2

3. What is the equation of the function shown in the graph?

(a) y = 3 sin(2x) + 2

(b) y = 3 cos(2x) + 2

(c) y = 3 sin(πx) + 2

(d) y = 3 cos(πx) + 2

(e) y = 3 sin( 1

πx) + 2

2

Q.(32.2:(Period(of(Trig(Func,on(

(c) π

(d) 2π

13. What is the amplitude of f(x) = −2 sin x?

(a) 1

(b) 2

(c) −2

14. What is the period of f(x) = −3 sin(2x)?

(a) 3

(b) -3

(c) π

(d) 2π

15. What is the period of f(x) =1

5tan(2x)?

(a)1

5(b) 2π

(c) π

(d)π

2

(e)π

4

16. Which of the basic trig functions below are odd functions?

(a) f(x) = sin(x).

(b) f(x) = cos(x).

(c) f(x) = tan(x).

(d) (a) and (b).

(e) (a) and (c).

(f) (b) and (c).

(g) (a), (b), and (c).

(h) None of the above.

7

Q.(33.1:(f(x)(=(sin(x)(

Do(you(think(that(f(x)(=(sin(x)(has(an(inverse?((A)  Yes,(and(I(feel(confident.(B)  Yes,(but(I(don’t(feel(confident.(C)  No,(but(I(don’t(feel(confident.(D)  No,(and(I(feel(confident.(E)  I(don’t(understand(the(ques,on.(

Q.(33.2:(Inverse(Trig(Func,ons(

Classroom Voting Questions: Precalculus

Inverse Trigonometric Functions

1. What is arcsin

!

1

2

"

?

(a) 0

(b)π

6

(c)π

4

(d)π

3

2. What is sin−1

#

−

√

3

2

$

?

(a)π

3

(b)5π

3

(c) −

π

3

(d)4π

3

3. What is arcsin(−1)?

(a)3π

2

(b)π

2

(c) −

π

2

(d) −

3π

2

4. What is arccos

!

1

2

"

?

1

Q.(33.3:(Inverse(Trig(Func,ons(

(a)π

6

(b)π

4

(c)π

3

(d)π

2

5. What is cos−1(−1)?

(a) 0

(b) π

(c) −π

(d) 2π

6. What is arccos!

−

√

2

2

"

?

(a)π

4

(b) −

π

4

(c)3π

4

(d)5π

4

7. What is tan−1(1)?

(a)π

2

(b)π

3

(c)π

4

(d)π

6

8. What is arctan(−√

3)?

(a)2π

3

2

Q.(34.1:(Law(of(Cosines(

(a)700 sin 43

sin 72meters

(b)700 sin 65

sin 72meters

(c)700 sin 47

sin 108meters

(d)700 sin 25

sin 108meters

14. Triangle ABC has sides of length a, b, and c, and angles of measure α, β, and γopposite those sides, respectively. If α = 73, b = 7.0, and c = 3.0, find a.

(a) 58 − 42 cos 73

(b)√

58 − 42 cos 73

(c) 58 − 21 cos 73

(d)√

58 − 21 cos 73

15. Triangle ABC has sides of length a, b, and c, and angles of measure α, β, and γopposite those sides, respectively. If a = 5, b = 4, and c = 2, find α.

(a) cos

!

5

16

"

(b) cos−1

!

5

16

"

(c) cos

!

−5

16

"

(d) cos−1

!

−5

16

"

4

If(A(=(73°,(b(=(7,(and(c(=(3,(find(a.(

Q.(34.2:(Law(of(Cosines(Given(a(=(6,(b(=(8,(and(c(=(11,(find(A,(B,(and(C.(

9. Given an oblique triangle with sides a = 3 and b = 8, and opposite angle A = 30, findthe measurement of opposite angle B.

(a) B ≈ 11

(b) B ≈ 42

(c) either (a) or (b)

(d) There is no solution.

10. A person leaves his house and walks 2 miles west and then 4 miles northwest. Approx-imately how far is he from home?

(a) 31.3 miles

(b) 2.95 miles

(c) 6 miles

(d) 5.6 miles

11. Given a triangle with sides a = 6, b = 8, and c = 11, find opposite angles A, B, andC.

(a) A ≈ 32, B ≈ 103, C ≈ 45

(b) A ≈ 45, B ≈ 32, C ≈ 103

(c) A ≈ 32, B ≈ 45, C ≈ 103

(d) There is no solution.

12. Triangle ABC has sides of length a, b, and c, and angles of measure α, β, and γopposite those sides, respectively. If α = 42, γ = 59, and b = 45, find a.

(a)45 sin 42

sin 79

(b)45 sin 42

sin 59

(c)45 sin 79

sin 42

(d)45 sin 59

sin 42

13. Alan at position A spots a deer bearing S43E. Bob at position B, 700 meters due eastof position A, spots the deer bearing S65W. How far from Alan is the deer?

3

Q.(34.3:(Systems(of(Equa,ons(True(or(False:(Every(system(of(equa,ons(has(a(solu,on.((A)  True,(and(I(feel(confident.(B)  True,(but(I(don’t(feel(confident.(C)  False,(but(I(don’t(feel(confident.(D)  False,(and(I(feel(confident.(E)  I(don’t(understand(the(ques,on.(

Q.(35.1:(Systems(of(Linear(Equa,ons(True(or(False:(All(systems(of(linear(equa,ons(have(solu,ons.((A)  True,(and(I(feel(confident.(B)  True,(but(I(don’t(feel(confident.(C)  False,(but(I(don’t(feel(confident.(D)  False,(and(I(feel(confident.(E)  I(don’t(understand(the(ques,on.(

Solve(the(system(of(equa,ons(below.((((A)  x(=(4/3,(y(=(0(B)  x(=(½,(y(=(S½(C)  x(=(0,(y(=(2(D)  Infinite(number(of(solu,ons.(E)  No(solu,ons.(

CLICKER QUESTION 5.1

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

x

y

Which of the following systems of equations can be represented by the graph below?

A. 3x + 3y = −6, x + 2y = 3

B. x − y = −5, 2x + y = 4

C. −8y + 2x = 4, 2x + 4y = −8

D. −x + 3y = 9, 2x− y = 4

CLICKER QUESTION 5.2What is the solution to the following linear system of equations?

−3x + 2y = 412x − 8y = 10

A. x = 4/3, y = 0

B. x = 1/2, y = −1/2

C. x = 0, y = 2

D. There are an infinite number of solutions to the system.

E. There are no solutions to the system.

CLICKER QUESTION 5.3

Which of the graphs below could represent the following linear system?

3x − y = 2−9x + 3y = −6

4

Q.(35.2:(Systems(of(Linear(Equa,ons(

Clicker(Ques,ons(and(Addenda( - Phillip Compeau | CMU...

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