Maddie, Rachel & Cameron Pre-Calculus 11 Sequences & Series

Trigonometry

Quadratics

Radicals

Rational Expressions & Equations

Absolute Value & Reciprocal Functions

Systems of Equations

Linear & Quadratic Inequalities A Course Summary ARITHMETIC SEQUENCES Sequences & Series A sequence is an ordered list of terms that follows a pattern or rule tn=t1+(n-1)d

d=tn-tn-1 A sequence in which the difference between consecutive terms is constant and we call this difference the common difference denoted by d A population of mice grows by 2000 every year. Suppose we have 4000 mice the first year, how long will it take for the population to grow to 20 000? d=2000 20 000 = 4000+(n-1)2000

t1= 4000 20 000 = 4000+2000n-2000

tn= 20 000 18 000 = 2000n

n=? 9 = n Sn: sum of the nth term

t1: first term

d: difference

tn: nth term

n: number of terms ARITHMETIC SERIES Sn=n/2[2t1+(n-1)d]

OR

Sn=n/2[t1+tn] A sum of numbers with a common difference Find the sum: (-15)+(-8)+(-1)+6+13...+146

t1: -15 tn=t1+(n-1)d Sn=n/2[t1+tn]

d: 7 146=-15+(n-1)7 S24=24/2[-15+146]

n: ? 146=-15+7n-7 S24=12[-15+146]

Sn: ? 168=7n S24=12[131]

tn: 146 24=n S24=1572 Mr. Smith GEOMETRIC SEQUENCES A geometric sequence is a list of numbers that have a common ratio r=tn/tn-1 A company stores 5kg of a radioactive material. After one year, 92% of this material remains. If the percentage of material that decays is constant per year, how much of the 5kg substance remains after 10 years? Initial amount: 5kg

After 1 yr: 5(0.92)=4.6

t1: 4.6

r: 0.92

n: 10 tn=t1(r)^n-1

tn=4.6(0.92)^10-1

tn=4.6(0.92)^9

tn=2.172kg GEOMETRIC SERIES A sum of numbers with a common ratio Sn=t1(r^n-1)/(r-1) A tennis tournament has 128 players. If players win their match, they go on to play in the next round. If they lose they are out. what is the total number of matches in the tournament?

64+32+16...+1

t1: 64 tn=t1(r)^n-1 S7=64(0.5)^7-1)

r: 0.5 1=64(0.5)^n-1 (0.5-1) 0.5-1 1/64=(0.5)^n-1 S7= 127

n: ? n=log(1/64) +1

tn: 1 log(0.5)

Sn: ? n= 7 INFINITE GEOMETRIC SERIES Convergent Series: has a finite sum

Divergent Series: does not have a finite sum If -1<r<1 then the series is convergent in the formula for its sum is S∞=t1/1-r 12+3+3/4+3/16+... -1<0.25<1 -> TRUE

r: 0.25 S∞=12/1-0.25

t1: 12 S∞=16 Trigonometry STANDARD POSITION

An 800m boom is used to move a bundle of piping from point A to point B. Find the EXACT horizontal displacement of the end of the boom if the operator raises it from 30° to 60°

COS(theta)=adj COS(theta)=adj DISP =(4√3-4)m

hyp hyp

COS60=b/8 COS30=a/8

1/2=b/8 √3/2=a/8

2b=8 2a=8√3

b=4 a=4√3 Quadrant II Quadrant I

90<theta<180 0<theta<90

Quadrant III Quadrant IV

180<theta<270 270<theta<360 special

triangles SIN30=1⁄2

COS60=1⁄2

TAN60=√3 SIN60=√3⁄2

COS60=1⁄2

TAN60=√3 SIN45=1⁄√2

COS45=1⁄√2

TAN45=1 Vertex is at (0,0)

Initial arm is on positive x-axis TRIGONOMETRIC RATIOS If the terminal arm of an angle lies on an axis, the angle is called a Quadrantal Angle The Quadrantal Angles are: 0 Degrees 90 Degrees 180 Degrees 270 Degrees 360 Degrees I All Positive II Sin Positive III Tan Positive IV Cos Positive Radicals Mixed Radical: The product of a monomial and a radical

-8³√45

Entire Radical: a radical with a coefficient of 1 or -1

√30, -√984³

Radical in Simplest Form: a radical where the radicand does not contain a fraction, the denominator does not contain a radical and the radicand does not contain a factor that can be removed √20

=√4√5

=2√5 coefficient index radicand sign radicand RESTRICTIONS

denominator cannot be zero

if index is even, the radicand must be positive

if the index is odd, the radicand can any real number Express as a mixed radical in simplest form

√192

• =√64√3 • =√2(2)(2)(2)(2)(2)(2)(3)

=8√3 =√2(2) √2(2) √2(2) √3

=(2)(2)2√3

=8√3 ADDING & SUBTRACTING Combine the coefficients

a^n√x±b^n√x = (a±b)^n√x

√27 + 2√12 = √9√3 + 2√4√3

= 3√3 + (2)2√3

= 3√3 + 4√3

= 7√3 MULTIPLYING & DIVIDING multiply coefficients & multiply radicands

(m^n√a)(j^n√b) = mj^n√ab

(3√5)(√10) = 3(1)√(5)(10)

= 3√(5)(2)(5)

= 3(5)√2

= 15√2 divide coefficients & divide radicands

m^n√a = m ^n√a

j^n√b = j ^n√b

25√40 = 25 √40

30√20 30 √20

= 5/6 √2

= 5√2

6 RATIONALIZING THE DENOMINATOR multiply numerator and denominator by a radical that will rationalize the denominator

8√5 = 8 (√5) (√3)

2√3 = 2 (√3) (√3)

= 4√15

3

multiply numerator and denominator by the conjugate of the denominator

Conjugates: a+b & a-b

4+√x & 4-√x

5√2+√7 & 5√2-√7 7 = 7(5√3+√2)

5√3-√2 (5√3-√2)(5√3+√2)

= 35√3+7√2

(25)(3)-2

= 35√3+7√2

73 (a+b)(a-b) = a²-ab+ba-b²

= a²-b²

(4+√x)(4-√x) = 16-4√x+4√x-x

= 16-x RADICAL EQUATIONS 1) Isolate one of the radicals

2) Eliminate a root

3) Check the answer/root

4) Check for extraneous roots: √4x -7=13

√4x =20 ³√4x =20

(√4x)² = 20² (³√4x)³ = 20³

4x = 400 4x = 8000

x = 100 x = 2000

left side right side

√4x -7 13

√4(100) -7 13

√400 -7 13

20-7 13

13 13

x ≠ (-) Quadratics Linear and Quadratic

Inequalities Absolute Values, Functions & Equations VERTEX FORM f(x) =a(x-p)²+q The shape is a parabola

VERTEX is the point (p,q)

AXIS OF SYMMETRY is the line x=p

SHAPE & DIRECTION OF OPENING is determined by “a”: if a>0 then the parabola is happy U, but if a<0 then the parabola is sad n AND if a<-1 or a>1 then the parabola is skinny, but if -1<a<1 then the parabola is wide. Sketch graph, state vertex, direction of opening, equation of axis of symmetry, maximum/minimum value of graph, and the domain & range Y=-3(x+4)^2+1

Vertex: (-4,1)

Equation of axis of symmetry: x=-4

Direction of opening: a=-3 sad opens down

Maximum value: Y=1

Domain: (-∞,∞) All real numbers

Range: (-∞,∞] Y≤1 COMPLETING THE SQUARE Convert to vertex form

y = -3x²-24x-19 1.factor “a” out of first 2 terms

Y = -3(x^2+8x)-19

2.Half the middle coefficient and square it:

Middle coefficient = 8

Half = 4

Squared = 4^2 16

3.Add square and balance equation

Y = -3(x^2+8x+16-16)-19 4.Take the negative square out of brackets

Y = -3(x^2+8x+16)-19+48

5.Factor and simplify

Y = -3(X=4)^2+29 Y = -2x²+12x+2

Y = -2(x²+6x)+2

Y = -29x²-6x+9x-9)+2

Y = -2(x²-6x+9)+2+18

Y = -2(x-3)²+20 The cost of admission to a hockey game is $3 per ticket, If 800 people attend. If the price per ticket is increased, the attendane will decrease by 200 people, for each $2 increase in price. Find the optimal ticket price.

Let x of price increase

Price = 3+x

Attendance = 800-200/2

R = (3+x)(800-200/2x)

R = 2400-300x+800x-100x²

R = -100x²+500x+2400

y int (0,2400) = -100(x²-5x)+2400

R = -100(x²-5x+6.25-6.25)+2400

R = -100(x²-5x+6.25)+2400+6.25

R = -100(x-2.5)²+3025

V = (2.5, 3025)

x y

Max revenue: $3025 R

Price increase: $2.50 X

Price: 3+x = 3+2.50

= $5.50 GRAPHICAL SOLUTIONS Quadratic equation: An equation with one term squared and no other term is raised to a higher power

Root: A solution to a quadratic equation, which is the same as an x intercept to the corresponding quadratic function.

Ex) x²+5x+6=0

Roots: x = -2, -3 1.when no points are touching the x axis there are zero real roots

2.when one point is touching the x axis there is one real root

3.when two points are touching the x axis there are two real roots FACTORING 1.X²-3x-40 = (x-8)(x+5)

-8 x 5 = -40

-8+5 = -3

Difference of squares: A²-B² = (A+B)(A-B)

1.X²/16 -25Y² = (X/4+5Y)(X/4-5Y)

Factoring ax²+bx+c:

2x²2-x-6 = 2x²-4x+3x-6

= 2x(x-2)+3(x-2)

Let a = x-2 2xa+3a

= a(2x+3)

= (x-2)(2x+3) 4(x+3) 2+8(x+3)-5 Let a = x+3

4a²+8a-5

= 4a²+10a-2a-5

= 2a(2a+5)-1(2a+5)

= (2a+5)(2a-1)

= (2[x+3]+5)(2[x+3]-1)

= (2x+6+5)(2x+6-1)

= (2x+11)(2x+5) Zero product property: If two factors have a product of zero then one or both factors must be zero

Ex: PQ = 0 P = 0 or Q = 0 QUADRATIC FORMULA We can solve any quadratic equation of the form ax^2+bx+c=0 using the quadratic formula: The nature of the roots:

b²-4ac<0 b²-4ac=0 b²-4ac>0

no real roots One real root Two real roots b²-4ac is called the Discriminant 5x^2-7x+4=0 Discriminant = b^2-4ac

a=5, b=-7, c=4 D = (-7)^2-4(5)(4)

D = 49 – 80

D = -31

D < 0 = NO REAL ROOTS Systems of Equations An expression that can be

written as a ratio of polynomials Rational Expressions

& Equations NON-PERMISSIBLE VALUES values that make the denominator 0

a rational expression is defined for all values of the variable except the non-permissible values x²+7x+6/x²-36

x²-36=0

x²=36

x=±6 NON-P= ±6 any polynomial divided by its opposite is -1

a-b/b-a=-1 MULTIPLYING & DIVIDING factor each denominator and numerator

determine non-permissible values

multiply numerators together & denominators together

simply / reduce NoNo YesYes

x-1/3x=-1/3 (x+4)(x-1)/3x(x+4) = x-1/3x A/B ÷ C/D = (A/B)(D/C)

B≠0 D≠0 C≠0 ADDING & SUBTRACTING factor each denominator if necessary

express rational expressions with a common denominator

add or subtract the numerators; denominator doesn't change

simplify / reduce (10a+5/ab) - (3a-2/ab)

10a+5-3a+2/ab

7a+7/ab

7(a+1)/ab

a≠0 b≠0 RATIONAL EQUATIONS factor all denominators & determine the LCD

identify all non-permissible values

multiply all terms on both sides of the equation by the LCD & simplify

solve for the variable

check that the solution is permissible and makes sense in the context of the question

verify that any remaining solutions are correct in the equation GRAPHING a system of equations is 2 or more different equations involving the same variable

determining the solution to a system of equations means determining the point(s) that are a called a point of intersection linear-quadratic quadratic-quadratic (also when the quadratic equations are the same, there is an infinite number of solutions) ALGEBRA Elimination x²-x-y=6

2x-y=2

1) x²-x-6=y

2) 2x-2=y

1)-2) : x²-3x-4=0 Substitution x²-x-y=6

y=2x-2

x²-x-(2x-2)=6

x²-3x-4=0 (x-4)(x+1)=0

x=4, x=-1

2(4)-2=y 2(-1)-2=y

8-2=y -2-2=y

6=y -4=y

(4,6) (-1,-4) Determine 2 integers that have the following relationships:

14 more than twice the first integer gives the second one. The second integer increased by one is the square of the 1st integer

2x+14=y y=2x+14

y+1=x² y=x²-1

2(5)+14=y 0=x²-2x-15

24=y 0=(x-5)(x+3)

2(-3)+14=y x=5, x=-3

8=y ABSOLUTE VALUE Absolute Value represents the distance from zero on a number line I-4I means absolute value of -4

ex) I-4I = 4 & I4I = 4 ex) I2I - I3(-4)I

= I2I - I-12I

= 2 - 12

= -10 On stock markets, individual stock and bond stock values fluctuate a great deal. A particular stock on the Toronto stock exchange opened the month at $13.55 per share, dropped to $12.70, increased to $14.05 and closed the month at $13.85. Determine the total change in the value for the stock for the month. Let the stock values be represented by

Vi = 13.55 V2 = 12.70 V3 = 14.05 V4 = 13.85 Each change in value = IVi + 1 - ViI Where i = 1,2,3 Total change = IV2 - V1I + IV3 - V2I + IV4 - V3I Total change = I12.70 - 13.55I + I14.05-12.70I + I13.85 - 14.05I Total change = I-0.85I + I1.35I + I-0.2I Total change = 0.85 + 1.35 + 0.2 Total change = $2.40 GRAPHING ex) y = IxI ex) y = Ix-3I ABSOLUTE VALUE FUNCTIONS To graph an absolute function:

use y = f(x) and y = -f(x) to sketch the function

for y = -f(x) all negative y values are reflected in the x axis The domain of an absolute fuction y = If(x)I is the same domain of y = f(x) The range of an absolute function, y = If(x)I depends on the range of f(x). For an absolute linear or absolute quadratic function, the range will usually, but not always, be [0, infiniti) An Invarient Point is a point that remains unchanged when a transformation is applied to it ex) y = I-x^2+2x+8I a) determine x & y intercepts xint:

0=-x^2+2x+8

x^2-2x-8=0

(x-4)(x+2)=0

x=4,-2

(-2,0) & (4,0) yint:

y = I-0^2+2(0)+8I

y = I8I

y = 8

(0,8) b) Sketch the graph Original: y= -x^2+2x+8

y= -(x^2-2x+1-1)+8

y= -(x^2-2x+1)+8+1

y= -(x-1)^2+9

v = (1,9) c) determine Domain & Range d) determine the piecewise function ABSOLUTE VALUE EQUATIONS To solve an absolute value equation by graphing:

1. Graph the left side and the right side

2. The x coordinate of the point of intersection is the solution(s) To solve an absolute value equation by Algebra:

1. consider two seperate cases:

CASE1

-The expression inside the absolute value is greater than or equal to zero

CASE2

-The expression inside the absolute value is less than zero

2. The roots in each are solutions

3. There may be Extraneous Roots that need to be restricted ex) solve

Ix+3I = 8

CASE1

if x+3>/=0

x+3=8

x=8-3

x=5

CHECK

I5+3I=8

I8I=8

8=8

true CASE2

if x+3<0

x+3= -8

x= -8-3

x= -11

CHECK

I-11+3I=8

I-8I=8

8=8

true ex) use graphing calc RECIPROCAL FUNCTIONS For any function f(x) the reciprocal function is 1 f(x) The reciprocal function is not defined when the denominator is 0, so f(x) cannot equal 0

ex) if f(x)=x then 1 = 1 f(x) x Graphing:

an asymptote is a straight line that the graph approaches but never touches

the general equation of a vertical asymptote is x=a, where "a" is a non permissable value of 1/f(x)

as If(x)I gets very large, the absolute value of the reciprocal function I1/f(x)I approaches zero

the x axis defined by the equation y=0 is a horizontal asymptote

an invarient point does not change when a transformation is applied

for a reciprocal function, invarient points occur when f(x)=1, -1

to fine the x values of the invarient points, solve the equation f(x)=1,-1

if a point (x,y) satisfies the function y=f(x) then the point (x,1/y) satisfies the reciprocal function

as the value of x approaches a non-permissable value, the absolute value of the reciprocal function, I1/f(x)I gets very large ex) y=1/x^2-1

a) find non-permissable values

x^2-1=0

(x+1)(x-1)=0

x cannot equal 1, -1 b) find equation of vertical asymptotes

x=1, -1

c) find invarient points d) draw graph

y=x^2-1

v= (0, -1) e) find equation of horizontal asymptotes

y=0

f)find domain & range Examples: Sine Law You can use the sine law for any type of triangle A B c a b c SinA = SinB = SinC a b c When solving a triangle, you must analyze the given information to determine if a solution exists. If you are given 2 angles and a side, then the triangle is uniquely defined. However, if you are given 2 sides and an angle opposite one of them, then we have the ambiguous case and 3 possible outcomes:

No solution

One Solution

Two Solutions Examples: Cosine Law You can use the cosine law when given 2 sides and the angle in between or when given 3 sides ex) LINEAR INEQUALITY IN TWO VARIABLES different linear inequalities:

Ax+By<C

Ax+By≤C

Ax+By>C

Ax+By≥C when graphing:

> or < results in dotted line

ex: 2x+3y≤6 - Because it is less than or equal to, use a solid line Convert to y = mx+b

2x+3y≤6

3y≤-2x+6

3y/3-2x/3+6/3

Y ≤ -2x/3+2 2x+3y=6

X int: (y=0) 2x+3(0) = 6

2x/2 = 6/2

x = 3

(3,0) and (0,2) QUADRATIC INEQUALITIES IN ONE VARIABLE Solving quadratic equations by graphing can also be used to solve quadratic inequalities in one variable. Instead of stating the solution, the graph is used to identify the intervals of x-values where the y-values of the graph are above or below the y-axis 1.Y>0

The solution set is the values of x of which the graph of f(x) lies above the x-axis. The solution set does not include the x-intercepts.

2.Y≥0

The solution set is the values of x for which the graph of f(x) lies on or above the x-axis. The solution set include the x-intercept

3.Y<0

The solution set is the values of x for which the graph of f(x) lies below the x-axis. The solution set does not include the x-intercepts

4.Y≤0

The solution set is the value of x for which the graph of f(x) lies on or below the x-axis. The solution set includes the x-intercepts. Find roots: 2x^2-x+6x-3≤0

x(2x-1)+3(2x-)≤0

(x+3)(2x-1)≤0

x= -3, ½ 1.x= -4 (-4+3)(2(-4)-1)≤0

9≤0

FALSE!

2.x= 6 (0+3)(2(0)-1)≤0

-3≤0

TRUE! 3.x=1 (1+3)(2(1)-1)≤0

4≤0

FALSE!

Solution is the true interval(s): [-3, ½] QUADRATIC INEQUALITIES IN TWO VARIABLES V: (3,2)

X: (1,5) (1,0) & (5,0)

Y: -2 Y= a(x-1)(x-5)

2= a(3-1)(3-5)

2= a(2)(2)

-1/2 = a Y>-1/2(x-1)(x-5)

Y>-1/2x^2+3x-5/2

Y>-1/2(x-3)^2+2 y<2x^2-x+6x-3

y<x(2x-1)+3(2x-1)

y<(2x-1)(x+3)

x int: ½ & 3

y int: -5/4 & -49/8

= (-1.25, -6.125)

y is <(2x-1)(x+30)

= shade below parabola

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# Copy of Pre-Calculus 11

A course summary

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