Functions - Sir Wilfrid Laurier School Board Materials...Functions Quadratic all three forms...
Transcript of Functions - Sir Wilfrid Laurier School Board Materials...Functions Quadratic all three forms...
Functions
Quadraticall three formsequation
Square root
Linear Exponential and LogarithmicLog lawsasymptotesCompound Interest
Step round to the leftlength of stepgapopenclosedclosedopen
a, b, h, ksketching them, domain, rangefinding the rulesolving for x and yProperties (increasing, decreasing)finding the inverse
Rationalasymptotes
absolute value
PiecewiseFinding the rule of different functions on one graph
composites
Transformed Linear Function
y = ax + b or f(x) = ax + b
Ax + By + C = 0
x + y a b
a = slope =
b = yint
ΔyΔx
xint = b a
slope =
yint =
x int =
slope =
yint = b
x int = a
A B
C BC A
b a
y2 y1x2 x1=
General form
Intercept form
A is usually a positive whole number
Standard form
= 1
Functional Form
B and C are positive or negative whole numbers
quadratic formula (solving x):
x = b ±√4ac + b22a
quadratics
f(x) = a(x h)2 + k
g(x) = a(x r)(x s)
h(x) = ax2 + bx + c
Basic Quadratic Function Quadratic Function (Standard Form)
y = x2 y = a(x h)2 + k
f(x) = x2 f(x) = a(x h)2 + k
Quadratic Function (General Form)
y = Ax2 + Bx + C
b2 + 4ac 4a
b2a
h = k =
vertex is (h, k)a tells us if parabola opens up or down
vertex is (h, k)A tells us if parabola opens up or downC is the yinterceptfactor to find the zeros or use theQuadratic formula.
quadratic formula: x = b±√4ac + b2 2a
Factored Form:y = a(x r)(x s)a tells us if parabola opens up or down. It is the same value in all three forms.
The zeros of the function are r and s.
The vertex is (h, k) in between the zeros.h = (r + s) ÷2Sub h into rule to calculate k
Basic Absolute Function Transformed Absolute Function
y = ΙxΙ y = aΙx hΙ + k
f(x) = ΙxΙ f(x) = aΙx hΙ + k
f(x) = a(bΙx hΙ) + k
Basic Exponential Function Transformed Exponential Function
y = cx y = a(c)(b(x h)) + k
f(x) = cx
a > 0, b > 0
a < 0, b > 0
a > 0, b < 0
a < 0, b < 0
f(x) = a(c)b(xh) + k, c > 1
c(t) = a(1 + )ntin
a: initial investment
i: interest rate
n: number of times per year the value of your investment is calculated
t: time (days, months, years)
c(t): the value of your investment after time t
Compound Interest
The logarithmic function is the inverse of the exponential function.
Transformed Logarithmic Function
a > 0, b > 0
a < 0, b > 0
a > 0, b < 0
a < 0, b < 0
y = alogc(b(x h)) + k , c > 1
Transformed Logarithmic Function
a > 0, b > 0
a < 0, b > 0
a > 0, b < 0
a < 0, b < 0
y = alogc(b(x h)) + k , 0< c < 1
The Laws of Log1. The Product Law: logc(mn) = logc(m) + logc(n)
2. The Quotient Law: logc( ) = logc(m) logc(n) 3. The Power Law: logc(m)n = nlogc(m)
4. The Change of Base Law: = logc(m)
mn
logn(m)logn(c)
OptimizationMaximum Minimumfunction to be optimizedSolutions inside and on edgessolid lines, dotted linesSymbols (≤, ≥, <, >)Adding a new constraintIncrease or decrease ofprofit or cost
Vectorsadding, subtractingworking backwardsopposite, equilibriumChasles relationscalar productcosine lawcalculating the angles
Trigonometric FunctionsIdentitiesradians to degreesFinding the ruleSolving
Properties of Trig Functionssine, cosine and tangentparameters: a, b, h, and kamplitude, radius, period, frequencyphase sift, mean level
coordinates of a point on trig circleexact values
trig identities
Trig stuff
amplitude= ΙaΙperiod = 2π ΙbΙ
mean level: y = kmaximum = k + ΙaΙ
b = 2π period
minimum = k ΙaΙ
frequency = 1 periodfrequency = b 2π
k = max + min 2amplitude = max min 2
phase shift = h
= radius of circle
Sine or Cosine
Distance from center of circle to the ground
y = sin x
P(θ)P(180o - θ)
When to use Sine:
When you are starting at the mean level.
When solving sine:
Don't forget the "sister" point: θ and 180o θ
y = cos x
When to use Cosine:When you are starting at the maximum or minimum.
When solving cosine:Don't forget the "cousin" point: θ and θ
P(θ)
P(-θ)
1 + cot2θ = csc2θ cot2θ = csc2θ 11 = csc2θ cot2θ
1 + tan2θ = sec2θ tan2θ = sec2θ 11 = sec2θ tan2θ
cos2θ + sin2θ = 1 sin2θ = 1 cos2θcos2θ = 1 sin2θ
Quotient Identities
tan x =
cot x =
sin xcos x
cos xsin x
The Pythagorean Identities.csc A =
sec A =
cotan A =
1sin A 1cos A 1tan A
Reciprocal Identities
sin A
cos A
tan A
csc A = 1
sec A = 1
cotan A = 1
sin A =
cos A =
tan A =
1csc A 1sec A 1cotan A
SUMMARY OF TRIGONOMETRIC IDENTITIES
Conic SectionsEllipsecenterverticesfociradiisum of focal radiimajor and minor axisequations (finding the rule and solving)relationship between a, b and cshading (inequalities)
Hyperbolacenterverticesfociradiidifference of focal radiifocal and conjugate axisequations (finding the rule and solving)relationship between a, b and cshading (inequalities)
Parabolavertexdirectrixfocusradiiequations (finding the rule and solving)properties of c direction it opensshading (inequalities)
Different conic sections overlappingFinding the length of a line througha conic section
Intersection of line and conic section(tangent or secant)
Circleequationcenterradiusfinding rule and solvingshading (inequalities)
Conicslocus of pointsfocus or focimajor axis, transversal axisminor axis, conjugate axisdirectrixrelationships with c
a = centre to left or rightb = centre to top or bottomc = centre to focus
The Ellipse:
(x h)2 + (y k)2 a2 b2where (h, k) is the centre a >b
=1
Sum of the focal radii = 2a =length of the major axis
c2 = a2 b2Four vertices(h + a, k) (h a, k)(h, k + b) (h, k b )
Foci(h + c, k) (h c, k)
a = centre to left or rightb = centre to top or bottomc = centre to focus
The Ellipse:
(x h)2 + (y k)2 a2 b2where (h, k) is the centre b >a
=1
Sum of the focal radii = 2b =length of the major axis
c2 = b2 a2Four vertices(h + a, k) (h a, k)(h, k + b) (h, k b )
Foci(h, k + c) (h, k c)
In general, the equation ofany hyperbola with centreat the origin and a horizontalfocal axis (xaxis):
c
a is distance from centre to vertex
c is the length of hypotenuseand distance from centre to focusb is distance from centre to top of box
Difference
asymptotes: y = ± xba
length of focal axis = 2a
centre (0, 0)
P
diagonalsare asymptotes
ba
In general, the equation ofany hyperbola with centreat the origin and a verticalfocal axis (yaxis):
a is distance from centre to side of box
c is the length of hypotenuseand distance from centre to focusb is distance from centre to vertexDifference
asymptotes: y = ± xba
length of focal axis = 2b
The Transformed Hyperbola:
(x h)2 (y k)2 a2 b2
where (h, k) is the centre
=1
y = ±b (x h) + ka
Asymptotes
The Transformed Hyperbola:(x h)2 (y k)2 a2 b2
where (h, k) is the centre
= 1
y = ±b (x h) + ka
Asymptotes
Foci are ALWAYS onthe same (transversal) axis as the vertices.
Difference of the focal radii
length of transveral axis
The Hyperbola
P
F2
F1
Conic Section FormulasParabola:
i) opens up ii) opens down (x h)2 = 4c(y k) (x h)2 = 4c(y k)
iii) opens right iv) opens left (y k)2 = 4c(x h) (y k)2 = 4c(x h)
one vertex (h, k),one focus (F), one directrix
directrix
directrix
directrix directrix
Each point on the parabola is the same distance from the focus as it is to the directrix
The Parabola
y = ax + b (Functional form)
DISTANCE
d =√(x2 x1)2 + (y2 y1)2
To find the distance between two points
To find the distance between a point (x, y) and a line Ax + By + C = 0
d(P, ) = A(x) + B(y) + C
√A2 + B2
l
For right triangles only:
S
CAH
TOA
sin θ 1
opposite sidehypotenuse
O H=
A H=
O A=
opposite sideadjacent side
adjacent sidehypotenuse
OH
CAH
TOA
SOH =
cos θ 1 tan θ 1
=
=
Cheat Sheet Suggestions
The sine law can be used on ANY triangle.
= = asin A
bsin B
csin C
A
B
C
To use this law,you need at least3 measures includinga side and the angleopposite it.
The cosine law can be used on ANY triangle.
A
B
C a2 = b2 + c2 2(b)(c)Cos(A)b2 = a2 + c2 2(a)(c)Cos(B)
c2 = a2 + b2 2(a)(b)Cos(C)
To use this law,you need at least3 measures.These will includethe measure of each side ORThe measure of two sidesand the contained angle.
A = cos 1 (a2 b2 c2)(2bc)( )