# MCR3U Grade 11 Functions Exam

Grade 11 University Functions Exam Study Notes

To Find X-Intercepts

1) Factoring to y=a(x-s)(x-r)

Finding Max/min (Vertex)

1) Factor to find X-int, then average them, and substitute

2) Complete the square to vertex form  y=a(x-h)2+k

=-3x2+12x+14

=-3(x2-4x)+14

=-3(x2-4x+(-2)2)-(-2)2+14

=-3(x-2)2-4+14

=-3(x-2)2-4(-3)+14

=-3(x-2)2+12+14

=-3(x-2)2+26

3) Partial Factoring

Families of Quadratics: sub x, y, s, and r into y=a(x-s)(x-r) to find a

1) Change it into one square able number and the least number that’s unsquarable

2) Square the rooted number and put it outside and the unsquarable number inside

1) Determine the step pattern according to a           a<1 -> compression, a>1 -> stretch

2) Determine vertical shift up/down and horizontal shift right/left (factor out h shifts)

3) Determine the vertex

Discriminant: 0= 1 root, >0 = 2 roots, <0 = no real roots

Factoring equivalent functions: Only cancel if multiplying, not if it’s added or subtracted

Exponential Functions: F(x) = abx

A = initial value, b = constant ratio

Horizontal Stretch: k>1 = compression, 0<k<1 = stretch

-Show stretches as reciprocals (ie 1/k)

Graphing exponential equations

1) Find first 5 main points

2) Apply changes in a table from left to right to each coordinate

3) Graph new points

Half Life: A(x) = Ao(1/2)t/h

A = amount remaining, Ao = initial amount, t = time, h = half life period

1) Take both equations and equal them (ie f(x)=g(x))

2) Isolate variables to one side

3) Simplify

4) Factor (as in like quadratic)

5) Find the 2 or 1 x-intersections

6) Sub back into equation to find y intersection

Trigonometric Ratios: Sine, Cosine, Tangent (SOH CAH TOA)

Cosecant: csc = 1/sin

Secant: sec = 1/cos

Cotangent: cot = 1/tan = cos/sin

Ambiguous cases

Height=c*sinA

If h=a, then there’s one triangle

If h<a, then there’s 2 triangles

If a<h, then there’s no triangles

Sine Law

SinA/a = SinB/b = SinC/c

Cosine Law

a2=b2+c2-2bc*cosA

Special Triangles + CAST Rule and quadrants

Identities

TanX=sinX/cosX

cotX = cosX/sinX

sin2X + cos2X = 1

sin2X = 1-cos2X

cos2X = 1-sin2X

sinX/(1+cosX) * (1-cosX)/(1-cosX)  <- completes the (1+cosX) to get 1-cos2X

Functions in Trigonometry: y=a sin[k(X-d)]+c

Amplitude = max-min/2

Axis of Symmetry = max+min/2

K=360/period

Period = 360/k

Interval = period/4

-In the context of time, we subtract to move to the right, and not pull time in from negative

-Horizontal Compressions, k, is described as a reciprocal (ie if k=3, then it was horizontally compressed by a factor of 1/3)

Graphic Trigonometric functions

1) Draw the axis

2) Mark the max and min points

3) Calculate period

4) Mark the middle points and connect the dots

Discrete Functions:

Explicit Formula: direct formula relating n and the term

Recursive Formula: find new terms by modifying the previous term

Pascal’s Triangle – (a+b)n– n- row number on Pascal’s triangle

(a-b)7=(a) 7+ 7(a)6(-b) + 21(a)5(-b)2 + 35(a)4(-b)3 + 35(a)3(-b)+ 21(a)2(-b)5 + 7(a)(-b)+ (-b)7

-Tn,r = tn-1 r-1, tn-1, r <- looking for terms in the Pascal’s triangle

-Rows start at row 0

Arithmetic Sequences = Tn=a(n-1)d

d= difference

a= initial number

n= term number

Geometric Sequences= Tn=a(r)n-1

a= initial number

r= difference ratio

n= term number

Arithmetic Sequences= Sum = n/2 [2a+(n-1)d]

Or  Sum = n/2 [a+tn]

Geometric Sequences= Sum = a(rn-1)/r-1

Financial Applications

Simple Interest: I=PrT

A=P+I

Compound Interest: A=P(1+i)

-compounding periods must be applied

Present Value of Compound Interest: PV = FV/(1+i)n

Annuities: A=R[(1+i)n-1] / i

Present Value of annuities: PV = R[1-(1+i)-n]/i