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Solving Rational Equations and Inequalities

To solve an equation

a) cross-multiply OR

b) multiply all terms by lowest common denominator

To solve an inequality

1) all terms to one side and set it equal to 0

2) use an interval table to find where the inequality is true

To graph rational functions, use interval table, using the factors of the function to find where f(x) is negative or positive in relation to the asymptotes and zeroes.

Solving Polynomial Inequalities and Equations

Solving Linear Inequalities and Equations

1) Move all terms to one side and use the remainder theorem and synthetic division to factor fully.

2a) If it is an equation (set equal to zero):

set each factor equal to zero to find the zeroes of the function

2b) If it is an inequality (set to greater than or less than zero):

Create an interval table to identify where the f(x) is greater or less than zero.

Solving linear equations

1) Bring all the numbers to one side, to isolate for x.

2) Cross-multiply, multiply by the same denominator, add, subtract from both sides as needed.

Solving linear inequalities is the same as solving linear equations except...

a) if you multiply or divide by a negative number, you MUST flip the sign.

b) linear inequalities could have multiple solutions.

Asymptotes

Synthetic Division

Characteristics

Definitions

Dividing Polynomials

If degree is odd...

If Leading Coefficient is (-), f(x) goes from 2nd quadrant to 4th quadrant.

If L.C. is (+), f(x) goes from 3rd quadrant to 1st quadrant.

have at least one zero (x-int)

even number of turning points between 0-n

If degree is even...

If LC is (-), f(x) goes from 3rd quadrant to 1st quadrant

If LC is (+), f(x) goes from 2nd quadrant to 1st quadrant.

can have up to n(degree) zeroes or none at all

odd number of turning points between 0-n

Factoring Polynomials

degree - the sum of the exponents of the variables in a term

degree of a function - the degree of the greatest-degree term

leading coefficient - the coefficient of the term with the highest degree in the polynomial

absolute max/absolute min - the greatest/least value attained by a function

Remainder theorem -

f(a) = 0 when (x-a) is a factor of f(x). Numbers that could make f(x) = 0 are of the form p/q, where p is a factor of the constant term of the polynomial, and q is a factor of the leading coefficient.

A sum of cubes -

A^3 + B^3 = (A+B)(A^2 - AB + B^2)

A difference of cubes -

A^3 - B^3 = (A-B)(A^2 + AB + B^2)

1) Standard Division

Follow standard long division rules, using a dividend, divisor, remainder and quotient.

2) Synthetic Division

Can only be used if divisor is linear (x-k) or (ax-k)

a) terms should be arranged in descending order of degree

b) to find divisor, set x-k = 0 and solve for x

c) zero must be used as the coefficient of any missing powers

What is a polynomial function?

f(x) = ax^n/(bx^m)

zero of the denominator = vertical asymptote

ratio of leading coefficients in numerator and denominator = horizontal asymptote

n>m:

horizontal asymptote at y=0

n=m:

H.A. at y=a/b

n>m by 1:

oblique asymptote

if numerator and denominator has a common factor of (x-a):

graph has a hole at a

Graphing a Factored Polynomial

A polynomial function is a function of the form f(x) = ax^n + ax^(n-1) . . . ax^2 + ax + a, where a is a real number, and n is a whole number.

If the degree of the factor is even, f(x) turns around at the x-int. The greater the even degree, the wider the turn.

If the degree of the factor is odd, f(x) bends at the x-int. The greater the odd degree, the greater the bend.

Chapter 4: Polynomial Equations and Inequalities

Secant line

Average Rate of Change

A secant line is a line that passes through two points on the graph of a relation.

Graphically, average rate of change over an interval between x1 and x2 is equivalent to the slope of a secant line through points (x1, y1) and (x2, y2).

Tangent line

A tangent line is a line that touches the graph at only one point, P, within a small interval of a relation.

The slope of the tangent line can only be estimated, not calculated, because only one point is known. The slope is equivalent to the instateanous rate of change at point P.

Reciprocal Functions vs Regular Functions

_________

Average rate of change is the change in y divided by the change in x, over an interval.

Average rate of change = Change in y

Change in x

= f(x2) - f(x1)

(x2 - x1)

_________

Chapter 3: Polynomial Functions

Rate of Change in Graphs

Methods to Calculate IRoC

Preceding and Following Intervals

Preceding interval - finding the average rate of change between (x-h, f(x-h)) and (x, f(x)), where h is a small positive number.

Following interval - finding the average rate of change between (x, f(x)) and (x+h, f(x+h)), where h is a small positive value.

Instantaneous Rate of Change

Methods to Calculate IRoC

Centered Interval

Centered interval - finding the average rate of change between x-h and x+h, where h is a very small number.

Instantaneous rate of change is the exact rate of change of a function y=f(x) at a specific value of the independent variable x = a.

It's estimated using average rates of change over very small intervals of the independent variable.

Chapter 5:

Rational Functions, Equations, and Inequalities

Chapter 2:

Understanding Rates of Change

Methods to Calculate IRoC

Difference Quotient

difference quotient - a formula to determine the rate of change between a and a+h, where h is a very small number.

Unit Circle

Graphing trick:

a point on a graph that underwent transformation is equal to (x/k + d, ay + c)

Additional Information about Trig Functions

|a| is the amplitude (distance between axis and a max/min point) and a = (max-min)/2

|k| is the number of cycles in 2pi radians, when period = 2pi/k

y = c is the equation of the axis

Trig Graphs

Info To Know

Main Attributes of Parent Functions

Attributes

What is a function?

Transformations of Functions (cont.)

Transformations of Functions

A function is a relation in which there is a unique output for each input. Each value of the domain (x-value) corresponds to only one y-value.

Grade 12 Advanced Functions Mind Map

radian to degrees:

multiply by (180/pi)

degrees to radians:

multiply by (pi/180)

csc x = 1/sin x

sec x = 1/cos x

cot x = 1/tan x

Special triangles:

Domain: set of all possible inputs for function

Range: set of all possible outputs of function

Intervals of Increase: points of the function where the function increase

Intervals of Decrease: points of the function where the function decreases

Discontinuity: a break in the graph

Even/odd function: if graph is reflected across y-axis (even); if not, odd

End behavior: behavior of f(x) as x approaches +/- infinity

Chapter 6:

Trigonometric Functions

Chapter 1: Functions and Characteristics

y = af(k(x-d)) + c

a:

if a>0, vertical stretch

if 0<a<1, vertical compression

if a<0, reflection in x-axis

k:

if k>0, horizontal compression by |1/k|

if 0<k<1, horizontal stretch

if k<0, reflection in y-axis

y = af(k(x-d)) + c

d:

if d>0, function is moved d units to the right

if d<0, function is moved d units to the left

c:

if c>0, function is moved c units up

if c<0, function is moved c units down

Graphs of Reciprocal Trig Functions

Not a function Is a function

Kevin Thevara

Table of Contents

Branch Colour

Chapter

Chapter 7:

Trigonometric Identities and Equations

Chapter 1: Functions: Characteristics and Properties

Chapter 2: Functions: Understanding Rates of Change

Chapter 3: Polynomial Functions

Chapter 4: Polynomial Equations and Inequalities

Chapter 5: Rational Functions, Equations, and Inequalities

Chapter 6: Trigonometric Functions

Chapter 7: Trig Identities and Equations

Chapter 8: Exponential and Logarithmic Functions

Cofunction Identities

Formulas

Compound angle formulas:

cos(a+b) = cosacosb - sinasinb

cos(a-b) = cosacosb + sinasinb

sin(a+b) = sinacosb + cosasinb

sin(a-b) = sinacosb - cosasinb

tan(a+b) = (tana + tanb)/(1 - tanatanb)

tan(a-b) = (tana - tan b)/(1 + tanatanb)

Double angle formulas:

cos(2x) = cos^2x - sin^2

= 1 - 2sin^2x

= 2cos^2x - 1

sin(2x) = 2sinxcosx

tan(2x) = 2tanx/(1 - tan^2x)

Pythagorean Identities:

sin^2x + cos^2x = 1

tan^2x + 1 = sec^2x

1 + cot^2x = csc^2x

Chapter 8:

Exponential and Logarithm Functions

Proving Trig Identities

a) Simplify the more complicated side until it is identical to the other side, or manipulating both sides to get the same expression

b) rewriting expressions using identities you know

c) using a common denominator or factoring, if possible

What is a logarithm function?

Solving Trig Equations

Rules to remember:

a) only find solutions within given interval.

b) use special triangles, the unit circle and graphs to find all solutions of x within the interval, if possible.

c) when solving a quadratic equation, remove sin/cos/tan/etc, to make it simpler to solve. But, remember to add it back once factored.

d) Use quadratic formula if quadratic equation is unfactorable.

The logarithm is the inverse of the exponential function (y = a^x).

y = alog(a)k(x-d) + c OR a^((y-c)/a) = k(x-d)

Base is a (which is equal to 10, unless stated otherwise), exponent is y, argument is x.

This function is used to find the exponent of certain equations.

Solving Equations

Log Laws

To solve exponential equations:

a) give both sides a common base and solve for the exponent

b) rewriting it in logarithm form and solving

product law: log(a)xy = log(a)x + log(a)y

quotient law: log(a)(x/y) = log(a)x - log(a)y

power law: log(a)x^r = r log(a)x

Solving Log Equations

Solving Exponential Equations

a) return the equation to exponential form to solve for the exponent.

b) use the log laws to simplify the equation to solve.

c) the argument and the base of a logarithm must always be positive (since products of powers cannot be negative).

a) If possible, set the bases equal to each other and solve for the exponent.

b) If not, take the log of both sides, and then simplify to isolate for the exponent.

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