MATHPart 2

Linear Functions- Graph is a line

Equation of a Line• Standard form: Ax + By = C

• Slope Intercept Form: y = mx + bm = slope

b = y – intercept or the value of y when x is zero

Equation of a Line• Point – slope form: y – y1 = m(x-x1)

- used when a point and slope are given• Two – point form:

or y – y1 = (

- given two points

Slope and Orientation of Lines

POSITIVE SLOPE

NEGATIVE SLOPE

UNDEFINED SLOPE

ZERO SLOPE

Parallel and Perpendicular Lines

Parallel Lines Perpendicular Lines Same slope

m1 = m2

m1 = -

x and y interceptsy -

intercept- Value of y when x is zero

x – intercept - Value of x when y is zero

Quadratic Equations• Equations dealing with

variables whose highest exponent is 2. • Standard form: y = ax2 + bx + c

Factoring• Get the Common Monomial Factor

first.• After getting the CMF, use the

techniques of factoring.

Example: 12x4 - 48x3 - 15x2

= 3x(4x3 -16x2 – 5x)

Trinomials

Factoring where a = 1x2 + bx + c = (x + m)(x + n)

Wherein:m + n = b

mn = c

Example:

x2 + 3x – 10What are the factors of c which

give a sum of b?(x – 2)(x + 5)m = -2; n = 5

m + n = 3mn = -10

TrinomialsFactoring where a ≠ 1

ax2 + bx + c = (mx + n)(px +q)

Wherein:mq +np = b

nq = c

Example:6x2 – 5x – 6

= (3x + 2)(2x – 3)m = 3, n = 2, p = 2, q = -3

3(-3) + 2(2) = -52(-3) = -6

Perfect Square Trinomialsx2 + 2xy + y2 = (x + y)2

Example:

x2 – 8x + 16= (x – 4)2

BinomialsDifference of Two Squares (DOTS)

x2 – y2 = (x + y)(x – y)

Example:36c2-144

= (6c + 12)(6c – 12)

Sum of Two Cubesx3 + y3 = (x + y)(x2 – xy +y2)

Example:y3+ 8

= (y + 2)(y2 – 2y + 4)

Difference of Two Cubesx3 – y3 = (x – y)(x2 + xy + y2)

Example:b3 – 64

= (b – 4)(b2 + 4b + 16)

Applications of Factoring• Simplifying rational algebraic

expressions or dividing polynomials• Getting the

solutions/roots/zeroes of quadratic equations

Quadratic Formula• An alternative way of solving for

the roots/zeroes of a quadratic equation

𝒙=−𝒃±√𝒃𝟐−𝟒𝒂𝒄𝟐𝒂

Discriminant• If b2 – 4ac < 0

- no real roots, imaginary• If b2 – 4ac = 0

- roots are real and equal• If b2 – 4ac > 0

- roots are real and unequal

√𝑏2−4𝑎𝑐

Laws of Exponents

1

Exponential Functions

F(x) =

One to One Correspondence of Exponential Functions

If xa = xb

Then a = b

Radicals• Exponents in fraction form

* Rules of exponents also apply to radicals

Rationalizing Radicals• Simplifying the radicals by

“removing” the radical sign from the denominator

Example:= =

Adding or Subtracting Radicals

• Treat radicals like variables and combine like terms/radicals

Example:

6

Logarithmic FunctionslogaN = x N = ax

Example:log232 = 5 32 = 25

Common Logarithm- No indicated base base is 10

Example:Log 10,000 = log1010000 = 4

Properties of Logarithm1.log xy = log x + log y2.log 3.log xn = n log x4.ln e = loge e = 1

Imaginary Numbers • Let:

thus: