Difference of two squares: x²?y²

x² – y² = (x-y)(x+y)

Perfect square: x²+2xy+y²

x² + 2xy + y² = (x+y)²

Difference of 2 cubes: x³-y³

(x-y)(x²+xy+y²)

Quadratic Formula

ax²+bx+c=0, where a?0

x= -b±?(b²-4ac) / 2a

Sum of 2 cubes: x³+y³

(x+y)(x²-xy+y²)

Properties of Inequality

For all real numbers a, b, and c:

1. If a

1. If a

**0, then ac** bc Solve: x²-x < 12

Set equation equal to 0:

x²-x-12 = 0

Factor: (x-4)(x+3) = 0

x = 4 and x = -3

these 2 pts represent changes in the graph – so find whether the factor will be positive or negative in each area.

Calculate for each factor:

(x-4)………..-.|..-..|..+

(x+3)……….-.|..+..|..+

(x-4)(x+3) .+.|..-..|..+

line…..(??..-3….4….+?)

This table represents how the factors change based on numbers in that area.

for x-4 between negative infinity and -3 i chose -5 – so -5-4 equals -9 and all #s in that area will result in x-4 being negative

and for (x+3) use -5, and all numbers between neg infinity and -3 will result in negative result for (x+3)

then for multiplying (x-4)(x+3) => – * – = positive, betwix -3 to 4, -*+=-, etc.

x²-x-12 = 0

Factor: (x-4)(x+3) = 0

x = 4 and x = -3

these 2 pts represent changes in the graph – so find whether the factor will be positive or negative in each area.

Calculate for each factor:

(x-4)………..-.|..-..|..+

(x+3)……….-.|..+..|..+

(x-4)(x+3) .+.|..-..|..+

line…..(??..-3….4….+?)

This table represents how the factors change based on numbers in that area.

for x-4 between negative infinity and -3 i chose -5 – so -5-4 equals -9 and all #s in that area will result in x-4 being negative

and for (x+3) use -5, and all numbers between neg infinity and -3 will result in negative result for (x+3)

then for multiplying (x-4)(x+3) => – * – = positive, betwix -3 to 4, -*+=-, etc.

Parallel lines

2 lines are parallel if and only if they have the same slope or if they are both vertical

Perpendicular Lines

2 lines are perpendicular if and only if the product of their slopes is -1, or if one is vertical and the other horizontal

Linear Function

a relationship f defined by

y = f(x) = mx + b,

for real #s m and b, is a linear function

y = f(x) = mx + b,

for real #s m and b, is a linear function

Linear Cost Function equation

C(x) = mx + b, the m is the marginal cost and b the fixed cost.

slope of a line formula given 2 pts

(y?-y?)/(x?-x?)=m

where x?,y? and x?,y? are points on the line

and m is the slope of the line

where x?,y? and x?,y? are points on the line

and m is the slope of the line

slope-intercept form

y=mx+b

m is the slope of the line

b is the y-intercept of the line

m is the slope of the line

b is the y-intercept of the line

point-slope form

y-y?=m(x-x?)

m is the slope

(x?,y?) is a point on the line

m is the slope

(x?,y?) is a point on the line

x=k

vertical line, undefined slope, i.e. 1/0

y=k

horizontal line, slope=0

least squares line: y intercept formula

Y=mx+b, slope is m, b is y-intercept:

b=y-mx

that is to say, given a point on the line, just put in the x, y into the equation and solve to get b – the y-intercept

given a point xsubn,ysubn the equation:

b=(?y – m(?x)/n

b=y-mx

that is to say, given a point on the line, just put in the x, y into the equation and solve to get b – the y-intercept

given a point xsubn,ysubn the equation:

b=(?y – m(?x)/n

least squares line: slope formula

m=(n(?xy)-(?x)(?y))/(n(?x²)-(?x)²)

break-even formula

Revenue = Cost

R(x)=C(x)

R(x)=C(x)

Profit P(x)=

R(x)-C(x)

Farenheit conversion to Celcuis

F = 9/5C +32

Celcius conversion to Farenheit

C=5/9(F – 32)

Correlation Coefficient

How well the original data fits a straight line:

r = (n(?xy)-(?x)(?y))/(?(n(?x²)-(?x)²) ×?(n(?y²)-(?y)²))

r = (n(?xy)-(?x)(?y))/(?(n(?x²)-(?x)²) ×?(n(?y²)-(?y)²))

function

a function is a rule that assigns to each element from one set exactly one element from another set

domain and range

the set of all possible values of the independent variable in a function is called the domain of the function (x), and the resulting set of possible values of the dependent variable is called the range (y)

vertical line test

a graph represents a function if and only if every vertical line intersects the graph in no more than one point

graph of a quadratic function

the graph of the quadfunc f(x) = ax²+bx+c is a parabola with its vertex at (-b/2a,f(-b/2a)). the graph opens upward if a>0 and downward if a<0

collnear points

3 or more points that are in a straight line

About r – the correlation coefficient

* the correlation coefficient measures the strength of the linear relationship between 2 variables

* r is between 1 and -1 or equal to 1 or -1

* if r = 1, the least squares line has a positive slope

* r = -1 gives a negative slope

* if r = 0 there is no linear correlation between the data points

(but some nonlinear function might provide excellent fit for the data)

* also a r=0 may also indicate that the data fit a horizontal line.

* the exact value of the linear relationship depends upon n – the number of data points

* r is between 1 and -1 or equal to 1 or -1

* if r = 1, the least squares line has a positive slope

* r = -1 gives a negative slope

* if r = 0 there is no linear correlation between the data points

(but some nonlinear function might provide excellent fit for the data)

* also a r=0 may also indicate that the data fit a horizontal line.

* the exact value of the linear relationship depends upon n – the number of data points

Suppose a positive linear correlation is found between 2 quantities. Does this mean that one of the quantities increasing causes the other to increase?

No. a positive correlation means that as one of the quantities increases, the other quantity also increases. It does not mean that one of the quantities increasing causes the other to increase. To prove that one of the quantities increasing causes the other to increase would require further research.

least squares line equations

Zero and Negative Exponents

a?=1

a??=1/a?

a??=1/a?

slope of horizontal line

m=0

slope of vertical line

m is undefined , i.e., 1/0

(??a)? =

a

??(a?) =

|a| if n is even

a if n is odd

a if n is odd

??a × ??b =

??(ab)

??a/??b =

??(a/b)

b?0

b?0

??(??a) =

(?*?)?a

If n is an even natural # and a>0, or n is an odd #, then:

a¹/? =

a¹/? =

??a

example 9¹/² = ²?9 = 3

example 9¹/² = ²?9 = 3

a^(m/n)

(a¹/?)^m

example 27^2/3 = (27^1/3)²=3² = 9

example 27^2/3 = (27^1/3)²=3² = 9

Effective rate for compound interest

If r is the annual stated rate of interest and m is the number of compounding periods per year, the effective rate of interest is:

r subE = ( 1 + r/m)^m -1

r subE = ( 1 + r/m)^m -1

Exponential growth and decay function

y=y?e^(kt)

Let y? be the amount or # of some quantity present at time t=0. The quantity is said to grow or decay exponentially if for some constant k, the amt present at time t .

Change of base theorem for exponents

For every positive real number a:

a^x = e ^((ln a)x)

a^x = e ^((ln a)x)

change of base theorem for logs

a & b > 0, a?1, b?1

log sub a x = (log sub b x) / (log sub b a)

y = log sub a x means

a^y = x

continuous compounding

A = Pe^(rt) dollars

If a deposit of P dollars is invested at a rate of interest r compounded continuously for t years, the compound amount is A.

If a deposit of P dollars is invested at a rate of interest r compounded continuously for t years, the compound amount is A.

Definition of e

As m becomes larger and larger, (1 + 1/m)^m becomes closer and closer to the number e whose approximate value is 2.718281828

Properties of Polynomial Functions. A polynomial function of degree n can have at most ________ turning points

n-1

If a graph has n turning points it must have a degree of

at least n + 1

In the graph of an even polynomial both ends

go up or down

In the graph of an odd polynomial …

one end goes up and one end goes down

If the graph goes up as x becomes a large positive # , the leading coefficient must be _____

positive

If the graph goes down as x becomes a large positive #, the leading coefficient must be _____

negative

Cummulative properties of real numbers

a + b = b + a

ab = ba

ab = ba

Associative property of real numbers

(a + b) + c = a + (b + c)

(ab)c = a(bc)

(ab)c = a(bc)

Distributative property of real numbers

a(b + c) = ab + ac

Given graph of quadratic function:

f(x) = ax²+bx+c

is a parabola with vertex:

f(x) = ax²+bx+c

is a parabola with vertex:

-b/2a is the x

y = f(-b/2a)

(-b/2a,f(-b/2a)) point of vertex

y = f(-b/2a)

(-b/2a,f(-b/2a)) point of vertex

Given f(x) =ax²+bx+c

what can you say about a if the parabola opens upward?

what can you say about a if the parabola opens upward?

a > 0

a^m * a^n =

a^(m+n)

a^m/a^n =

a^(m-n)

(a^m)^n =

a^(mn)

(ab)^m =

a^m * b^m

(a/b)^m =

a^m/b^m

log_a xy=

log_a x + log_a y

log_a (x/y) =

log_a x – log_a y

log_a x^r =

r * log_a x

log_a a =

1

log_a 1=

log_a a^r =

r

Completing the square: ax²+bx+c=0

a(x+d)²+e=0

d = b/2a

e = c – (b²/4a)

d = b/2a

e = c – (b²/4a)

completing the square step-by-step

ax²+bx+c=0 into form

y=a(x-h)²+k

ax²+bx+c=0 into form

y=a(x-h)²+k

1. First, factor a from 1st and 2nd term

a(x²+(b/a)x) + c

2. Then take half of the coefficient of x inside the parans and square it. Subtract a times this number from c

a(x²+(b/a)x+[(b/2a)²]) + c -(a*(b²/4a²)

3. simplifying:

a(x+b/2a)²+c-(b²/4a)=y

Note: so h = -b/2a and k = c-(b²/4a)

a(x²+(b/a)x) + c

2. Then take half of the coefficient of x inside the parans and square it. Subtract a times this number from c

a(x²+(b/a)x+[(b/2a)²]) + c -(a*(b²/4a²)

3. simplifying:

a(x+b/2a)²+c-(b²/4a)=y

Note: so h = -b/2a and k = c-(b²/4a)

Solve 3x²+9x+1=0 by completing the square into the form

y=a(x-h)²+k

and find the vertex.

y=a(x-h)²+k

and find the vertex.

1. 3(x²+3x)+1

2. 3(x²+3x+9/4)²+1-(3*9/4)

3. 3(x+3/2)²+1-(27/4)

4. y=3(x+3/2)²-(23/4)

2. 3(x²+3x+9/4)²+1-(3*9/4)

3. 3(x+3/2)²+1-(27/4)

4. y=3(x+3/2)²-(23/4)

vertex of this formula is (-3/2,-23/4)

What is horizontal asymptote if degree of denominator is greater than degree of numerator? x/x²

horizontal asymptote is y=0

what is horizontal asymptote if degree of numerator is greater than degree of denominator? x²/x

no horizontal asymptote

what is horizontal asymptote if the highest degree of the numerator is equal to highest degree of denominator? 2x²/x²

horizontal asymptote is equal to coefficients of highest degree

for this example: y=2

for this example: y=2