sides of a 45, 45, 90 triangle

x, y, (√2 *x)

sides of a 30, 60, 90

short side= x

longer side = √3*x

hyootenuse = 2x

what is a real number?

A real number is a number whose square is a positive number.

standard equation form of a hyperbola opening right/left

((x-h)^{2}/a^{2}) - ((y-k)^{2}/b^{2}) = 1

center of hyperbola (asymptote intersection) at (h,k)

standard equation form of a hyperbola opening up/down

what is the asymptote slope

((y-k)^{2}/b^{2}) - ((x-h)^{2}/a^{2}) = 1

center of hyperbola (asymptote intersection) at (h,k)

a is distance from center to vertex

asymptote slope = b/a

Given a hyperbola, if the negative sign is in front of the x variable, it opens ____/______, if the negative sign is in front of the y variable, it opens ____/______

x= up/down

y = right/left

hyperbola asymptote equation when opens

- left/right

- up.down

left/right y = k±(b/a)(x-h)

up/down y = k±(b/a)(x-h)

they are the same

what is a conic section?

a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane

general formula for an ellipse

((x-h)^{2}/a2) + ((y-k)^{2}/b2) = 1

Given eelipse std equation, the coordinates for the left-most point, the right-most point, upper most point, and lower most point

right most= (h+a, k)

left most= (h-a,k)

upper most= (h, k+b)

lower most= (h, k-b)

parabolic functions,

standard form, others

f(x) = ax^{2} + bx +c

f(x) = a(x-h)^{2}+k

(h,k) is vertex, ±a is open up/down

how to find h,k (vertex) with parabola standard form equation

h = ((-b/2a), (f(-b/2a))

when completng the square, what do you do to the "b" term to get the new (temp) c term that will allow you to complete the square

divide by 2 then square

formula for the distance between two points in cartesian plane and circle standard form equation

the distance formula

d = ((x_{1}-x)^{2} + (y_{1}-y)^{2})^{(1/2)}

for a circle:

r = ((x_{1}-h)^{2} + (y_{1}-k)^{2})^{(1/2) }

=

r^{2} = (x-h)^{2} + (y-k)^{2}

where (h,k) is the center

If trying to find the equation for a circle, but given this, what are the steps to get this into standard form

x^{2} + y^{2} + 8x + 7

get constant on other side

group the x and y terms together

complete square for relevant variable(s)

linear equations/lines adhere to 4 rules

1) variables x and y are raised only to the 1st power

2) variables may be multiplied only by real numbers

3) any real # term may be added or subtracted

4) Nothing else is permitted (1/x, e^{x}, x^{2})

what is needed to define a line equation?

1) 2 points or;

2) one point and a slope

3 different ways an equation may be written for a line

1) point intercept y=mx+b

2) point slope form y-y_{1 }= m(x-x_{1})

3) two-point form y-y_{1} = ((y_{2}-y_{1})/(x_{2}-x_{1}))(x-x_{1})

4 methods to solve quadratic equation

1) factoring

2) quadratic equation

3) completing the square

4) factor by grouping

what is the discriminant in a quadratic equation

b^{2} - 4ac

what the discriminant tells you about the solution to a quardratic equation

- if <0, then there are 2 distinct real solutions

- if = 0, then one real # solution

- if > 0, then 2 distinct complex imaginary number solutions

when is a something a function

for all values of x there is only one value for y

law of sines

Sin A/a = Sin B/b = Sin C/c

when to use the law of consines (2)

1) When the 3rd side of a triangle is needed, you know the other two sides, and the angle between

2) When the angles of a triangle are needed and you know all three sides