what does the number system consist of?
real numbers and imaginary numbers
what do real numbers consist of?
rational and irrational
what do rational numbers consist of?
integers, whole numbers, and natural
what is an example of an irrational number?
What is the order of operations?
PEMDAS (parenthesis, exponents, multiplication and division, addition and subtraction)
y = mx + b
Ax + By = C
y - y(1) = m (x - x(1))
What are some ways to solve a system of equations?
substitution, elimination, and graphing
What is the point of solving a system of equations?
to find the point of interception of two lines
What happens if two lines have the same slope but different y-intercepts?
the lines will be parallel and there will be no solution
What happens if two lines have the same slope and the same y-intercept?
the lines will be overlapping and the solution will be infinite (all real numbers).
What happens if two line have opposite reciprocal slopes and different y-intercepts?
the lines will be perpendicular (forms 90˚ angle) and the lines will intersect at one point.
less than or equal to
greater than or equal to
On the number line, when do you use a closed circle?
if the point the circle is on is included (less than or equal to OR greater than or equal to)
On the number line, when do you use an open circle?
if the point the circle is on is not included (less than OR greater than)
When does the inequality flip?
when it is multiplied or divided by a negative
What point is the easiest to check when shading a graph?
b^m * b^n = ?
b^m/b^n = ?
(b^m)^n = ?
b^-n = ?
1/b^-n = ?
b^0 = ?
conjugate of (3 + √11)
how do you check a rational expression?
plugging in any number for y in both the original problem and the simplified answer (they should be the same).
difference of squares
a^2 -b^2 = (a+b)(a-b)
what does a negative slope mean?
it means opposite direction
What does a distance and time graph tell you?
the speed or velocity
what does a velocity and time graph tell you?
why do you need to flip the inequality when dividing or multiplying by a negative?
for the statement to remain true
what are the rows of a matrix?
(100) (010) (001)
what the equation for projectile motion?
what are four ways to solve a system of 3 equations?
substitution, elimination, matrices, and determinants
what is the pythagorean theorem?
side^2 + side^2 = hypotenuse^2
What does SOH CAH TOA stand for?
sin opposite hypotenuse, cos adjacent hypotenuse, tan opposite adjacent
what is the universal conversion factor from degrees to radians or radians to degrees?
360 degrees = 2π radians
what is the radius of a unit circle?
it is always 1
what is the reciprocal of sin?
what is the reciprocal of cos?
what is the reciprocal of tan?
What is positive in the first quadrant?
What is positive in the second quadrant?
What is positive in the third quadrant?
what is positive in the forth quadrant?
what is the first pythagorean identity?
sin^2Θ + cos^2 Θ = 1
what is the second pythagorean identity?
1 + tan^2 Θ = sec^2 Θ
what is the third pythagorean identity?
cot^2 Θ + 1 = csc^2 Θ
What are the special right triangles?
30-60-90 triangles and 45-45-90
the side opposite the 30˚ angle will always be ____ of the hypotenuse?
what is the formula for a 30-60-90 triangle?
across from 30 is x, across from 60 is x√3, across from 90 is 2x
what is the formula for a 45-45-90 triangle?
the sides are x and the side across from the hypotenuse is x√2.
sin of 0
sin of 90
sin of 180
sin of 270
cos of 0
cos of 90
cos of 180
cos of 270
free falling objects formula
d = 1/2gt^2 (g=9.8)
how do you verify identities?
you choose one side of the equation to manipulate, find a common denominator, factor, and use any of the pythagorean identities.
sin, cos, and tan of 30˚
1/2, √3/2, and 1/√3
sin, cos, and tan of 45˚
√2/2, √2/2, 1
sin, cos, and tan of 60˚
√3/2, 1/2, √3
what does the inverse help find?
the degree of an angle
what side of a triangle is the longest?
what does a^2 equal in law of cosines?
b^2 + c^2 - 2bc Cos A
what does b^2 equal in law of cosines?
a^2 + c^2 - 2ac Cos B
what does c^2 equal in law of cosines?
a^2 + b^2 - 2ab Cos C
what is the standard form of a quadratic function?
ax^2 + bx + c
what is the vertex form of a quadratic function?
a(x-h)^2 + k
how do you complete the square?
take half of the coefficient, square that quantity, balance the equation, factor, and simplify.
what does completing the square help you find?
what is the quadratic formula?
-b +or - √b^2 - 4ac / 2a
what is the discriminant?
how many solutions and what type do you get when the discriminant is equal to 0?
one solution; crosses the x axis once
how many solutions and what type do you get when the discriminant is more than 0 (positive?
two solutions; crosses the x-axis twice
how many solutions and what type do you get when the discriminant is less than 0 (negative)?
two complex solutions; does not cross the x axis at all
how do you graph inequalities?
factoring or simplifying to find the x-intercept, plugging in 0 to find the y intercept, and then shading by checking the origin point.
set of x,y
set of all x values.
set of all y values
for every x-value in the domain, there is exactly one y-value for it in the range.
what is exponential form?
B^c = A
what is log form?
log(base b)A = C
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