Algebra II - Midterm Flashcards
how to find domain from a fraction
set denominator not equal to 0
how to find domain from a radical
set radical greater than or equal to 0
what is set notation
{x / 1 < x < 100}
what is interval notation
(1,100)
when the negative is on the outside of the equation, what transformation occurs
reflect over x (outside=x)
when the negative is on the inside of the equation, what transformation occurs
reflect over y (inside=y)
when a in f(x)=af(x-c)+d is a fraction, what transformation occurs
vertical compression
when a in f(x)=af(x-c)+d is >1, what transformation occurs
vertical stretch
how do you write a piecewise function
f(x)= {function domain}
what is the axis of symmetry in vertex form? f(x)=a(x-h)^2+k
h
what is the axis of symmetry in intercept form? f(x)=a(x-p)(x-q)
p+q/2 (halfway between p and q)
how do you find the vertex in intercept form? f(x)=a(x-p)(x-q)
plug in axis of symmetry
how do you find the axis of symmetry in standard form? f(x)=ax^2+bx+c
-b/2a
how do you find the x-intercepts in standard form? f(x)=ax^2+bx+c
quadratic formula
how do you find the y-intercept in standard form? f(x)=ax^2+bx+c
c
how do you go from vertex/intercept form to standard form
FOIL
how do you go from standard form to intercept form?
factor
how do you go from standard form to vertex form
complete the square
how do you do difference of squares
take sqrt of 1st term = a
take sqrt of 2nd term =b
(a+b) (a-b)
how do you factor a trinomial
cross method
how do you rationalize the denominator?
ex. 8/sqrt2
multiply the top and bottom by the radical
ex. 8sqrt2/2
what is the conjugate of a binomial?
the inverse of a binomial
ex. 2+sqrt7 2-sqrt7
what is i^2
-1
what do you do to add/subtract complex numbers?
ex. (2+5i)-(3-2i)
combine like terms
ex. 2-3=-1 5i+2i=7i -1+7i
what do you do to multiply complex numbers
ex. (2+3i)(3-5i)
distributive property and treat i as variables. not fully simplified if i has an exponent >1
what is the discriminant
b^2-4ac (inside of quadratic equation)
when the discriminant is 0 how many solutions are there
1 rational solution
when the discriminant is >0 how many solutions are there
2 solutions, rational if a perfect square
when the discriminant is <0 how many solutions are there
2 imaginary solutions
how do you complete the square
ax^2+bx+c
isolate x^2+bx on one side and c on the other
take (b/2)^2 and add to both sides of the equation
factor for a perfect square
take the sqrt of both sides
what are the different ways to solve a quadratic equation?
ex. x^2+2x=48
- square root
- factoring
- completing the square
- quadratic formula
what makes a non-polynomial
- exponents that are not whole numbers
- sqrt of x
- negative exponents
- fractions with x underneath
- x in the exponent
how do you divide a polynomial
- long division (treat like regular long division)
- synthetic division
what is the equation for difference of squares
a^2-b^2
(a+b)(a-b)
what is the equation for the sum of cubes
a^3+b^3
(a+b)(a^2-ab+b^2)
what is the equation for the difference of cubes
a^3-b^3
(a-b)(a^2+ab+b^2)
how do you factor a polynomial with 4 terms
factor by grouping
what is the remainder theorem
if the polynomial P(x) is divided by x-c, the remainder is P(c)
use synthetic division to divide and the remainder is the solution
what is the factor theorem
if the remainder comes out to be 0, then x-c is a factor of P(x)
what is the rational zero test
factors of p/factors of q
p=constant term
q=leading coefficient
use synthetic division to see which zeros are correct
how many real zeros does a function have
whatever its degree is
how many relative max/min does a function have
degree-1
what will the end behavior of an even degree and positive leading coefficient look like
ex. x^4-11x^3+42x^2-64x+32
^ ^
I I
|
what will the end behavior of an odd degree and a positive leading coefficient look like
ex. x^7+2x^3-5x^2+2
I ^
v I
what will the end behavior of an even degree and negative leading coefficient look like
ex. -2x^4+3x^2-5
I I
V V
what will the end behavior of an odd degree and negative leading coefficient look like
ex. -x^7+2x^2+3x-4
^ I
I V
what is the multiplicity of a zero
the exponent on its factor
ex. (x+1)^2 is 2
determines how the graph will cross the x axis
if the multiplicity is 1 how will it pass the x axis
pass through
if the multiplicity is odd how will it pass the x axis
squiggle (cubic)
if the multiplicity is even how will it pass the x axis
bounce (parabola)
if a+sqrt b is a zero, what is the other zero
a-sqrt b
