algebra 2H formulas + concepts to know(t):
slope intercept form: y=mx+b
point-slope form: y-y1=m(x-x1)
slope formula: y2-y1/x2-x1
vertical translation: y=f(x)+K
horizontal translation: y=f(x-h)
reflection in x-axis: y=-f(x)
reflections in y-xis: y=f(-x)
vertical stretch/shrink: y= a*f(x) stretch: a>1 shrink: 1>a>0
horizontal stretch/shrink: y=f(ax) by a factor of 1/a stretch: 1>a>0 shrink: a>1
< dashed ≤ solid boundary
open circle: (
closed circle: [
vertex form: a(x - h)2 + k
aos: x=h
vertex: (h,k)
standard form: y = ax2 + bx + c
aos: x= -b/2a
vertext: f(-b/2a)
Intercept Form:f(x) = a(x - p)(x - q)
aos: x= p+q/2
vertex: p+q/2 , f(p+q/2)
i | √-1 |
i2 | -1 |
i3 | -i |
i4 | 1 |
If r is a positive real number, then √-r = i√r
a = real part, bi = imaginary part
standard form of a complex number: a+bi
perfect square thing: c=(b/2)²
quadratic formula: x = − b ± b 2 − 4ac/2a
long division answer format: quotient + remainder/divisor
If a polynomial f(x) is divided by x - k, then the remainder is r = f(k)
p= factor of a constant term / q= factor of a leading coefficient
The Fundamental Theorem of Algebra:
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one solution in the set of complex numbers.
Corollary:
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as two solutions, each solution repeated three times is counted as three solutions, etc.
-An nth degree polynomial function f has exactly n zeros.
Descartes rule of signs:
The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number.
The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.
zero: k is a zero of the polynomial function f
factor: x-k is a factor of the polynomial
solution: k is a solution (or root) of the polynomial equation f(x) = 0
x-intercept: if k is a real number, then k is an x-intercept of the graph of the polynomial function f. the graph of f passes through (k,0)
Location Principle:
If f is a polynomial function and a and b are two real numbers such that f(a) < 0 and f(b) > 0 then f has at least one real zero between a and b
the graph of every polynomial function has at most n-1 turning points
if a polynomial of degree n and n distinct real zeros then its graph has exactly n-1 turning points
a function is even when f(-x)=f(x)
a function is odd when f(-x)=f(-x)
if a polynomial function y=f(x) has degree n, then nth differences of function values for equally-spaced-x-values are nonzero and constant
conversely, if the nth differences of equally-spaced data are nonzero and constant, then the data can be represented by ba polynomial function of degree n
slope intercept form: y=mx+b
point-slope form: y-y1=m(x-x1)
slope formula: y2-y1/x2-x1
vertical translation: y=f(x)+K
horizontal translation: y=f(x-h)
reflection in x-axis: y=-f(x)
reflections in y-xis: y=f(-x)
vertical stretch/shrink: y= a*f(x) stretch: a>1 shrink: 1>a>0
horizontal stretch/shrink: y=f(ax) by a factor of 1/a stretch: 1>a>0 shrink: a>1
< dashed ≤ solid boundary
open circle: (
closed circle: [
vertex form: a(x - h)2 + k
aos: x=h
vertex: (h,k)
standard form: y = ax2 + bx + c
aos: x= -b/2a
vertext: f(-b/2a)
Intercept Form:f(x) = a(x - p)(x - q)
aos: x= p+q/2
vertex: p+q/2 , f(p+q/2)
i | √-1 |
i2 | -1 |
i3 | -i |
i4 | 1 |
If r is a positive real number, then √-r = i√r
a = real part, bi = imaginary part
standard form of a complex number: a+bi
perfect square thing: c=(b/2)²
quadratic formula: x = − b ± b 2 − 4ac/2a
long division answer format: quotient + remainder/divisor
If a polynomial f(x) is divided by x - k, then the remainder is r = f(k)
p= factor of a constant term / q= factor of a leading coefficient
The Fundamental Theorem of Algebra:
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one solution in the set of complex numbers.
Corollary:
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as two solutions, each solution repeated three times is counted as three solutions, etc.
-An nth degree polynomial function f has exactly n zeros.
Descartes rule of signs:
The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number.
The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.
zero: k is a zero of the polynomial function f
factor: x-k is a factor of the polynomial
solution: k is a solution (or root) of the polynomial equation f(x) = 0
x-intercept: if k is a real number, then k is an x-intercept of the graph of the polynomial function f. the graph of f passes through (k,0)
Location Principle:
If f is a polynomial function and a and b are two real numbers such that f(a) < 0 and f(b) > 0 then f has at least one real zero between a and b
the graph of every polynomial function has at most n-1 turning points
if a polynomial of degree n and n distinct real zeros then its graph has exactly n-1 turning points
a function is even when f(-x)=f(x)
a function is odd when f(-x)=f(-x)
if a polynomial function y=f(x) has degree n, then nth differences of function values for equally-spaced-x-values are nonzero and constant
conversely, if the nth differences of equally-spaced data are nonzero and constant, then the data can be represented by ba polynomial function of degree n
