FRONT OF CARD
Quadratic Function
 General Form
 Vertex Form
BACK OF CARD
 General form:
o y  ax 2  bx  c
 Vertex form:
o y  a( x  h)2  k
Graph Quadratic
1) Vertex  h, k 
2) a  0  face up
a  0  face down
3) a  1  narrow
a  1  wide
Find intercepts of quadratic fct
Application:
 quadratic function
 find max or min
Graphs of polynomials are …
Definition of a zero of a function.
Equivalency Theorem
4) Axis: x  h
5) Find point:
a) narrow – one unit from vertex
b) wide – more than one unit
6) Label: vertex, axis, point
 x-intercepts
o y=0, solve, ordered paris
 y-intercept
o f(0), ordered pair
y-value of vertex.
 y-value is max/min
 x-value is where the max/min
occurs (for example, at time t)
 Smooth – no sharp points
 Continuous – no breaks
x  a is a zero of function f if f ( a )  0.
Given polynomial function f
1) x  a is a zero of fct f
2)
 a,0 is an x-intercept on the graph
of f
3) x  a is a solution to f  x   0
4)
Only change the sign when …
End behavior – Leading Coefficient Test
f  x   ax n  ...
Multiplicities of zeros
 x  a
is a factor of f
going to or from a factor.
1) a  0, n  even  both up
2) a  0, n  even  both down
3) a  0, n  odd  left down, right up
4) a  0, n  odd  left up, right down
 even  bounce
 odd  through
 higher degree  flatten
Intermediate Value Theorem (IVT)
A polynomial function of degree n has …
turning points
Graphing a polynomial
Rational zero theorem
Find all zeros of a polynomial fct
Find all solutions to a polynomial equation
Imaginary zeros always …
A poly of degree n has … zeros
Rational fct: find vertical asymptotes
Rational fct: find horizontal asymptotes
ax m  ...
f ( x)  n
bx  ...
Slant Asymptote
If:
1) f is a polynomial function
2) a  b
3) f (a ) and f (b) are opposite in sign
Then:
f has at least one zero in  a, b
at most n  1 turning points
1) End behavior
2) Zeros:
a) x=zero
b) multiplicity
c) means …
3) Find points as needed
4) Draw graph
5) Label
f ( x )  an x n  ...  a0
a
factors of 0 are possible rational zeros
an
Graph - x-int
Divide Synthetically
Quadratic formula
Graph - x-int
Divide Synthetically
Quadratic formula
… come in conjugate pairs
exactly n
Set the denominator equal to zero and solve
1) deg num < deg denom  y  0
a
2) deg num = deg denom  y 
b
deg num > deg denom  no HA
When the degree of the numerator is
exactly one more than the degree of the
denominator.
1) Divide denominator into numerator
2) Throw away the remainder
3) 3) SA is y  quotient
Solve polynomial inequalities graphically
Solve polynomial inequalities algebraically
Solve rational inequalities algebraically
Direct variation
Inverse variation
Joint variation
Combined variation
Solve a variation problem
1) If poly > 0, then choose intervals where
the graph is above the x-axis.
2) If poly < 0, then choose intervals where
the graph is below the x-axis.
3) ,  use parenthesis
,  use brackets
1) Write as equation and solve to find
boundary points.
2) Put boundary points on a number line
and use test points from each interval in the
original inequality.
3) Include all “yes” intervals in the
solution.
Same process as solving polynomial
inequalities, except:
1) When finding boundary points, include
any undefined values (denom = 0)
2) Can’t include in the solution any
boundary points that are undefined.
y  kx
k
y
x
y varies as the product of two or more
quantities:
y  kxz
y varies both directly and inversely
x
yk
z
1) Write variation model
2) Plug in values and solve for constant k
3) Re-write var model with value for k
4) Plug in values and solve for y