Dana L.
Math Analysis Honors
Mrs. Kirch
"The most important facts, terms, and tips I need to remember
are... "
Unit A
Unit A Concepts 1-3
-Positive slope
-Negative slope
-Zero slope-horizontal
-Undefined slope-vertical
-Horizontal-Yeah! (y=#)
-Vertical-X-Ray Vision (x=#)
-Slope-Intercept Form: y=mx+b
-Point-slope Form: (y-y1)=m(x-x1)
-m=slope, (0,b)= y-intercept
-Opposite reciprocal of ""undefined"" is 0. Opposite reciprocal of 0 is
"undefined"
Unit A Concepts 4-5
A function is a very special relationship.
-with ordered pairs, every x-value is associated with only one y-value in a
function, but it is okay if two different x's are matched with the same y's
(like in a zero slope)
-functions past the vertical line test; slushies fail
-EVALUATING expressions-plug in and simplify
-order of operations"
"Linear models: set up 2 ordered pairs, find slope, find equation, plus in
time values into equation to find amounts.
-x-value always represents time
-y-value always represents amount
Unit A Concepts 6-7
Cost: C(x)=fixed cost + (variable costs)x
fixed: paid monthly, is constant; variable: how much it costs to make each
item, and varies monthly based on how many items you decide to produce
Revenue R(x)= __(x) how much you charge for each item
Profit: Revenues-Costs. PRiCe P(x)=R(x)-C(x)
Break-Even Point (BEP)- point at which business makes no/loses no
money. must at least break even to gain profit. P(x)=0 or R(x)=C(x)
Unit A Concepts 8-10
Piecewise functions: pieces divided out by certain x-values that work for
each piece of the function.
-decided which piece x fits with
-plug that x in to that piece ONLY.
-the inequalities are fences/transition points
Domain=x
If you get an imaginary number or an undefined answer, those x's are
restrictions on the domain.
Parenthesis: not included in domain
Brackets: included in domain
Union: domain has multiple parts; you want to connect the answers into
one.
-notations are read left to right
PREO: Polynomial functions, rational functions, even-rooted radical
functions, odd-rooted radical functions
Polynomial and Odd-rooted functions: domain is ALL REAL NUMBERS.
Even rooted function: radicand greater than or equal to 0.
Rational function: set denominator equal to 0 to find restrictions
Difference quotient: f(x+h)=f(x) all divided by h.
Unit B
Unit B Concepts 1-2
Extrema-extreme points; highest/lowest points
Relative Minimums-valleys; Relative Maximums-mountaintops
HOW TO USE GRAPHING CALCULATOR: hit y=; clear old function; type in
new function; hit graph; determine minimums and maximums; from left to
right, determine the minimum and maximum values: hit 2nd; hit TRACE;
select minimum or maximum; move spidey guy to left bound-enter; move
spidey guy to right bound-enter; move spidey guy to guess-enter; write
down ordered pair, rounding to nearest 10th; repeat for all other extrema.
-In between each interval of increase/decrease is an extrema.
-All intervals are OPEN intervals (parenthesis) because we never actually
reach the extrema.
-Intervals defined Left to Right. We only care about x-values
Unit B Concept 3
-Fences break up the graph into different pieces.
-No graphs can overlap!
-No two graphs can OWN the same fence (like a regular function that
passes the vertical line test)
-Open circle is only RENTING the fence, not owning it
-Closed circle owns the fence
-Fences can only have one closed circle; if it has 2-->not a function
-Fences CAN have two OPEN circles, since they're only renting.
-You can graph the piecewise functions on a graphing calculator: go to
"y=" and type in all the parts of the function, then graph.
Unit B Concepts 4-5
f(x)=a(x-h)+k is the parent graph equation.
a is negative=x-axis reflection, |a|>1 vertically stretched/skinny, |a|<1
vertically shrunk/fat
-h is positive-move left h units; h is negative-move right h units
-k is positive-move up k units; k is negative-move down k units
-for every function, the starting point (sq root)/vertex (parabola, abs
value)/point of inflection (cube root)is always the origin
-Root Graphs: |a|>1, stretched in height, |a|<1, shrunk in height
Unit C
Unit C Concepts 1-3
-When subtracting functions, make sure to distribute the negative to the
entire second function.
-When multiplying functions, you can FOIL, distribute, or "box" method.
Box method is more helpful when multiplying longer functions.
-Imaginary=no domain restrictions!! (-∞,∞)
-remember rational functions!"
Unit C Concept 4
-f(g(x)) and "fog" are the same notation. f(x) is outside function; g(x) is
inside function.
Regular composition: Plug the inside function in for the variable in the
outside function.
Double Evaluation (My favorite (:) Evaluate inside value. Take answer and
evaluate with outside function
Double Composition: Evaluate inside expression. Take answer and
compose it with outside function.
Unit C Concepts 5-7
"There are so many things..I'll feel bad for making the box big.
Inverse Functions:
-Numerically: on tables, x and y values are switched!
Inverse of f = f^-1(x)
-Graphically: Inverse functions reflect across identity line (y=x)
-Algebraically: When you switch the x and y values in the equation of f(x)
and solve for y, you get g(x).
-To verify that two functions are inverses, you compose them!
-Do BOTH fog and gof!!! step-by-step. label each step.
-The answers for both fog and gof will be ""x"" if they are inverses.
Horizontal Line Test:
-passes through original graph ONCE-inverse is a function
-passes through original more than once-inverse not a function.
-BOTH function and inverse are functions (original passes VLT and HLT),
original function is ONE-TO-ONE
-If orig function doesn't pass HLT, we restrict the domain to allow the
function to pass HLT.
Unit D
Unit D Concepts 1-2
"Why completing the square??
-puts equations in more usable form (to graph)
-rearranges terms to create a perfect square trinomial
-easier to solve that using quadratic formula.
-when putting into parent graph/vertex form: move all to one side, do not
divide by ""a"" term
-to solve, set y=0
-when solving, divide by the ""a"" term first.
-after taking the square root of something, you get a positive and a
negative answer.
Helpful Calculator tip: ""MATH""-> 1 Frac. to turn into fraction.
Magic Number to add in the blanks; forms perfect square trinomial:
(b/2)^2
Unit D Concepts 3-7
"FACTORING tree:
--First, look for a GCF!
-2 terms: diff of 2 squares, sum of two cubes, diff of two cubes
-3 terms: regular factoring (x-box, grouping, spiderman)
-4 terms: grouping
Sum of two cubes: (a+b)(a^2-ab+b^2) "+ sandwich"
Diff of two cubes: (a-b)(a^2+ab+b^2) "- + +"
*Qualifications for quartic trinomials*:Powers of trinomial must be a
certain pattern (i.e. 4-2-0, 2-1-0, 6-3-0) All multiples of ax^2+bx+c.