Formulas - Evaluation 2 Flashcards
Revenue and profit quadratics
R = xp
rearrange equation to make for only p
multiply equation by p to make solvable equation
find vertex for price that maximises revenue
multiply that price by f(x) or quantity to find maximum revenue
Quadratic inequalities
f(x) < 0 , below x-axis
f(x) > 0, above x-axis
Polynomial function standard form
f(x) = axⁿ
Polynomial specification
all n values must be positive integers
Leading term
leading term is full term with highest degree power eg. 2x³, leading coefficient would be 2
End behaviours
end behaviours: even positive coefficient has same end behaviour as positive parabola, same as negative parabola for negative. odd positive coefficient has same end behaviour as x³, same as negative x³ for negative.
Graph shape
greater n value means flatter curvature for graph near y-axis
Zeroes (multiplicity)
zeroes are the x-ints for the graph, multiplicity is the power they have eg x - odd (crosses x-axis), x² - even (touches x-axis)
Turning points
if a polynomial has degree n the max number of turning points for the graph is n-1
Tip for determining equation of polynomial functions based on graph or description
if question doesn’t specify multiplicity keep it simple and as 1
find ‘a’ value if specific point is given
Graphing polynomial function
- find zeros/x-int
- draw number line and test values within each range (not the x-int) to see if graph is positive or negative between x-intercepts
- determine where graph touches and crosses x-axis based on multiplicity of zeros (odd crosses, even touches)
- check the turning points agree with the degree
- check end behaviours with the entire equations multiplicity ( eg. x(x-1)(x-2)², multiplicity is 4 ), (even/odd positive/negative function)
- evaluate points that are next to the intercepts or between to assist accuracy while graphing
Rational functions
R(x) = P(x) / Q(x)
P(x) & Q(x) are polynomials where Q(x) ≠ 0
Always use number line to solve
Vertical Asymptote
simplify function to lowest term (simplest form), where x makes denominator 0 is the V.A
form: x = __ V.A
not to be confused with domain, function should be in factored quadratic form to determine both zeros
Horizontal Asymptote
R(x) = P(x) / Q(x)
P(x) - degree n
Q(x) - degree m
if n is less than m:
y = 0 , H.A
if n is equal to m:
y = (leading coeff of numerator / leading coeff of denominator) H.A
if n is greater than m:
No H.A
if n is greater than m by 1 (n = m + 1):
Oblique asymptote (O.A) found by dividing numerator by denominator (long algebraic division) to find equation for O.A (eg. y = 3x + 2 , O.A)
Asymptote notation
Asymptotes:
ex.
{ x = 3 , x = -3 V.A
y = -2 H.A
no O.A }
y =
x =
DO NOT FORGET IT !!
Graphing a rational function
- function in lowest term
- Domain
- V.A and multiplicity (odd goes away, even goes together)
- H.A (graph won’t touch on one side)
- x-int
- y-int
- find where graph is above and below x-axis, by evaluating with number line
- evaluate points to confirm shape
Polynomial and Rational Inequalities
move both sides of the equation to the left side and equate the other side to zero
eg.
x - 1 < x²
should be solved as..
- x² - 1 < 0
when multiplying both sides of the equation by a negative number reverse the sign
y = |x|
y = coefficient, x, degree of equation
eg
y = 3x(x-4)(x+7)
y = 3x^3
a^3 - b^3 simplification
(a-b)(a²+ab+b²)
Parabola vertex
-b/2a
Quadratic word problems
2W + 2L = (number given in question)
W x L = A
use this to create a quadratic equation and solve
