Formulas - Evaluation 2

Question 1

Revenue and profit quadratics

Answer 1

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

Question 2

Quadratic inequalities

Answer 2

f(x) < 0 , below x-axis
f(x) > 0, above x-axis

Question 3

Polynomial function standard form

Answer 3

f(x) = axⁿ

Question 4

Polynomial specification

Answer 4

all n values must be positive integers

Question 5

Leading term

Answer 5

leading term is full term with highest degree power eg. 2x³, leading coefficient would be 2

Question 6

End behaviours

Answer 6

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.

Question 7

Graph shape

Answer 7

greater n value means flatter curvature for graph near y-axis

Question 8

Zeroes (multiplicity)

Answer 8

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)

Question 9

Turning points

Answer 9

if a polynomial has degree n the max number of turning points for the graph is n-1

Question 10

Tip for determining equation of polynomial functions based on graph or description

Answer 10

if question doesn’t specify multiplicity keep it simple and as 1

find ‘a’ value if specific point is given

Question 11

Graphing polynomial function

Answer 11

1. find zeros/x-int
2. draw number line and test values within each range (not the x-int) to see if graph is positive or negative between x-intercepts
3. determine where graph touches and crosses x-axis based on multiplicity of zeros (odd crosses, even touches)
4. check the turning points agree with the degree
5. check end behaviours with the entire equations multiplicity ( eg. x(x-1)(x-2)², multiplicity is 4 ), (even/odd positive/negative function)
6. evaluate points that are next to the intercepts or between to assist accuracy while graphing

Question 12

Rational functions

Answer 12

R(x) = P(x) / Q(x)
P(x) & Q(x) are polynomials where Q(x) ≠ 0

Question 13

Vertical Asymptote

Answer 13

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

Question 14

Horizontal Asymptote

Answer 14

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)

Question 15

Asymptote notation

Answer 15

Asymptotes:
ex.
{ x = 3 , x = -3 V.A
y = -2 H.A
no O.A }

y =
x =
DO NOT FORGET IT !!

Question 16

Graphing a rational function

Answer 16

1. function in lowest term
2. Domain
3. V.A and multiplicity (odd goes away, even goes together)
4. H.A (graph won’t touch on one side)
5. x-int
6. y-int
7. find where graph is above and below x-axis, by evaluating with number line
8. evaluate points to confirm shape

Question 17

Polynomial and Rational Inequalities

Answer 17

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

Question 18

y = |x|

Answer 18

y = coefficient, x, degree of equation
eg
y = 3x(x-4)(x+7)
y = 3x^3

Question 19

a^3 - b^3 simplification

Answer 19

(a-b)(a^2+ab+b^2)

Question 20

Parabola vertex

Answer 20

-b/2a