Rational Functions
(Same Degree)
(Same Degree)
Where will the HA of a rational function occur when both functions have the same degree?
It will occur at a/c
What are all the key features of a rational function with varied degrees?
-X-Intercept (s)
-Y-Intercept
-Holes
-Vertical Asymptote (s)
-Horizontal Asymptote
(x+2)/(x+3)(x-4)
HA: Y= 0
VA: X= -3, X= 4
Y-int = -1/6
X-int = -2
x+4/x-3 ≥ x+5/x+2
-2, 1/2 ] U [3, ∞ )
On the 42-km go-cart course at Sportsworld, Arshia drives 0.4 km/h faster than Sarah, but she has engine trouble part way around the course and has to stop to get the go-cart fixed. This stop costs Arshia one-half hour, and so she arrives 15 min after Sarah at the end of the course. How fast did each girl drive and how long did each girl take to finish the course? Answer to one decimal place
Arshia’s speed: 8.0 km/h; Sarah’s speed: 8.4 km/h; Arshia’s time:5.5 h; Sarah’s time: 5.25 h
The HA will be at x = -3/2
What is the Y-Intercept of f(x) = x-1 / (x+3) (x2 -9).
Y-Intercept: y = 1/27
Does the function Y= 1/x have a HA of Y=1. Justify your awnser
Y= 0
If degree of numerator is smaller than degree of denomintor, then Y= 0 for Horizontal asymptote
3x-2 < x+4/x-2
XE(-∞,0) U (2,3) X ≠ 2
Working together Jake and Ben painted a fence in 8 hours. Last month Jake painted the fence by himself and The month before, then painted by himself but took 12 hours less than Jake took. How long did Jake and then take, one each was painting the fence alone
It takes Jake 24 hours to paint the fence and it takes Ben 12 hours
When does an Oblique Asymptote occur in a rational function ?
When the degree of the numerator is greater than the degree of the denominator by exactly one degree.
Is there a hole in the function f(x) = (x-5)(x+1) / (x-2)(x+2)2(x+1)? Explain why there is or isn't.
There is a hole in this function because (x+1) is both in the denominator and the numerator, which cancels them out making a hole.
Graph f(x)= x2+4x+3 and reciprocal
State the key properties
1/(x2+4x+3)
=1/(x+3)(x+1)
HA: Y=0
VA: X= -1,-3
Y=+1. -1
x3+6x2-2x/x2 +4 ≥ 2
xe(-1,3) U (3, ∞)
x=3,-1
An economist for a sporting goods company estimates the revenue and cost functions for the production of a new snowboard. These functions are R(x) = -x^2 + 10x and C(x) = 4x + 5, respectively,where x is the number of snowboards produced, in thousands. The average profit is defined by the function AP(x) = P(x)/x, where P(x) is the profit function. Determine the production levels that make AP(x) > 0.
(-∞ ,1) U (5, ∞)
What is the Oblique Asymptote of f(x) = x3 - 16x / -4x2 + 4x + 24 ?
The Oblique Asymptote is at y = -1/4x + 1/4
List all key features of f(x) = (x-5) / (x+1)(x-2)
Y-Intercept: y= 5/2
VA: x = 2, x = -1
HA: y = 0
f(x)= 4x+5/x-8
State key properties
VA:8
HA:4
X-int: -5/4
Y-int: -5/8
The function P(t)= 20t/t+1 Models the population, and thousands, of Georgetown, T years after 1997. The population and thousands of Milton is modeled by Q(t)= 240/t+8. Determined to the values of Q (t) Are greater than the values of P(T)
TE(-8,-2)U (-1,6)
Over a distance of 120 km, the average speed of a car is 35 km/h slower than that of a train and The train covers the same distance and 45 minutes less time. Find the speed of the car.
Speed of the car is 59.4 km/h.
If the formula for HA is y = a/c , then what is the formula for VA? Hint: use f(x) = ax+b / cx+d to solve.
The formula for VA is x = -d/c
List all key features of f(x) = (x-5)(x+1) / (x-2)(x+2)2(x+1).
X-Intercepts: (5,0)
Y-Intercept: y = 5/8
Hole: (-1,2)
VA: x = 2, x = -2
HA: y = 0
Has hole x= -2
zeros: at x= -6, x=4
VA: X= +20.-20
HA: Y=14
Y=14/-2x-20
The equation f(t)=5t/t^2 +3t+2 models the bacteria count, in thousands, for a sample of tap water that is left to sit over time, t, in days. The equation g(t) =15t/t^2+ 9 models the bacteria count, in thousands, for a sample of pond water that is also left to sit over several days. In both models, t > 0. Will the bacteria count for the tap water sample ever exceed the bacteria count for the pond water?
t=0.31 at t > 0
The fletchers football team bought Pizza for $900 to sell at the game. They kept 10 pizzas to feed the players after the game and saw the rest for $1040. If their profit was $4 per pizza how many pizzas were in the original order and what was the original price of each pizza
75 Pizzas at $12 each