Transcript Rational Functions - Matrix Mathematics

RATIONAL FUNCTIONS
Elizabeth Brown, Arely Velazquez, and Dylan Brown
RATIONAL FUNCTIONS
• A rational function is formed when a polynomial
is divided by another polynomial
-From this kind of function you can find domain
and range, end behavior, vertical, horizontal,
and slant asymptotes, zeros (x-intercepts), and
y-intercepts
*ex: F(x)=x
DOMAIN AND RANGE
• The domain of a function is the set of all possible x values which
will make the function “work” and will output real y-values (most
of the time this is written in interval notation)
• To find the domain of a rational function, set the denominator
equal to 0 and solve for the variable, that answer will not give
you an output, so it is not part of the domain
F(x)=1/2x-5  2x-5=0  x=5/2
 (-infinity,5/2)U(5/2,+infinity)
• Once you graph the function, then you can find the range (this is
also written in interval notation most of the time)
END BEHAVIOR
• End behavior is what y value the graph
approaches as it goes to negative infinity or
positive infinity and is described by limits
*F(x)=1/x
Lim F(x)
x  -∞=0
Lim F(x)
x  ∞=0
ZEROES (X-INTERCEPT) & YINTERCEPT
• A zero is where the function crosses the x-axis
• To find the x-intercept, change the y to zero and solve for x
*F(x)=5x+3/10
0=5x+3/10
0=5x+3
 the zero of this function is -3/5
• To find the y-intercept of a rational function, plug in 0 for any x’s, then
solve for y
*F(x)=5(0)+3/12
F(x)=3/12
F(x)=1/4
• Now, you can plot these on the X&Y axis
VERTICAL ASYMPTOTES
• A vertical asymptote happens where the zeros
occur in the function-this would happen when
the denominator equals zero because you
cannot divide by zero
• To find the VA, set the bottom equal to zero, and
those numbers will be where an asymptote
occurs because you can’t divide by 0!
VERTICAL ASYMPTOTES
• F(x)=x²+2x-3/x²-5x-6
x²-5x-6=0
(x-6)(x+1)=0
x=6 and x=-1
^these are your vertical asymptotes
HORIZONTAL ASYMPTOTES (HA)
• HOBO- if the exponent is higher in the
denominator, then the HA is 0
• HOT- if the exponent is higher on the top, then
there is no HA
• If the exponents are equal, then ratio of the
leading terms is the HA
HORIZONTAL ASYMPTOTES
• F(x)=2x²-11/x²+9
since the exponents are the same, take the
ratio of the leading terms: 2 and 1-the HA is
2
SLANT ASYMPTOTES
• If there is no HA, then there is a slant
asymptote (end behavior asymptote)
• To find this, divide the numerator by the
denominator and that equation is the slant
asymptote
SLANT ASYMPTOTES
• F(x)=-3x²+2/x-1
This function is higher on top, so there is no HA,
therefore there must be a slant asymptote, so
divide the function
The asymptote will end up being -3x-3
TABLE OF SIGNS
• Another name for the table of signs is a number
line analysis
• Once you have found the zeros and the vertical
asymptotes of the function, you can do this
analysis to figure out where it will be positive
and negative
• F(x)=x²+4x+4/x²-4
x-intercept: (-2,0) VA: x=2
y-intercept: (0,-1) HA: y=1
TABLE OF SIGNS ( F(X)=X²+4X+4/X²-4)
• To make the analysis easier, factor the original
function  (x+2)(x+2)/(x+2)(x-2)
• Now, you put the zero and the VA on a number
line
-2
2
• Now, you will take numbers less than -2
and plug them into the function is positive
or negative, the do the same between -2
and 2 and for numbers greater than 2
TABLE OF SIGNS ( F(X)=X²+4X+4/X²-4)
F(x)=(x+2)(x+2)/(x+2)(x-2)
(-3+2)(-3+2)/(-3+2)(-3-2)
positive
(3+2)(3+2)/(3+2)(3-2)
(0+2)(0+2)/(0+2)(0-2)
-2
negative
2
positive
TABLE OF SIGNS ( F(X)=X²+4X+4/X²-4)
• So, now you know that from -∞ to -2 the function
is positive; from the x-intercept at -2 to the VA at
2 the function will be negative; from the VA to ∞
the function will be positive
• To help yourself out even more in the graphing
process, you can make a table with other X and
Y values to guide the graph
X AND Y VALUE TABLE ( F(X)=X²+4X+4/X²-4)
• For this, you will
make a chart with
different X values,
plug them into the
function to get the
corresponding Y
value, and plot the
points on the graph
X (x+2)(x+2)/(x+2)(x- Y
2)
-6 (-6+2)(-6+2)/(-6+2) 1/2
(-6-2)
-4 (-4+2)(-4+2)/(-4+2) 1/3
(-4-2)
0 (0+2)(0+2)/(0+2)(0 -1
-2)
4 (4+2)(4+2)/(4+2)(4 3
-2)
F(X)=X²+4X+4/X²-4
GRAPHING
• Now, you can put everything together to graph the
rational function
• 1st-find the X and Y-intercepts
• 2nd-find all of the asymptotes :vertical, horizontal, and
slant
• 3rd-do the table of signs
• 4th-make a chart with other X and Y values to guide the
graph
RATIONAL INEQUALITIES
•
A rational inequality is just like a rational function, except you are figuring out
where it is less than, greater than, equal to, etc., the number given
•
X^2-x-11/x-2 < 3
x^2-x-11/x-2 -3/1 < 0
x^2-x-11-3x+6/x-2 < 0
(x-5)(x+1)/x-2 < 0
•
the zeros are 5 and -1 and there is a vertical asymptote at 2, so these are the
numbers that need to be used in the number line analysis to see where the
function is < 3
•
Instead of going back to the problem at the very beginning, you can use the
simplified version and find where it is < 0 and it will still give you the same
answer
RATIONAL INEQUALITIES
(0-5)(0+1)/(0-2)
(-2-5)(-2+1)/(-2-2)
(3-5)(3+1)/(3-2)
(6-5)(6+1)/(6-2)
negative
-1
positive
2
negative
positive
5
• The rational inequality will be less than 3 from negative
infinity to -1 and from 2 to 5; this should be written in
interval notation
• (-∞,-1)U(2,5)
RATIONAL INEQUALITIES
•
Graphing a rational inequality is basically like graphing a normal rational function
•
Find the intercepts, asymptotes, do the table of signs, and find other points using a table
•
What is different is the way the line looks: if the function is just less than or greater than,
then the line is dotted, if it is less than or equal to or greater than or equal to, then the line
will be solid
•
Then the graph must be shaded in the appropriate area; say that it is less than or equal to
3, you will have a solid line and it will be shaded under 3
REAL WORLD PROBLEM
• There are many different ways that rational functions
are used in the real world-doctors can use them to find
the concentration of different drugs in the blood system,
they could be used in finances, to find different rates at
which things can happen (scientists use them all the
time!)
• Say that 4x/.12x²-2.6 will determine the concentration
of a certain drug in a patient’s blood stream when the
drug is taken each day. Doctors can use this equation
to figure out if a dosage needs to be raised or lowered
and how it should be helping the patient.
REAL WORLD PROBLEM
• Now, with this graph, doctors can monitor
patients and figure out what dosage of certain
medicines need to be taken and what might be
too much. With this equation you can still find X
and Y intercepts, asymptotes, end behavior, and
do a number line analysis.
REAL WORLD PROBLEM
•
X-intercept: (0,0)
•
Y-intercept: (0,0)
•
Domain: (-∞,-4.6)U(-4.6,4.6)U(4.6,∞)
•
Range: (-∞,∞)
•
Vertical Asymptotes: x=-4.6 and x=4.6
•
Horizontal Asymptote: y=1
•
End Behavior: lim F(x)
x -∞ =
-1
lim F(x)
x∞ =
1
REAL WORLD PROBLEM
Number Line Analysis
4(-5)/.12(-5)²-2.6
negative
F(x)=4(-2)/.12(-2)²-2.6
-4.6
positive
F(x)=4(3)/.12(3)²-2.6
0
negative
F(x)=4(5)/.12(5)²-2.6
4.6 positive
REAL WORLD PROBLEM
F(x)=4x/.12x²-2.6