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Section 5.1 Model Inverse and Joint Variation
Remember a Direct Variation Equation y  kx has a y-intercept of (0, 0).
Different Types of Variation
Relationship
a) y varies directly with x.
b) y varies inversely with x.
Equation
y  kx
k
y
x
z  kxy
k
y 2
x
kx
z
y
c) z varies jointly with x and y.
d) y varies inversely with the square of x.
e) z varies directly with x and inversely with y.
The nonzero constant k is called the constant of variation.
Example: 1) The variables x and y vary inversely, and y  8 when x  3 .
a. Write an equation that relates x and y.________________
b. Find y when x  4 ________________
Example: 2) Do x and y show direct variation, inverse variation, or neither?
a. xy  4.8 ________________
y
________________
1.5
y  x  3 ________________
b. x 
c.
Example: 3) Does the following table of data show inverse variation? If so, fins a model.
W
H
2
9
4
4.5
6
3
8
2.25
10
1.8
Example: 4) the ideal gas law states that the volume V (in liters) varies directly with the number of
molecules n (in moles) and temperature T (in Kelvin) and varies inversely with the pressure P (in
kilopascals). The constant of variation is denoted by R and is called the universal gas constant.
a. Write an equation for the ideal gas law. ________________
b. Estimate the universal gas constant if V  251.6 liters; n  1 mole; T  288 K; P  9.5 kilopascals.
109
Section 5.2 & 5.3 Graphing Rational Functions
Rational functions are quotients of polynomial functions. This means that rational functions can be
p( x)
expressed as:
f ( x) 
q( x)
where p(x) and q(x) are polynomial functions and q(x)  0. The domain of a rational function is the set
of all real numbers except the x-values that make the denominator zero.
Vertical asymptotes of a rational function: To find a vertical asymptote, set the denominator equal to
0 and solve. These values become your vertical asymptote(s). Caution: If you can factor the
numerator and the denominator and cancel a factor, it becomes a point of discontinuity, (hole) and
not a vertical asymptote.
Parent rational function:
Graph
y
1
x
Graph
y
1
x
Domain: ________________
Domain: ________________
Range: ________________
Range: ________________
Asymptotes: ________________
Asymptotes: ________________
The transformations that we have used all year to graph functions will work with rational functions.
a
 k a is our “outside” coefficient and will vertically stretch
xh
or compress the equation or if it is negative, it will reflect it over x. The denominator x  h is
If we look at the rational function y 
considered the “inside” and will move the graph left/right. The k value will move the graph up/down.
110
Example:
Graph
1
y
2
x 3
x
y
x
y
x
y
x
y
x
y
x
y
Domain: ________________
Range: ________________
Asymptotes: ________________
Graph
y
3
1
x
Domain: ________________
Range: ________________
Asymptotes: ________________
Facts to know and use when graphing rational functions in the form:
y
p( x)
q( x)
1) The x-intercepts of the graph are the solutions of the numerator p( x) . p( x)  0 and solve.
2) To find the y-intercept, plug 0 in for x and simplify.
Discontinuities in the graph:
3) Vertical asymptotes are equations of vertical lines where the denominator is zero. x  c
4) Holes are found using the binomials that are factored and canceled when simplifying the
rational function. Set the binomial equal to zero and solve to get the x-value. To find the yvalue of the hole, plug the x-value into the simplified rational function. Holes are always
ordered pairs!
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5) Horizontal Asymptote of a rational function: There are three scenarios to look for when finding
horizontal asymptotes.
a) If the degree of the numerator is less than the degree of the denominator, then
automatically y = 0 (the x-axis) is the horizontal asymptote.
b) If the degree of the numerator equals the degree of the denominator, then keep the two
“matching” degree terms and simplify them to 𝑦 =
𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
.
c) If the degree of the numerator is greater than the degree of the denominator, then there
are no horizontal asymptotes.
SPECIAL CASE: When the numerator is exactly one degree more than the denominator, they
are called slant asymptotes and can be found using long division (disregarding the remainder).
It will be in the form of y  mx  b .
Graphing Rational Functions by Hand
Let f ( x)  p( x) define a rational function in lowest terms. To sketch its graph, follow these steps.
q( x)
STEP 1:
Find all vertical asymptotes.
STEP 2:
Find all horizontal asymptotes or slant asymptotes.
STEP 3:
Find the y-intercept, if possible, by evaluating f (0) .
STEP 4:
Find the x-intercepts, if any, by solving f(x) = 0. (These will be the zeros of the numerator p(x).)
STEP 5: Determine whether the graph will intersect its non-vertical asymptote y = b by setting the
rational equation equal to the horizontal asymptote value and solve for x. If x is a real number, the graph
crosses the asymptote at this x-value.
STEP 6: Plot selected points as necessary. Choose an x-value in each interval of the domain determined by
the vertical asymptotes and x-intercepts.
STEP 7: Complete the sketch.
Graph: f ( x) 
3x  2
2x  4
Domain: ________________
Range: ________________
Equation(s) of Vertical Asymptotes: ____________
Equation(s) of Horizontal Asymptotes: __________
Equation(s) of Slant Asymptotes: ______________
x-intercepts: ________________
y-intercepts: ________________
hole: __________
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Graph: f ( x) 
x3
( x  2)( x  5)
Domain: ________________
Range: ________________
Equation(s) of Vertical Asymptotes: ____________
Equation(s) of Horizontal Asymptotes: __________
Equation(s) of Slant Asymptotes: ______________
x-intercepts: ________________
y-intercepts: ________________
hole: __________
x2  x
Graph: f ( x) 
x 1
Domain: ________________
Range: ________________
Equation(s) of Vertical Asymptotes: ____________
Equation(s) of Horizontal Asymptotes: __________
Equation(s) of Slant Asymptotes: ______________
x-intercepts: ________________
y-intercepts: ________________
Graph: f ( x) 
hole: __________
x 1
x x2
2
Domain: ________________
Range: ________________
Equation(s) of Vertical Asymptotes: ____________
Equation(s) of Horizontal Asymptotes: __________
Equation(s) of Slant Asymptotes: ______________
x-intercepts: ________________
y-intercepts: ________________
Graph: f ( x) 
3x 2  5 x  2
x2  x  2
hole: __________
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Domain: ________________
Range: ________________
Equation(s) of Vertical Asymptotes: ____________
Equation(s) of Horizontal Asymptotes: __________
Equation(s) of Slant Asymptotes: ______________
x-intercepts: ________________
y-intercepts: ________________
Graph: f ( x) 
hole: __________
4
x 1
2
Domain: ________________
Range: ________________
Equation(s) of Vertical Asymptotes: ____________
Equation(s) of Horizontal Asymptotes: __________
Equation(s) of Slant Asymptotes: ______________
x-intercepts: ________________
y-intercepts: ________________
hole: __________
Section 5.4 Multiply and Divide Rational Expressions.
A Rational expression is in simplified form provided its numerator and denominator have no common
factors (other than 1 ). To simplify a rational expression, apply the following property:
Let a, b, and c be nonzero real numbers or variable expressions. Then
ac a

bc b
Simplifying a rational expression usually requires 2 steps.
1. Factor the numerator and denominator.
2. Divide out common factors to both the numerator and the denominator.
Example: 1) Simplify the ration expression.
x2  5x  6
x2 1
x2  5x
Example:
x2
114
Example: 2) Multiplying monomial rational expressions.
6 x 2 y 3 10 x3 y 4
2 x 2 y 2 3x
Example: 3) Multiplying polynomial rational expressions.
3x  27 x3 3x 2  4 x  1
3x 2  2 x  1
3x
Example: 4) Multiplying rational expressions by a polynomial.
x2
9x2  6 x  4
3
27 x  8
To divide one rational expression by another, multiply the first expression by the reciprocal of the
second expression.
a c a d ad
ad
 

then simplify
if possible.
b d b c bc
bc
Example: 5) Dividing polynomial rational expressions.
3
x 2  3x
 2
4x  8 x  x  6
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Example: 6) Dividing polynomial rational expressions.
8 x 2  10 x  3
 (4 x 2  x)
2
4x
Example: 7) Multiplying and dividing polynomial rational expressions.
x
4 x2  9
(2 x  3) 
x2
x2
Section 5.5 Add and Subtract Rational Expressions.
Like Bases:
Basically, you just keep the same denominator and add/subtract the numerators. (BE VERY CAREFUL
WHEN SUBTRACTING AND THE SECOND NUMERATOR IS a polynomial.) If this is the case, be sure to
distribute the negative sign to all that follow it.
Example: 1) With Like bases.
a)
3
7

2x 2x
x0
b)
3x
6

x4 x4
x4
c
x
3x  2

x 3 x 3
x3
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Unlike Bases:
Steps to add/subtract ration expressions with unlike bases.
Step 1: Factor all of the denominators.
Step 2: Look for Common Binomials, first. (Take the highest power where appropriate.)
Step 3: Look for Variable Coefficients. (Take the highest power where appropriate.)
Step 4: Look for Numerical Coefficients. (Take the LCM)
Step 5. Multiply the top and bottom of the fractions by the “Missing” parts of the LCD so that all
fractions have the same denominator.
Step 6: Add/subtract tops and keep the common denominator.
Example: 2) With Unlike bases.
a)
4
x
 3
3
3x 6 x  3x 2
b)
x 1
1
 2
x  6x  9 x  9
2
To simplify Complex Rational Expressions, you will need to find the LCD of all fractions contained in
the complex fraction. Then use that value to “Clear” the fractions. From there, just simplify whatever
is left.
117
Example: 3) Simplifying a Complex Fraction.
2
x2
a)
1
2

x2 x
You Try:
3
2

xy 2 x 2 y
b) 1
2
 3
2
2 x y xy
3
x4
1
3

x  4 x 1
Section 5.6 Solve Rational Expressions.
The trick is to “clear” the fractions using the least common denominator!
Example: 1) An equation with ONE Solution
3 1 12
 
x 2 x
118
5x
5

4

Example: 2) An equation with an Extraneous Solution
x 1
x 1
Example: 3) An equation with TWO Solution
3x  2
6
 2
1
x2 x 4
2
1

Example: 4) Solving an equation by Cross Multiplying
x2  x x 1
Example: 5) Application
The recommended percent p of oxygen (by volume) in the air that a diver breathes is given by
p
660
, where d is the depth (in feet) of a diver. At what depth is air containing 5% oxygen
d  33
recommended?
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Extension: Rational Inequalities.
Steps to solve a Rational inequality:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Get it set to zero
Combine all terms involved as one fraction with a “common denominator” and simplify.
Factor, if necessary.
Determine what values will make both the top expression and the bottom expression
equal zero. . These will be our “critical values”
Place the critical values on a real number line and use the open or closed circles to
represent the inequality being observed. REMEMBER THAT THE VALUES FROM THE
DENOMINATOR WILL ALWAYS BE OPEN CIRCLES REGARDLESS OF THE ORIGINAL
INEQUALITY.
Use the number line with the critical values to set up “test intervals” using interval
notation.
Choose a value that falls in the test interval and plug it in to the “factored” version of
your inequality. All you need to determine here is the sign (positive or negative) that
would happen once the test value is plugged in. This is called sign analysis.
Your answer will be the intervals that satisfy the inequality that is set to zero, if it is <
𝑜𝑟 ≤ you are looking for the interval was negative. If it is > 𝑜𝑟 ≥ you are looking for
the interval was positive.
You may be asked to graph the solution and also put it in interval notation.
Example: 1)
a.
𝑥−2
𝑥+3
≤1
b.
4𝑥+5
𝑥2
≥
4
𝑥+5