Conclusion of 2.6
Rational Functions and Their Graphs
The graph of a rational function has a slant asymptote
if the degree of the numerator is one more than the
degree of denominator. The equation of the slant
asymptote can be found by division. It is the equation
of the dividend with the term containing the remainder
dropped.
Example 12
x2  6 x  2
Find the slant asymptote of the function f(x)=
.
x
Example 13
x3  1
Find the slant asymptote of the function f(x)= 2
.
x  2x  2
Section 2.7
Polynomial & Rational Inequalities
Polynomial Inequalities
• Deal with finding the regions of a graph where a
given polynomial is above, below, or on the 𝑥axis.
Example 1
Solve the polynomial inequality and graph the
solution set on a real number line. Express the
solution in interval notation.
𝑥 2 − 11𝑥 + 28 > 0
Example 2
(𝑥 − 2)(3𝑥 + 1) ≤ 0
Example 3
7𝑥 ≤ 15 − 2𝑥 2
Example 4
𝑥 2 ≤ 4𝑥 − 2
Example 5
a.
9𝑥 2 > −6𝑥 − 1
2
b.
9𝑥 ≥ −6𝑥 − 1
c.
9𝑥 2 ≥ −6𝑥 − 1
Example 6
−5𝑥 2 + 6𝑥 < −𝑥 3
Example 7
𝑥 3 + 2𝑥 2 − 4𝑥 − 8 ≥ 0
Example 8
𝑥 3 − 𝑥 2 + 9𝑥 − 9 > 0
Rational Inequalities
• We can do the same thing with rational functions.
• Process is very similar, except we can have sign
changes at vertical asymptotes as well.
• Never put a bracket where there is an asymptote.
Example 9
−𝑥 − 3
≤0
𝑥+2
Example 10
(𝑥 + 3)(𝑥 − 2)
≤0
𝑥+1
Example 11
1
<1
𝑥−3
Questions???
Bring review questions Tuesday!!!