Polynomial & Rational Functions
Sections 3.1 – 3.4
For questions 1 and 2, graph each quadratic by determining whether the graph opens up or down,
finding the vertex, axis of symmetry, y –intercept, and x-intercept(s), if any. Determine the domain,
range, increasing intervals, and decreasing intervals for the function.
1. 𝑓(𝑥) = −3𝑥 2 + 3𝑥 − 2
2. 𝑓(𝑥) = 2𝑥 2 + 2𝑥 − 3
3. Write the equation of the quadratic function in standard form whose vertex is at (1, 4) and passes
through the point (-1, -8).
4. Write an equation for a polynomial whose zeros are given below:
Zero: -2, multiplicity 2
Zero: 4, multiplicity 1
For questions 5 & 6, determine:
a) the degree of the polynomial
c) the x & y intercepts (if any)
e) sketch the graph using the information
5. 𝑓(𝑥) = 𝑥 2 (𝑥 − 3)(𝑥 − 1)
6. 𝑓(𝑥) = −𝑥 2 (𝑥 2 − 1)(𝑥 + 1)
b) the end behavior
d) if the graph touches or crosses at the zero
For questions 7 - 12, use our eight steps from class to graph the rational function.
7. 𝑓(𝑥) =
8. 𝑓(𝑥) =
𝑥 3 −1
𝑥 2 −9
𝑥2
𝑥 2 +𝑥−6
9. 𝐻(𝑥) =
𝑥
𝑥 2 −4
10. 𝑅(𝑥) =
−4
(𝑥+1)(𝑥 2 −9)
11. 𝐹(𝑥) =
12. 𝑅(𝑥) =
4(𝑥 2 −1)
𝑥 4 −16
𝑥 2 +𝑥−12
𝑥−4
13. The current I in a circuit is inversely proportional to its resistance R measured in ohms. Suppose that
when the current in a circuit is 30 amperes the resistance is 8 ohms. Find the current in the same circuit
when the resistance is 10 ohms.
14. Jane has 100 feet of fencing available to enclose a rectangular field. Express the area A of the
rectangle as a function of x where x is the length of the rectangle. For what value of x is the area the
largest? What is the maximum area?
15. A parabolic arch has a span of 120 feet and a maximum height of 25 feet. Find the equation for the
parabola. Then calculate the height of the arch at points 10 feet, 20 feet and 40 feet from the center.