Assignment
Answer all 21 questions and show thorough work in this document. Work must be typed and not hand written. An asterisk indicates an exercise for which a graph needs to be provided (#2, #14, and #15).
1. Given an equation and the graph of a quadratic function, do each of the following.
a. Give the domain and range.
b. Give the coordinates of the vertex.
c. Give the equation of the axis.
d. Find the y-intercept.
e. Find the x-intercepts.
2. Graph each quadratic function*. Give the (a) vertex, (b) axis, (c) domain, and (d) range. Then determine (e) the interval of the domain for which the function is increasing and (f) the interval for which the function is decreasing.
a. f(x) = -3 (x - 2)2 + 1
b. f(x) = -3x2 + 24x - 46
3. Accident rate-According to data from the National Highway Traffic Safety Administration, the accident rate as a function of the age of the driver in years x can be approximated by the function
f(x) = 0.0232x2 - 2.28x + 60.0,
for 16 ≤ x ≤ 85. Find both the age at which the accident rate is a minimum and the minimum rate.
4. Use synthetic division to perform each division.
a. (x3 + 7x2 + 13x + 6) / (x+ 2)
b. (x4 - x3 - 5x2 - 3x ) / (x+ 1)
5. Express f(x) in the form f(x) = (x -k) q (x) + r for the given value of k.
f(x) = 2x3 + 3x2 - 16x + 10; k = -4
6. For each polynomial function, use the remainder theorem and synthetic division to find f(k).
a. f(x) = x2 - 4x - 5; k = 5
b. f(x) = x4 + 6x3 + 9x2 + 3x - 3; k = 4
7. Use synthetic division to decide whether the given number k is a zero of the given polynomial function. If not, give the value of f(k).
f(x) = 2x3 + 9x2 - 16x + 12; k = 1
8. Use the factor theorem and synthetic division to decide whether the second polynomial is a factor of the first.
2x3 + x + 2; x + 1
9. Factor f(x) into linear factors given that k is a zero of f(x).
f(x) = 2x3 - 3x2 - 5x + 6; k = 1
10. For the polynomial function, one zero is given. Find all others.
f(x) = x3 + 4x2 - 5; 1
11. For the polynomial function, find all zeros and their multiplicities.
f(x) = (x + 1)2 (x - 1)3 (x2 - 10)
12. Find a polynomial function f(x) of degree 3 with real coefficients that satisfies the given conditions.
Zeros of 2, -3 and 5; f(3) = 6
13. Find a polynomial function f(x) of least degree having real coefficients with zeros as given.
7 - 2i and 7 + 2i
14. Sketch the graph of each polynomial function.* Determine the intervals of the domain for which each function is (a) increasing or (b) decreasing.
a. f(x) = -(5/4)x5
b. f(x) = (1/3) (x + 3)4 - 3
15. Graph the polynomial function.* Factor first if the expression is not in factored form.
f(x) = x(x + 1) (x - 1)
16. Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given.
f(x) = 3x2 - x - 4; 1 and 2
17. Show that the real zeros of the polynomial function satisfy the given conditions.
f(x) = 2x5 - x4 + 2x3 - 2x2 + 4x - 4; no real zero greater than 1
18. Solve the following variation problem.
If m varies jointly as z and p, and m = 10 when z = 2 and p = 7.5, find m when z = 6 and p =
19. Current in a circuit-The current in a simple electrical circuit varies inversely as the resistance. If the current is 50 amps when the resistance is 10 ohms, find the current if the resistance is 5 ohms.
20. Long-distance phone calls-The number of long-distance phone calls between two cities in a certain time period varies directly as the populations p1 and p2 of the cities and inversely as the distance between them. If 10,000 calls are made between two cities 500 mi apart, having populations of 50,000 and 125,000, find the number of calls between two cities 800 mi apart, having populations of 20,000 and 80,000.
21. Suppose y is inversely proportional to x, and x is tripled. What happens to y?