Directions: For each problem show your work and circle/box your answers. Calculators are allowed but no TI-89/TI-92 or cell phones.
1. What is the midpoint between P(7, -3) and Q(-4, 5)?
What is the distance between P(7, -3) and Q(-4, 5)?
2. Rewrite the following equation in standard form to find the center and radius (So you should have two answers) for the circle defined by the equation: ??2 - 4?? + ??2 = 5
3. Find the equation (in slope-intercept form) for the line perpendicular to ?? =1/2?? + 3 that goes through the point (1,2).
4. Graph the set of points described: { (x,y) | -2 < y < 1 }
5. Test the following equation for all three kinds of symmetry (show the tests and state results for x-axis, y-axis and origin symmetry):
?? = (??2 + 3)/ x
6. Solve the following and give its solution using interval notation: (?? - 3)2(?? +1/2) > 0
(Hint: Be sure to use test points)
7. Solve the following and give its solution in interval notation: 2/?? - 1 ≤ ??
8. Solve the following and give its solution in interval notation: |2 - ??| > 5
9. Re-write the following expressions in a+bi form:
a)2/(3 + i)
b) (3i)3
10. For (a)-(e) only an answer is needed (no partial credit on these):
a) If f(x) is 'one to one' and f(3) = -2 then f-1(-2) = _______.
b) Circle the function that does not have an inverse function among the following:
f(??) =1/??
f(??) = |??|
f(??) = -??3 + 5
f(??) = 2??
c) True or False: The domain of f(??) = | - ?? - 1| is ‘all real numbers'.
d) Give an example of an ‘even' function: f(??) = ___________________________________________ .
e) If f(??) has an inverse, then we can simplify (f-1° f)(??) to just _________________.
11. If f(??) = 2??2 and g(??) = 2?? then what is the domain of (f/g) (??) ?
12. Why does the Vertical Line Test say that a graph can't represent a function if a vertical line touches it more than once (think about the definition of a function)?
13. Sketch the graph of the function f(??) = {??2 if ?? ≤ 1
{x/2 - 1 if ?? > 1
Be sure to label some points!
Using the piecewise defined function above: What is f(4)?
Using the piecewise defined function above: What is (f ° f)(-2)?
14. If f(??) = 1/(??+2):
a) Find a formula for the inverse function, f-1(??):
b) What is the domain and range for f(??) and f-1(??)? (This question has 4 separate answers)
c) Show (in steps) that (f.f-1)(??) = ?? :
15. Starting with the graph of ?? = |??| use transformations (and describe the ones you use below) to graph the new function f(??) = -|??| + 1 Use as many of the grids below as you need.
16. If f(??) = 2?? - 4 and g(??) = √?? find the following:
a) Find (f ° g)(??) and its domain.
b) Find (g ° f)(??) and its domain.
17. Synthetically divide P(??) = 4??3 - 2??2 + 6?? - 5 by ?? -1/2 to find the value of P(1/2) according to the Remainder Theorem. You must show the synthetic division to get credit.
Synthetic Division:
So, P (1/2) = ________________
18. Finding the difference quotient f(x + h) - f(x)/h if f(x) = 2x2 in 3 steps:
a) First using f(??) = 2??2, find just f (x + h):
b) Then find and simplify f(x+h)-f(x):
c) Finally, divide that result by h and simplify:
19. Here is a graph of f(x):
a) What is the domain of f(x) in interval notation?
b) What is the range of f(x) in interval notation?
c) When is f(x) increasing (in interval notation)?
d) When is f(x) decreasing (in interval notation)?
e) Give the (x,y) point(s) that f(x) attains a relative maximum (if any)?
f) Give the (x,y) point(s) that f(x) attains a relative minimum (if any)?
20. P(??) = 2??2 + 8?? + 6
(a) What is the vertex of the graph of P(??)?
(b) Write this quadratic function in standard form, P(??) = a(?? - h)2 + k :
(c) Graph P(??) :
21. For the polynomial f(??) = 2??3 + 3??2 - 2??:
(a) Factor f(??) into linear factors.
(b) List all of the zeroes of f(??).
(c) Use f(??)'s end behavior, zeroes and possibly a couple extra points to graph f(??):
22. Let P(??) = ??4 - 2??3 - ??2 - 4?? - 6
a. List all of the possible rational zeroes for P(??):
b. Find all of the complex zeroes of P(??) (not just the real zeroes!). [You might want to use part (a) and synthetic division to help start finding these zeroes as a way to start, or try to factor it first.]
c. Write P(x) in the completely factored form of P(??) = a(x - c1)(?? - c2)... (?? - cn)
where the ci values are its zeroes.
23. Consider the rational function R(??) = (2??2+4??-6)/ (??2-??-6)
(a) Find all of the intercepts of R(??):
(b) Find all of the different asymptotes of R(??):
(c) Use (a) and (b), extra points and a calculator if needed to graph R(??):
Survey (free points for completing - don't skip!) :
How were you placed into College Algebra?
Circle one: Took Pre-Requisite Course COMPASS test ACT or SAT Score Other
How many hours did you prepare for this exam, and do you think it was enough?
How often do you review College Algebra material throughout the week (or do homework)?
Circle one: a couple times a week several times a week every day