Math 1090
Practice Test 1
29 September, 2010
1. Find the domain of each function:
√
(a) f (x) = x − 3
(b) g(x) =
1
x+2
2
(c) h(x) = x − 4
2. Solve the system of equations, if solutions exist.
(a)
1
y
3
= −x − 19
x + 31 y = 5
(b) 2x + y = 9
3x = y + 11
(c) y + 3z = −7
2x − y + 4z = −21
3y + 2z = 7
3. Solve the linear inequality and graph the solution set on a number line:
x+5
2x + 4
−2=
+1
3
4
4. Describe the end behavior of the graphs of the following polynomials (e.g., up on the
left, down on the right)
(a) y = x2 + 2
(b) y = −5x5 − x3 + 6x2 − 3
(c) y = −4x12 − 4x11 + 6x7 − 8x4 + x2 − 13
(d) y = x3 − 1
5. Which of the following are functions?
(a) .
(b) (Ignore colors)
(c) (1, 0), (2, 4), (1, 3), (5, 7)
(d) (0, 4), (3, 6), (5, 8), (4, 9)
(e) x2 + y 2 = 1
(f) 3x + y = 6
6. Let f (x) = x2 − 2x − 24.
(a) Find the vertex of this parabola.
(b) Find the axis of symmetry.
(c) Find the roots.
(d) Is the parabola concave up or concave down?
(e) Is the vertex a maximum or a minimum?
7. You are offered a job with two salary options: either (a) a fixed salary of $ 2000 per
month or (b) a salary of $ 1000 per month plus an 8% commission on sales. For what
range of slaes is plan (a) better?
8. Fill in the table and sketch the graph:

 x+3
x2
f (x) =

4
x
-3 -2 -1 0
f(x)
1 2 3 4
x ≤ −1
−1<x≤2
x>2
9. Graph the solution set for this system of inequalities:
x − 3y < 16
2x + 3y < −4
4x − 3y > 10
10. Solve:
2(x − 7) = 5(x − 3) − x
11. Are these functions polynomials?
(a) f (x) = 3x2 + 1
(b) g(x) = πx4 − 1
(c) h(x) = 3xπ − 2
1
(d) k(x) = x 2 + 1
(e) p(x) = x3 − x5 + 1
12. Match the polynomials to their graphs. (Hint: Pay attention to degree, leading coefficient, and end behavior.)
(a) y = x4 − 2x3 + 1
(b) y = −x4 + x2
(c) y = 12 x3 − 2x2
(d) y = −x3
(i).
(ii).
(iii).
(iv).
13. Solve the following quadratic equations. You may use any method, but one is suggested
for each.
(a) (square root)
(x + 2)2 = 3
(b) (factoring)
x2 − 2x + 63 = 0
(c) (completing the square)
x2 − 4x − 9 = 0
(d) (quadratic formula)
3x2 + 4x = 3
14. Graph the solution to this inequality:
5x − 2y ≥ 3
15. Let
f (x) =
3x2 + 6x + 3
x2 − 3x − 10
(a) Find the domain.
(b) Find the vertical asymptotes, if any.
(c) Find the horizontal asymptotes, if any.
(d) Find the y-intercept, if any.
(e) Find the x-intercepts, if any.
(f) Sketch the graph.
16. Three less than four times a number is 25. What is the number?
17. Write the equation of each line described below.
(a) slope 45 , y-intercept −7
(b) slope
−2
,
3
passes through (4, 3)
(c) passes through (−4, 3) and (8, 10)
(d) passes through (2, 1), perpendicular to y = 32 x + 4
(e) passes through (8, 3), parallel to y = 14 x − 1
18. Find the number of units needed to break even with the following revenue and cost
functions. Remember there may be more than one break-even point!)
R(x) = −x2 + 1600x
C(x) = 1500x + 1600
19. Find the maximum and minimum of the objective function y − 2x given the following
constraints:
x≥0
y≥0
1
y ≤− x+4
2
2x − 2y ≥ −6
(a) Draw the feasible region.
(b) Find the corners.
(c) Evaluate the objective function at each corner.
20. Find the slope of the line that passes through the points (14, 3) and (2, 12)
21. Give the degree and leading coefficient of each polynomial.
(a) 2x2 − 3x3
(b) 2x4 − 6x2 + 12
22. If a company’s profit function is given by P (x) = −5(x − 35) + 10000, what is the
company’s maximum possible profit?
23. Find the slope and y-intercept of each line.
(a) y = 3x − 2
(b) 8x + 4y = 2
24. A craftsman sells leather belts for $15 each. If his tools required $300 in one-time startup costs and his variable costs are $9 per belt, how many belts must he sell to break
even?
25. Evaluate each function at the given value.
√
(a) f (x) = x2 + 3 at x = −1
(b) g(x) =
1
x−6
+ 2 at x = 4
26. The perimeter of a basketball court is 104 feet. The length is 15 feet more than the
width. What are the dimensions of the court?
27. Let g(x) = −2(x + 3)2 + 6
(a) Find the vertex.
(b) Find the axis of symmetry.
(c) Find the roots.
(d) Is the parabola conave up or concave down?
(e) Is the vertex a minimum or a maximum?
28. Are lines with the following slopes parallel, perpendicular, or neither?
(a) m1 = − 21 , m2 = 2
(b) m1 = 3, m2 =
(c) m1 = 45 , m2 =
1
3
4
5
(d) m1 = −2, m2 = 2
Answers
1. (Section 3.2)
(a) x > 3
(b) x 6= −2
(c) all real numbers
2. (Section 1.4)
(a) no solution
(b) x = 4, y = 1
(c) x = 0, y = 5, z = −4
3. (Section 1.2)
x<7
0
7
4. (Section 3.5)
(a) up on the left, up on the right
(b) up on the left, down on the right
(c) down on the left, down on the right
(d) down on the left, up on the right
5. (Section 3.2) (a),(d),(f)
6. (Section 3.3)
(a) (−1, 21)
(b) x = −1
(c) x = 6, x = −4
(d) concave up
(e) minimum
7. (Section 1.2) s < 12500
8. (Section 3.5)
x
-3 -2 -1 0
f(x) 0 1 2 0 1
1 2 3 4
4 4 4 4
(Note that the vertical line at −1 is an error and should not be there.)
9. (Section 1.6)
(Triangle should be shaded; all lines should be dashed)
10. (Section 1.1)
x = 12
11. (Section 3.5)
(a) yes
(b) yes
(c) no
(d) no
(e) yes
12. (Section 3.5)
(a) (i)
(b) (iv)
(c) (iii)
(d) (ii)
13. (Section 3.1)
(a) x = 2 ±
√
3
(b) x = −7 or x = 9
√
(c) x = 2 ± 13
√
(d) x = − 23 ± 31 13
14. (Section 1.6)
(Should be shaded below the line)
15. (Section 3.6)
(a) x 6= 5, −2
(b) x = 5 and x = −2
(c) y = 3
(d) (0, −3
)
10
(e) (−1, 0)
(f) .
(Note that the vertical lines are asymptotes and should therefore be dotted)
16. (Section 1.1) 7
17. (Section 1.3)
(a) y = 45 x − 7
(b) y − 4 =
(c) y − 3 =
(d) y − 1 =
−2
(x − 3)
3
7
(x + 4)
12
−3
(x − 2)
2
(e) y − 3 = 14 (x − 8)
18. (Section 3.4) 20 or 80 units
19. (Section 1.7)
(a) .
(Note that the quadrilateral bounded by the two lines and the two axes should be
shaded)
)
(b) (0, 0), (8, 0), (0, 3), and ( 23 , 11
3
(c) maximum is 73 at ( 23 , 11
)
3
minimum is −16 at (8, 0)
20. (Section 1.3) − 43
21. (Section 3.5)
(a) degree: 3 Leading coefficient: -3
(b) degree: 4 Leading coefficient: 2
22. (Section 3.4) $10000
23. (Section 1.3)
(a) slope: 3 y-intercept: -2
(b) slope: -4 y-intercept:2
24. (Section 1.5) 50 belts
25. (Section 3.2)
(a) f (x) = 2
(b) g(x) =
3
2
26. (Section 1.1) width: 18.5 feet length:33.5
27. (Section 3.3)
(a) (−3, 6)
(b) x = −3
√
(c) −3 ± 3
(d) concave down
(e) maximum
28. (Section 1.3)
(a) perpendicular
(b) neither
(c) parallel
(d) neither