Algebra III Fall 2008 Diagnostic Review
Name___________________________________
Use the distance formula to find the distance between the pair of points.
1) (-6, -2) (3, -5)
Decide whether or not the points are the vertices of a right triangle.
2) (-6, 10), (0, 12), (-1, 7)
Solve the problem.
3) Find all the points having an x-coordinate of 9 whose distance from the point (3, -2) is 10.
4) City B is located at 100 miles east and 50 miles north of cityA. City C is located at 75 miles west and 150 miles
south of city A. Find the distance between city B and city C. You can choose city A as the origin of the
rectangular coordinate system. Write your answer rounded to two decimal places, if necessary.
Find the indicated point(s).
5) Find the midpoint of the line segment joining the points (7, 1) and (-16, -16).
Graph the equation by plotting points.
6) y = -x -2.
10
y
10 x
-10
-10
Solve the problem.
7) If (3, b) is a point on the graph of 3x - 2y = 17, what is b?
1
Sketch the graph of the equation by hand.
8) Sketch the graph of y = -x - 3
5
y
4
3
2
1
-5
-4
-3
-2
-1
-1
1
2
3
4
5 x
-2
-3
-4
-5
Using a graphing utility, find the intercepts of the graph of the equation given. Approixmate your answers rounded to
three decimal places.
9) Find the x- and y-intercepts of y = x3 - 3x2 - 45x - 16.
Determine whether the function is symmetric with respect to the y-axis, symmetric with respect to the x-axis, symmetric
with respect to the origin, or none of these.
10) y = 3x2 - 5
11) y = -5x3 + 2x
Determine whether the given equation is symmetric to the x-axis, y-axis, origin, or none of these.
12) 5x3 - 6y5 = 3
Write the standard form of the equation for the circle.
13) Give the equation for a circle.
Center at (5, 8), radius 7
List the intercepts of the graph.Tell whether the graph is symmetric with respect to the x-axis, y-axis, origin, or none of
these.
14)
y
10
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10 x
-4
-6
-8
-10
2
Find the center (h, k) and radius r of the circle. Graph the circle.
15) Find the center and radius of x2 + y2 + 4x + 12y + 15 = 0. Then graph the equation.
10
y
10 x
-10
-10
Use a graphing utility to approximate the real solutions, if any, of the equation rounded to two decimal places.
16) x3 - 6x + 3 = 0
17) x4 - 3x2 + 4x + 15 = 0
Write each expression in interval notation. Graph each interval.
18) x > 7
19) -6 < x ≤ 3
Using the variable x, write each interval as an inequality.
7
20) (-«, )
9
Write the interval in inequality notation.
21) (-5, 6]
Calculate and interpret the slope of the line.
22) For the line containing the points (1, -3) and (7, 8),
(a) find the slope, and (b) interpret the slope.
Find the slope of the line.
23)
y
x
3
Solve by factoring.
24) 3k2 - 23k - 8 = 0
25) 6m 2 - 5m = 0
Solve the equation by factoring.
26) 4x 3 - 8x2 = 12x
Write an equation for the line.
27) Through (-4, -6), parallel to -8x + 9y = -4
28) Through (9, 4) perpendicular to 5x + 8y = 13
Use the quadratic formula to solve the equation.
29) 4x2 + 12x = - 2
Solve the equation.
30)
2x + 3 -
x+1=1
31) |2m + 7| = 5
32) (5x + 3)1/2 = 3
Solve the inequality. Graph the solution set.
33) 18x - 9 > 3(5x + 3)
Solve the problem.
34) Express that the distance on the number line between x and the origin is 2 as an equation with absolute value.
Solve for x.
Solve the inequality.
35) |h - 2| - 1 ≤ 8
Solve the problem.
36) In a college algebra class, Jane has four test scores of 75, 81, 69, and 93. To get a grade of B, the average of the
four tests and the final exam must be greater than or equal to 80 and less than 90. The final exam weighs twice
as much as a test and has 100 points as the maximum available points. Solve an inequality to find the range of
the score Jane needs on the final exam to get a grade of B. Let x represent her score in the final exam.
Give the domain of the function.
37) f(x) = 3x2 + 2x + 5
38) f(x) =
14 - x
Solve the problem.
39) Let P = (x, y) be a point on the graph of y =
x.
x. Express the distance d from P to the point (1, 0) as a function of
4
Find the requested function.
40) Given the functions f(x) = x - 2 and g(x) = 3x - 3, find (f + g)(x), (f - g)(x), (f œ g)(x), and (f/g)(x).
Decide whether the relation defines a function. Explain why or why not.
41) {(-3, -8), (1, 4), (6, -2), (9, 2), (11, 3)}
A) Not a function
B) Function
Determine whether or not the graph represents a function. Explain why or why not.
42)
A) Not a function
B) Function
Evaluate the function.
43) Find f(k - 1) when f(x) = 5x2 + 4x - 4.
Solve the problem.
44) A wire of length 6x is bent into the shape of a square. Express the area of the square as a function of x.
45) Suppose that a school has just purchased new computer equipment for $15,000 .00. The school chooses to
depreciate the equipment using the straight line method over 5 years. (a) Write a linear function that expresses
the book value of the equipment as a function of its age. (b) Graph the linear function. (c) What is the value of
the machine after 2 years?
25000
y
22500
20000
17500
15000
12500
10000
7500
5000
2500
2.5
5
x
Determine whether the equation is a function. Explain why or why not.
46) y =3x2 - 3x - 1
A) No
B) Yes
47) x = y2 + y - 1
A) Yes
B) No
5
Answer Key
Testname: ALGEBRA III DIAGNOSTIC - FALL 2008.TST
1) Answer: 3 10
2) Answer: No
3) Answer: (9, 6), (9, -10)
4) Answer: 265.75 miles
9
15
5) Answer: - , 2
2
6) Answer:
7) Answer: -4
8) Answer:
5
y
4
3
2
1
-5
-4
-3
-2
-1
-1
1
2
3
4
5 x
-2
-3
-4
-5
9) Answer: (-5.144, 0), (-0.366, 0), (8.509, 0), (0, -16)
10) Answer: y-axis only
11) Answer: origin only
12) Answer: neither
13) Answer: (x - 5)2 + (y - 8)2 = 49
14) Answer: (0, 2) and (0, -2); symmetric to x-axis, y-axis, and origin
1
Answer Key
Testname: ALGEBRA III DIAGNOSTIC - FALL 2008.TST
15) Answer: center (-2, -6); r = 5
10
y
10 x
-10
-10
16) Answer: {2.15, 0.52, -2.67}
17) Answer: no solution
18) Answer: (7, «)
3
4
5
6
7
8
9
10 11
19) Answer: (-6, 3]
-6
20) Answer: x <
-5
-4
-3
-2
-1
0
1
2
3
7
9
21) Answer: -5 < x ≤ 6
11
22) Answer: (a)
(b) If x increases by 6 units, y increases by 11 units.
6
23) Answer:
1
2
24) Answer: 25) Answer:
1
3
,8
5
,0
6
26) Answer: {-1, 0, 3}
27) Answer: -8x + 9y = -22
28) Answer: -8x + 5y = -52
29) Answer:
-3 ±
2
7
30) Answer: {3, -1}
31) Answer: {- 1, - 6 }
6
32) Answer: x =
5
2
Answer Key
Testname: ALGEBRA III DIAGNOSTIC - FALL 2008.TST
33) Answer: (6, «)
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
34) Answer: |x| = 2; x = -2 or x = 2
35) Answer: -7 ≤ h ≤ 11
36) Answer: 81 ≤ x ≤ 100
37) Answer: All real numbers
38) Answer: x ≤ 14
39) Answer: d =
x2 - x + 1
40) Answer: (f + g)(x) = 4x - 5, (f - g)(x) = -2x + 1, (f œ g)(x) = 3x 2 - 9x + 6, (f/g)(x) =
41) Answer: B
42) Answer: B
43) Answer: 5k2 - 6k - 3
9
44) Answer: A(x) = x2
4
45) Answer: f(x) = - 3000 x + 15,000
value after 2 years is $9000 .00
25000
y
22500
20000
17500
15000
12500
10000
7500
5000
2500
2.5
5
x
46) Answer: B
47) Answer: B
3
x-2
3x - 3