Coordinate Algebra
Final Exam Review with Key
Spring, 2015
Unit 1: Relationships Between Quantities
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There are 5280 feet in one mile
There are 0.034 ounces in one milliliter
There are 0.454 kg in one pound
There are 1.6 kilometers in one mile
There are 73 gallons in 2 barrels
There are 1.05 quarts in one liter
There are 4 quarts in one gallon
There are 16 ounces in a pound.
1. What range of measurement is equivalent to 35 mm ± 15%?
35 ± 5.25 gives a range of 29.75-40.25
35 x 0.15 = 5.25
2. A cyclist travels 45 miles in 4 hours. What is her speed in feet per second?
45 miles 5280 ft
1 hour
1 min
4 hours
60 min
60 sec
1 mile
45×5280
= 4×60×60 =
237,600
14,400
= 16.5𝑓𝑡/𝑠
5
9
3. Solve the formula for F: C    F  32 .
A.
9
F    C  32
5
4. Solve for x:
5
F    C  32
9
B.
5
F    C  32
9
C.
D.
9
F    (C  32)
5
4𝑥+2 = 16𝑥
Once the bases are equal, the powers are then equal. We can rewrite 16 to have a base of 4.
4𝑥+2 = (42 )𝑥
4𝑥+2 = 42𝑥
𝑥 + 2 = 2𝑥
2=𝑥
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
5. Convert 75 ounces to milliliters.
A . 2.55 mL
B. 2205.88 mL
C. 220.59 mL
D. 25.5 mL
Unit 2: Reasoning with Equations and Inequalities
6. A rectangle has a length of 6 meters and a width of 2 meters. A larger, similar rectangle has a
length of 26 meters. What is the width of the larger rectangle?
6
26
Using proportions, we have that
2
2
= 𝑥. After cross multiplication, 6𝑥 = 52, so 𝑥 = 8 3
7. When solving an equation, how do you determine if there is no solution? Infinite solutions?
When solving an equation algebraically, we know that there is no solution when we arrive at an
untrue statement. For example, 0 = 4 indicates no solution. We know that there are infinite
solutions when we arrive at a true statement. For example 2=2 or -8 = -8. In both cases (no
solution and infinite solutions), the variables are eliminated from the problem.
8. Find three consecutive integers whose sum is 450.
x + (x + 1) + (x + 2) = 450
3x + 3 = 450
3x = 447
x = 149
The three consecutive integers are 149, 150, and 151.
9. Write and graph (one a number line) the solutions of -2(1 – x) < 3(x – 2).
-2 + 2x < 3x – 6
-2 < x – 6
4<x
On your number line, you should have a shaded circle at 4, with your arrow pointing right.
10. Fill in the chart below.
Graph
Intersecting Lines
Same Line
Parallel Lines
Solutions
Exactly One
Infinite
No Solution
Classification
Consistent, Independent
Consistent, Dependent
Inconsistent
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
Unit 3: Functional Relationships
11. Which relation is not a function?
a. {(1, -5), (3, 1), (-5, 4), (4, -2)
b. {(1, -5), (-1, 6), (1, 5), (6, -3)
c.
d.
{(2, 7), (3, 7), (4, 7), (5, 8)}
{(3, -2), (5, -6), (7, 7), (8, 8)}
12. Which equation models an exponential decay?
a. 𝑦 = −0.12(1.05)𝑡
b. 𝑦 = 0.12(1.05)𝑡
b. 𝑦 = 1.05(0.12)𝑡
d. 𝑦 = 0.12(1 + 0.5)𝑡
13. Brian calculates the charge for each lawn that he mows using the function y = 4x + 5.5, where x
is the number of hours spent mowing the lawn. Brian never takes more than 2 hours to mow a
lawn. Write an inequality that best represents the reasonable range for the function.
We must first know the domain in order to provide a range. The domain consists of the x-values,
which in this case represent the number of hours Brian spends mowing the lawn. The domain
here is 0 < 𝑥 ≤ 2. When we plug in 0 for x, we get that y = 5.5. When we plug in 2 for x, we get
that y = 13.5. Therefore, a reasonable range is 5.5 < 𝑦 ≤ 13.5.
14. A line passes through the point (5, -1). The slope of the line is -1/2. Write the equation of the
line in slope-intercept form. __________________________
1
−1 = − 2 (5) + 𝑏
y = mx + b
5
3
−1 = − 2 + 𝑏
𝑏=2
So we have that the slope is -1/2 and the y-intercept is 3/2. Therefore, our equation is
1
3
𝑦 = − 2 𝑥 + 2.
15. Perpendicular lines have slopes that are ____opposite____________ _reciprocals_________.
16. Parallel lines have the same __slope________ but different __y-intercepts_______________.
17. What equation is used to find a term in an arithmetic sequence? __𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑______
18. What does 𝑎𝑛 stand for? __The 𝑛𝑡ℎ _term _in the sequence_ 𝑎1 ?__The 1st terms in the
sequence__
19. What equation is used to find a term in a geometric sequence? __𝑎𝑛 = 𝑎1 (𝑟)𝑛−1 ___________
20. Write a rule to define the sequence -3, 15, -75, 375, … __𝑎𝑛 = −3(−5)𝑛−1 ________________
1
1
1
1
21. Find the next three terms in the sequence −36, 6, −1, 6, … __− 36 , 216, − 1296______________
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
𝑟
Compound Interest _𝐴 = 𝑃(1 + 𝑛)𝑛𝑡 __________ Exponential
Write the following formulas:
Growth __𝐴 = 𝑃(1 + 𝑟)𝑡
Exponential Decay _𝐴 = 𝑃(1 − 𝑟)𝑡 __________
22. The value of a gold coin picturing the head of the Roman Emperor Vespasian is $105. The value
is increasing at a rate of 10% per year. Write an exponential function to model this situation.
Then find the value of the coin in 11 years.
𝐴 = 105(1 + 0.1)11 = $299.58
23. Two-thousand dollars is invested at a rate of 4.5% compounded monthly. Write a function to
model the situation and find the amount after 3 years.
. 045 12∙3
𝐴 = 2000(1 +
)
= $2288.50
12
* The formula to calculate half-life is A = P(0.5)𝑡 , where A represents the final amount, P represents the
original amount, and t represents the number of half-lives in a given period. *
24. Flourine-20 has a half-life of 11 seconds. Find the amount of Flourine-20 left from a 40-gram sample
after 44 seconds.
25. Which function is not an exponential decay model?
a.
1
𝑦 = 5(3)𝑥
b.
1
𝑦 = −5(3)𝑥
c.
𝑦 = 5(3)−𝑥
d.
𝑦 = 5(3−1 )𝑥
Unit 4: Data Distributions
1. The number of people at a caterer’s last 12 parties are given below.
16, 18, 17, 19, 15, 25, 18, 17, 18, 16, 17, 19
a. Use the data to make a frequency table with intervals (answers may vary).
# of
people
15-16
17-18
19-20
21-22
23-24
25-26
Frequency
3
6
2
0
0
1
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
b. Use your frequency table to make a histogram.
2. A bookshop surveys its customers about their magazine-buying habits, summarized in the table.
Reads Look Around
Reads
Super
News
Yes
No
Yes
62
21
No
15
136
a. Make a table of the joint relative frequencies and the marginal relative frequencies.
Yes
62
21
83
No
15
136
151
Total
77
157
234
Yes
No
Total
62/234≈ 15/234≈ 77/234≈
.26
.06
.33
No
21/234≈ 136/234≈ 157/234≈
.09
.58
.67
Total 83/234≈ 151/234≈ 234/234 =
1
.35
.65
b. Given that a customer reads Super News, what is the probability that he or she also reads Look
Yes
No
Total
Yes
62
Around? 77 customers read Super News. 62 of those 77 also read Look Around. 77 ≈ .81
3. What are the three types of distributions? _Uniform______________
__Symmetric______________ __Skewed______________
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
4. Fill in the table below:
Type of Distribution
Uniform
Description
All data points have an approximately equal
frequency
Symmetric
A vertical line can be drawn and the result is a
graph divided in two parts that are approximate
mirror images of each other.
Skewed
The data is not uniform or symmetric. The data
may be skewed to the right (most data values fall
to the less than the mean) or skewed to the left
(most data values are greater than the mean).
5. The data table shows the number of miles run by members of two track teams during one day.
Make a dot plot and determine the type of distribution for each team. Explain what the
distribution means for each.
Miles
Team A
Team B
3
2
1
3.5
3
2
4
4
2
4.5
4
3
5
3
4
5.5
2
6
6
0
5
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
Tell whether the situation is best characterized by a positive negative, or no correlation.
6. The height of a candle and the amount of time it stays lit __Negative____________________
7. The price of a pizza and the number of toppings added ___Positive___________________
8. The temperature of a cup of hot chocolate and the length of time it sits
____Negative_________________
9. Which of the following data sets has the greatest interquartile range?
a. 3, 4, 6, 5, 7, 5, 7
b. 8, 8, 8, 8, 8, 8, 8 c. 10, 3, 11, 11, 12, 11, 11
d.
2, 1, 2, 6, 4, 5, 6
Unit 5: Transformations in the Coordinate Plane
10. A figure has vertices at X(1, 1), Y(3, 1), and Z(3, 4). After a transformation, the image of the
figure has vertices at X’(-1, -1), Y’(-3, -1), and Z’(-3, -4). Identify the transformation.
180 degree rotation
11. The image of point A under a 90 degree rotation about the origin is A’(10, -4). What are the
coordinates of point A? (-4, -10)
12. The point G(6, 7) is rotated 90 degrees about point M(-9, -3) and then reflected across the line
y = 9. What are the coordinates of the image G’? OMIT
13. How many lines of symmetry does a regular hexagon have? 6
14. How many lines of symmetry does a regular pentagon have? 5
15. The composition of two reflections across two parallel lines is equivalent to a
___translation_________________.
16. The composition of two reflections across two intersecting lines is equivalent to a
_rotation__________.
17. Any translation or rotation is equivalent to a composition of two
____reflections___________________.
Unit 6: Connecting Algebra and Geometry Through Coordinates
18. Write the following formulas:
𝑦2 −𝑦1
Slope Formula: _
𝑥2 −𝑥1
________________
Slope-Intercept Form: ___y = mx + b________________
Point-Slope Form _𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )_____ Distance Formula: _√(𝑦2 − 𝑦1 )2 + (𝑥2 − 𝑥1 )2 ________
MidPoint Formula ___(
𝑥1 +𝑥2 𝑦1 +𝑦2
2
,
2
)________________
19. The slope of a vertical line is _undefined___________________.
20. The slope of a horizontal line is ___zero________________.
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
21. The equations of four lines are given. Identify which lines are parallel.
1
8
Line 2: 𝑦 − 6 = (𝑥 − 4)
Line 1: y = 8x – 3
Line 3: y = 3x + 4
1
3
Line 4: 𝑥 − 𝑦 = −4
________________________________
Parallel lines have the same slope. We must first put all equations in the form y = mx + b to determine
the slopes. Line 1 has a slope of 8, line 2 has a slope of 1/8, line 3 has a slope of 3, and line 4 has a slope
of 3. So lines 3 and 4 are parallel .
22. Identify the lines that are perpendicular.
1
Line 2: 𝑦 = 5 𝑥 − 5
Line 1: y = 4
Line 3: x = 8
Line 4: 𝑦 + 5 = −5(𝑥 + 1)
Perpendicular lines have slopes that are opposite reciprocals-meaning their signs are opposite and the
number is flipped. Line 1 has a slope of 0 (horizontal line), line 2 has a slope of 1/5, line 3 has an
undefined slope (vertical line), and line 4 has a slope of -5. So lines 2 and 4 are perpendicular, and lines 1
and 3 are perpendicular.
23. Willy plots the points (0, 0), (4, 0), and (0, 7). Then he connects the points to form a right
triangle. What is the length of the hypotenuse? _______________________
Using the Pythagorean theorem, 42 + 72 = 𝑥 2
7
𝑥 = √65 ≈ 8.06
4
24. Write the equation of the line that is perpendicular to the line -3x + 12y = 4 and passes through
the point (2, -3). ________________________
Write the equation in slope-intercept form in order to identify the slope.
-3x + 12y = 4
12y = 3x + 4
1
4
𝑦= 𝑥+
1
3
Since the slope is 1/4, the slope of the line perpendicular to this line will be -4. Use the point (2,
-3) and the slope -4 in the equation y = mx + b and solve for b.
-3 = -4(2) + b -3 = -8 + b
b = 5 So the equation is y = -4x + 5
25. Which is the equation of a line whose slope is undefined?
a. x = -5
b. y = 7
c. x = y
d.
x+y=0
26. What is the distance between the points (7, -3) and (-5, 6)? __√(6 − (−3))2 + (−5 − 7)2 =
√81 + 144 = √225 = 15__________________________
Coordinate Algebra
Final Exam Review with Key
Spring, 2015
̅̅̅̅. What is VX if VW = 2x + 5 and WX = 4x – 3?
27. W is the midpoint of 𝑉𝑋
__________________________
̅̅̅̅, then VW = WX. So 2x + 5 = 4x – 3. Solving for x gives that x = 4.
If W is the midpoint of 𝑉𝑋
VX = (2x + 5) + (4x – 3). Substituting 4 for x gives that VX =26
28. A directed line segment begins at F(-2, -2), ends at H(8, 8), and is divided in the ratio 4 to 2 by G.
What are the coordinates of G? Round to the nearest hundredth if necessary.
____________________________
First find the vector components of the segment using < 𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 >
< 8 − (−2), 8 − (−2) >
< 10, 10 >
Since the point G divides the segment in the ratio 4 to 2, G is 4/6 of the way from F to G.
Multiplying the vector components by 4/6 (or 2/3) gives <
20 20
,
3 3
>. We now add these units to
the F coordinates in order to find the point G.
(−2 +
20
20
, −2 + 3 )
3
14 14
)
3
= (3 ,
What is the perimeter of the triangle below?
Use the distance formula to find the lengths of the three sides
2
2
𝐴𝐵 = √(6 − (−2)) + (−1 − (−6)) = √64 + 25 = √89 ≈ 9.4
Coordinate Algebra
Final Exam Review with Key
2
𝐵𝐶 = √(1 − 6)2 + (2 − (−1)) = √25 + 9 = √34 ≈ 5.8
𝐴𝐶 = √(1 − (−2))2 + (2 − (−6))2 = √9 + 64 = √73 ≈ 8.5
9.4 + 5.8 + 8.5 = 23.7
Spring, 2015