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Algebra II Items to Support Formative Assessment
Unit 6: Radical and Rational Functions
Understand solving equations as a process of reasoning and explain the reasoning.
A.REI.A.2 Solve simple rational and radical equations in one variable, and give examples
showing how extraneous solutions may arise.
A.REI.A.2 Task
As a new element is heated, its temperature can be modeled with the function
T(x)= 100x 2 -800x +3500 where x represents the time, in minutes, from when the element is
placed on top of a Bunsen burner, and T(x) is the temperature, in degrees Celsius. When will the
temperature be 100C?
Answer: x =13, -5
However, in the context of this problem, x represents time, so -5 is an extraneous solution.
A.REI.A.2 Item 1
On her last quiz, Sally solves the following equation and finds two solutions. When she gets her
quiz back, she notices that she did not receive full credit for this problem. Explain where and
why Sally’s answer is incorrect.
(
9x +10 = x
)
2
()
9x +10 = x
2
9x +10 = x 2
x 2 - 9x -10 = 0
( x -10)( x +1) = 0
x = 10,x = -1
Answer:
Sally did not check for extraneous solutions. If she had, she would have seen that x = -1 is
extraneous, thus the only appropriate solution is x = 10.
A.REI.A.2 Item 2
Solve. Check for extraneous solutions.
6x +7 - 3x +3 = 1
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Answer:
1
3
x = -1
x=
A.REI.A.2 Item 3
Determine all restrictions of the following expression. Then, solve.
x -1
x -1
¸ 2
= 17
x -5 x -7x +10
Answer:
Restrictions: x ¹ 5,1,2
Solution: x =19
A.REI.A.2 Item 4
Working alone, Kyle can clean his room in 15 minutes. Ally can clean the same room in 11
minutes. If they work together, how long would it take them to the clean the room? Write a
rational equation that models the situation. Round your answer to the nearest thousandth.
Answer:
1 1 1
+ =
15 11 x
Together, they can clean the room in 6.346 minutes.
Represent and solve equations and inequalities graphically.
A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations
y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions
approximately, e.g., using technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational,
absolute value, exponential, and logarithmic functions.* (Cross-cutting)
A.REI.D.11 Task
For the following situation, find the length of the pendulum based on each of the times given
using your graphing calculator. Find the point of intersection for part a, part b, and part c. Then,
solve by hand to verify your answer.
The period of a pendulum is the time T (in seconds) it takes for a pendulum of length (in feet) to
go through one cycle. The period is given by
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
T = 2p
L
32
Given each period of a pendulum, find the length. Round your answer to two decimal places.
a. T = 1 second
b. T = 0.5 seconds
c. T = 2 seconds
Answer:
a.
Length: 0.811 feet
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
b.
Length: 0.203 feet
c.
Length: 3.242 feet
Verify:
a.
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
L
32
1 = 2p
1
L
=
2p
32
2
æ 1ö æ L ö
çè 2p ÷ø = ç 32 ÷
è
ø
2
2
æ 1ö
L
çè 2p ÷ø = 32
2
æ 1 ö
32ç ÷ = L
è 2p ø
L = 0.811 feet
b.
L
32
0.5 = 2p
0.5
L
=
2p
32
2
æ 0.5 ö æ L ö
çè 2p ÷ø = ç 32 ÷
è
ø
2
2
æ 0.5 ö
L
çè 2p ÷ø = 32
2
æ 0.5 ö
32ç
=L
è 2p ÷ø
L = 0.203 feet
c.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
2 = 2p
L
32
2
L
=
2p
32
2
æ 2ö æ L ö
çè 2p ÷ø = ç 32 ÷
è
ø
2
2
æ 2ö
L
çè 2p ÷ø = 32
2
æ 2ö
32ç ÷ = L
è 2p ø
L = 3.242 feet
A.REI.D.11 Item 1
The velocity of a free falling object is given by V = 2gh where h is the distance (in feet) the
object has fallen and g is the acceleration due to gravity (in feet per second squared). The value
of gravity depends on your altitude. If an object hits the ground with a velocity of 25 feet per
second, from what height was it dropped in each of the following situations?
a. You are standing on earth, so g = 32 ft/s2
b. You are on a space shuttle, so g = 29 ft/s2
c. You are on the moon, so g = 0.009 ft/s2
Answer:
a. 9.766 feet
b. 10.776 feet
c. 34,722.222 feet
A.REI.D.11Item 2
Determine all restrictions of the following expression. Then solve without the use of technology.
Verify your answer by finding where the two functions intersect using your graphing calculator.
1
x
4
+
= 2
x - 6 x - 2 x - 8x +12
Answer:
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Restrictions: x ¹ 6,2
Solution: x = -1 (x ¹ 6 because 6 is a restriction)
A.REI.D.11Item 3
The speed that a tsunami (tidal wave) can travel is modeled by the equation
S = 356 d
where S is the speed in kilometers per hour and d is the average depth of the water in kilometers.
a. What is the speed of the tsunami when the average water depth is 0.75 kilometers? Round
your answer to the nearest thousandth.
b. Solve the equation for d.
c. A tsunami is found to be traveling at 205 kilometers per hour. What is the average depth of the
water? Round your answer to the nearest thousandth.
Answer:
a. 308.305 kph
b.
S = 356 d
S
= d
356
2
æ S ö
çè 356 ÷ø =
( )
d
2
2
æ S ö
çè 356 ÷ø = d
c. 0.332 km
A.REI.D.11 Item 4
State the restrictions. Then, solve graphically.
x +2
= 2x +7
x -3
Answer:
Restrictions: x ¹ 3,x ³ -
7
2
Solutions: x = 4.647,-3.474
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.