Name: ________________________

Class: ___________________

Date: __________

ID: A

Analytic Geo B - Benchmark #1 Study Guide Multiple Choice Identify the choice that best completes the statement or answers the question. ____

____

2

1. Solve 8x = 135. a.

x=±

b.

x=±

3

30 4

3

15

2

2

2. Solve 16x2 – 81 = 0. 9 a. ± 4 b.

No real solution

c.

x = ±8.4275

d.

x = ±16.875

c.

± 81

d.

±9

16 4

2

____

3. Find the minimum or maximum value of the function g ( x) = − ( x − 2 ) − 3. a. maximum of 2 c. maximum of −3 b. minimum of −2 d. minimum of −3

____

4. What ordered pair corresponds to the minimum or maximum of the function y = a.

ÊÁ 3, −4 ˆ˜ Ë 

b.

ÊÁ −3, −4 ˆ˜ Ë 

c.

1

ÊÁ 3, 4 ˆ˜ Ë 

d.

ÊÁ −3, 4 ˆ˜ Ë 

2 3

2

( x − 3 ) − 4?

Name: ________________________ ____

ID: A

5. Use a table to graph the quadratic function f(x) = 4x2 – 5x – 3. a.

c.

b.

d.

2

Name: ________________________ ____

ID: A

2

6. Graph y = x − 3x + 4. Find the axis of symmetry and the vertex. a. c.

3

3

The axis of symmetry is x = − 2 . The Ê 3 7ˆ vertex is ÁÁÁÁ − 2 , 4 ˜˜˜˜ . Ë  b.

The axis of symmetry is x = 2 . The Ê3 7ˆ vertex is ÁÁÁÁ 2 , 4 ˜˜˜˜ . Ë  d.

The axis of symmetry is x = 4. The vertex is (4, 0).

The axis of symmetry is x = 0. The vertex is (0, 4).

3

Name: ________________________ ____

7. The function h ( t) gives the height in feet of a ball t seconds after it is thrown upward from the roof of a 64-foot tall building. How many seconds after the ball is thrown does it reach its maximum height? What is the ball’s maximum height?

a. b. c. d. ____

____

ID: A

The The The The

ball ball ball ball

reaches reaches reaches reaches

a a a a

maximum maximum maximum maximum

height height height height

of 64 feet 0 seconds after it is thrown. of 96 feet 1 second after it is thrown. of 100 feet 1.5 seconds after it is thrown. of 104 feet 1.5 seconds after it is thrown.

2

8. Write c ( x) = x + 6x − 11 in vertex form. 2

c.

c ( x) = ( x − 6 ) − 11

2

d.

c ( x) = ( x − 6 ) − 11

a.

c ( x) = ( x − 3 ) − 20

b.

c ( x) = ( x + 3 ) − 20

2 2

9. What does the imaginary number i represent? a. b.

−1 1

c. d.

−1 −

−1

____ 10. Which of the following is equivalent to a. b. c. d.

−121 ?

11 −11 11i 121i

____ 11. Simplify ( 9 − 2i) ( 3 + i) . a. 12 + 4i b. 25 + 3i

c. d.

4

27 + i 29 + 3i

Name: ________________________ ____ 12. Add. Write the result in the form a + bi. (7 – 9i) + (–6 + 5i) a. 12 – 15i b. 1 – 4i

ID: A

c. d.

–2 – i 13 – 14i

____ 13. Which of these expressions is equal to 4 − 7i? a. b. c. d.

( 6 − i) ( 6 + i) ( 6 + i) ( 6 − i)

− ( 2 − 8i) − ( 2 + 8i) + ( 2 + 8i) + ( 2 − 8i)

____ 14. Write the expression as a complex number in standard form. 8 + 7i 3 − 4i 4 53 52 53 a. + i c. + i 25 25 7 7 52 11 4 11 b. − i d. − i 7 7 25 25 1 6

____ 15. Write y as a radical expression. a.

y 6

b.

6

c.

y

d.

6

y 1

6

y

____ 16. Write the radical expression in rational exponent form. 3

k

7

7

a.

k

3 3

b. c. d.

7

k 4 k 10 k 2

____ 17. Consider h ( x) = 3x + 18x + 4. What is its vertex and y-intercept? a. vertex: (−2, −20), c. vertex: (−3, −23), y-intercept: 4 y-intercept: 4 b. vertex: (−2, 20), y-intercept: −4 d. vertex: (−3, −23), y-intercept: −4

5

Name: ________________________

ID: A

____ 18. The graph shows the height h ( t) of a model rocket t seconds after it is launched from the ground at 48 feet per second. Where is the height of the rocket increasing? Where is it decreasing?

a. b. c. d.

The The The The

height height height height

of of of of

the the the the

rocket is always increasing. rocket is always decreasing. rocket is increasing when 0 < t < 3 and decreasing when 3 < t < 6. rocket is increasing when 3 < t < 6 and decreasing when 0 < t < 3. 2

2

____ 19. What is the transformation of the graph of f ( x) = x that yields f ( x) = 2 ( x − 4 ) − 1? a. vertical stretch of 2, shift 3 units right and 1 unit up b. shift 8 units right and 1 unit down c. vertical stretch of 2, shift 4 units left and 1 unit up d. vertical stretch of 2, shift 4 units right and 1 unit down

6

ID: A

Analytic Geo B - Benchmark #1 Study Guide Answer Section MULTIPLE CHOICE 1. ANS: B PTS: 1 NAT: NT.CCSS.MTH.10.9-12.A.REI.4.b STA: MCC9-12.A.REI.4b DOK: DOK 1 2. ANS: A PTS: 1 REF: 12305066-4683-11df-9c7d-001185f0d2ea OBJ: Using Square Roots to Solve Quadratic Equations NAT: NT.CCSS.MTH.10.9-12.A.REI.4.b STA: MCC9-12.A.REI.4b LOC: MTH.C.10.06.04.01.002 TOP: Solving Quadratic Equations by Using Square Roots KEY: quadratic DOK: DOK 1 3. ANS: C PTS: 1 NAT: NT.CCSS.MTH.10.9-12.A.SSE.3.b STA: MCC9-12.A.SSE.3b DOK: DOK 1 4. ANS: A PTS: 1 REF: A1.10.EN.ST.02 NAT: NT.CCSS.MTH.10.9-12.A.SSE.3.b STA: MCC9-12.A.SSE.3b LOC: NCTM.PSSM.00.MTH.9-12.ALG.1.c KEY: quadratic equation DOK: DOK 1 5. ANS: D PTS: 1 REF: a03dcc5b-9631-11dd-8a40-001185f11039 OBJ: Graphing Quadratic Functions NAT: NT.CCSS.MTH.10.9-12.F.IF.7.a STA: MCC9-12.F.IF.7a LOC: MTH.P.06.03.012 | MTH.C.10.07.06.005 | MTH.C.10.07.06.01.001 TOP: Quadratic Functions KEY: quadratic function | graph DOK: DOK 1 6. ANS: C PTS: 1 REF: 121ab41a-4683-11df-9c7d-001185f0d2ea NAT: NT.CCSS.MTH.10.9-12.F.IF.7.a STA: MCC9-12.F.IF.7a LOC: MTH.C.10.07.06.01.001 | MTH.C.10.07.06.01.006 | MTH.C.10.07.06.01.007 TOP: Graphing Quadratic Functions KEY: quadratic DOK: DOK 2 7. ANS: C The ball achieves its maximum height at the vertex of the parabola that represents h ( t) . The vertex of the parabola is ÊÁË 1.5, 100 ˆ˜ . So, the ball reaches a maximum height of 100 feet 1.5 seconds after it is thrown. Feedback A B C D

You found the h ( t) -intercept. You need to find the vertex of the parabola. You need to find the vertex of the parabola. That’s correct! You need to find the vertex of the parabola.

PTS: STA: KEY: DOK: 8. ANS: STA:

1 NAT: NT.CCSS.MTH.10.9-12.F.IF.7.a* | NT.CCSS.MTH.10.K-12.MP.4 MCC9-12.F.IF.7a graph of a function | function | quadratic function | vertex | maximum | minimum DOK 1 B PTS: 1 NAT: NT.CCSS.MTH.10.9-12.F.IF.8.a MCC9-12.F.IF.8a DOK: DOK 1

1

ID: A 9. ANS: C The imaginary number i is defined to be

−1 .

Feedback 2

A

Recall that i = −1. When you square −1, do you get −1?

B C

Recall that i = −1. When you square That’s correct!

D

−

2

1 , do you get −1?

−1 represents −i, not i.

PTS: 1 NAT: NT.CCSS.MTH.10.9-12.N.CN.1 KEY: the imaginary number i DOK: DOK 1 10. ANS: C −121 = (121)(−1) = 121 • −1 = 11i

STA: MCC9-12.N.CN.1

Feedback

ab =

a •

b.

B

Factor −121 as 121(−1) and then apply the rule That’s correct!

C

Factor −121 as 121(−1) and then apply the rule

ab =

a •

b.

D

Factor −121 as 121(−1) and then apply the rule

ab =

a •

b.

A

PTS: KEY: 11. ANS: STA: 12. ANS: OBJ: STA: TOP:

1 NAT: NT.CCSS.MTH.10.9-12.N.CN.1 STA: MCC9-12.N.CN.1 imaginary numbers DOK: DOK 1 D PTS: 1 NAT: NT.CCSS.MTH.10.9-12.N.CN.2 MCC9-12.N.CN.2 DOK: DOK 1 B PTS: 1 REF: 159db4a6-4683-11df-9c7d-001185f0d2ea Adding and Subtracting Complex Numbers NAT: NT.CCSS.MTH.10.9-12.N.CN.2 MCC9-12.N.CN.2 LOC: MTH.C.10.03.01.002 Operations with Complex Numbers DOK: DOK 1

2

ID: A 13. ANS: B ( 6 + i) − ( 2 + 8i) = ( 6 − 2 ) + ( i − 8i) = 4 + ( −7i) = 4 − 7i The other expressions produce different results: (6 − i) − (2 + 8i) = 4 − 9i (6 + i) + (2 + 8i) = 8 + 9i (6 − i) + (2 − 8i) = 8 − 9i Feedback

To subtract two complex numbers, subtract the real parts and subtract the imaginary parts. That’s correct! To add two complex numbers, add the real parts and add the imaginary parts. To add two complex numbers, add the real parts and add the imaginary parts.

A B C D

PTS: KEY: DOK: 14. ANS: STA: TOP: KEY: 15. ANS: STA: DOK: 16. ANS:

1 NAT: NT.CCSS.MTH.10.9-12.N.CN.2 STA: MCC9-12.N.CN.2 adding complex numbers | subtracting complex numbers DOK 1 A PTS: 1 REF: MAL20640 NAT: NT.CCSS.MTH.10.9-12.N.CN.3 MCC9-12.N.CN.3 LOC: NCTM.PSSM.00.MTH.9-12.NOP.1.b Perform Operations with Complex Numbers complex | imaginary | divide | standard form DOK: DOK 2 B PTS: 1 NAT: NT.CCSS.MTH.10.9-12.N.RN.1 MCC9-12.N.RN.1 KEY: rational exponents DOK 1 A m

n

a

m

=a

n

7

, so

3

k

7

3

=k .

Feedback A

That’s correct! n

B C D

It seems you rewrote the expression It seems you rewrote the expression It seems you rewrote the expression

PTS: KEY: 17. ANS: NAT: STA:

n n n

a

m

as a

m

a

m

as a

m−n

.

a

m

as a

m+n

.

.

1 NAT: NT.CCSS.MTH.10.9-12.N.RN.1 STA: MCC9-12.N.RN.1 radical expressions | properties of rational exponents DOK: DOK 1 C PTS: 1 NT.CCSS.MTH.10.9-12.F.IF.8.a | NT.CCSS.MTH.10.9-12.F.IF.4 MCC9-12.F.IF.8a DOK: DOK 2

3

ID: A 18. ANS: C

The graph of h ( t) is increasing from t = 0 to t = 3. The rocket is traveling upward when 0 < t < 3 and its height is increasing. The graph of h ( t) is decreasing from t = 3 to t = 6. The rocket is traveling downward when 3 < t < 6 and its height is decreasing. Feedback A B C D

Notice that the graph of h ( t) decreases for some values of t. Notice that the graph of h ( t) increases for some values of t. That’s correct! Carefully examine the graph to determine where h ( t) increases and where it decreases.

PTS: STA: KEY: DOK: 19. ANS: STA:

1 NAT: NT.CCSS.MTH.10.9-12.F.IF.4* | NT.CCSS.MTH.10.K-12.MP.4 MCC9-12.F.IF.4 quadratic function | graph | increasing | decreasing | modeling DOK 1 D PTS: 1 NAT: NT.CCSS.MTH.10.9-12.F.BF.3 MCC9-12.F.BF.3 DOK: DOK 1

4