high school algebra ii semester - hcde.org

High School Algebra II Semester

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Applications of Quadratic Functions and EquationsNO HORSING AROUND

Embedded Assessment 1Use after Activity 9

1. Kun-cha has 150 feet of fencing to make a corral for her horses. Thebarn will be one side of the partitioned rectangular enclosure, as shownin the diagram above. The graph illustrates the function that representsthe area that could be enclosed. a. Write a function, A(x), that represents the area that can be enclosed

by the corral. b. What information does the graph provide about the function? c. Which ordered pair indicates the maximum area possible for the

corral? Explain what each coordinate tells about the problem. d. What values of x will give a total area of 1000 ft2? 2000 ft2?

2. Critique the reasoning of others. Tim is the punter for the BitterrootSprings Mustangs football team. He wrote a function h(t) = 16t2 + 8t + 1that he thinks will give the height of a football in terms of t, the numberof seconds after he kicks the ball. Use two different methods to determinethe values of t for which h(t) = 0. Show your work. Is Tim’s functioncorrect? Why or why not?

3. Tim has been studying complex numbers and quadratic equations. Histeacher, Mrs. Pinto, gave the class a quiz. Demonstrate yourunderstanding of the material by responding to each item below. a. Write a quadratic equation that has two solutions,

x = 2 + 5i and x = 2 − 5i. b. Solve 3x2 + 2x − 8 = 0, using an algebraic method.

c. Rewrite 43 2

+−

ii in the form a + bi, where a and b are rational

numbers.

200

400

600

800

1000

1200

Area

(ft2 )

1400

1600

1800

10 20 30 40 50

A(x)Horse Corral Enclosure

x

Width (ft)

Barn

Corral Diagram

x x x

Unit 2 • Quadratic Functions 151

Algebra 2 Semester 1

Quadratics 1. Which of the following are the x-intercepts of 𝑓(𝑥) =6𝑥2 − 4𝑥 − 10. Show your work.

A. -5 and 2 B. -10 and 0B. -1 and 5/3 D. 2 and 3

2. Which is the vertex of 𝑓(𝑥) = 2𝑥2 + 20𝑥 + 30? Show your work.

A. (20, 30) B. (-5,-20)C. (20,5) D. (5, -20)

3. Which function has a maximum to the left of the y –axis? Justify analytically.

A. 𝑓(𝑥) = 𝑥2 + 8𝑥 + 10B. 𝑓(𝑥) = 𝑥2 − 6𝑥𝑥 + 10C. 𝑓(𝑥) = 𝑥2 − 2𝑥 + 3D. 𝑓(𝑥) = −𝑥2 + 4𝑥 + 8

4.𝑓(𝑥) = 𝑥2 is translated 1 unit down, shrunk by afactor of ½ and reflected over the x-axis. Without theuse of technology, which is the equation of thetransformation? Explain your reasoning.

A. 𝑔(𝑥) = (1

2𝑥 − 1)

2

B. (𝑥) = − (1

2𝑥 − 1)

2

C. 𝑔(𝑥) = − 1

2𝑥2 − 1

D. 𝑔(𝑥) =1

2𝑥2 − 1

5. Solve the quadratic equations below Show your work. Use different methods to solve the equations.A. A. 6𝑥2 + 5 = 11𝑥 B. 4(𝑥 + 1)2 − 9 = 27

C. −8𝑥 = 𝑥2 − 1 D. 10𝑥2 + 1 = 5𝑥


Writing a Polynomial Task

Consider the graph of the polynomial function below.

1. Lisa claims that, since the point (0, 6) is on the graph, (x – 6) is a factor of thispolynomial. Explain why you agree or disagree with Lisa’s claim. Identify all the zeroesof the function and use that information in your explanation.

2. Write a function in factored form to represent this graph, using 𝑎 =1

12. Explain how

you used key characteristics of the graph to form your equation.


Polynomials 1. The volume of a package V is a function of w, the width of the square ends of the package such that 𝑉(𝑤) =(180 − 4𝑤)𝑤2. Which of the following is the domain of the function? Justify your reasoning.

A. 0 ≤ 𝑤 ≤ 45 B. 0 < 𝑤 < 45

𝐶. 0 ≤ 𝑤 ≤ √453

D. 0 < 𝑤 < √453

2. Determine the expression(s) which is equivalent to (𝑥 − 2)3? Show your work.

A. 3𝑥2 − 6B. 𝑥6 − 8C. 3𝑥6 − 6𝑥2 − 8D. (𝑥2 + 4𝑥 + 4)(𝑥 − 2)E. 3𝑥3 − 6𝑥2 + 12𝑥 − 8F. 3𝑥6 − 6𝑥2 − 8G. (𝑥2 − 4𝑥 + 4)(𝑥 − 2)

3. Make use of structure.The product of two polynomials is𝑥3 + 11𝑥2 + 13𝑥-10. One factor is 𝑥 + 2. Determinethe other factor. Show you work.

4. Factor & show your work.

A. 25𝑥4 − 81

B. 2𝑥3 − 6𝑥2 + 5𝑥 − 15

C. 3𝑥4 + 2𝑥2 − 5

D. 𝑥4 + 6𝑥2 − 9

5. Sketch the graph of a polynomial function that increases as 𝑥 → ∞, decreases as 𝑥 → −∞, and has zeros at x = -5, 0, and 2.

6. The graph of 𝑚(𝑥) has an x-intercept at (-3,0). Whichof the following is NOT true? Justify.

I. 𝑚(−3) = 0II. 𝑥 + 3 is a factor of 𝑚(𝑥)III.𝑚(𝑥) also has an x-intercept at (3,0).

A. I only B. II onlyC. III only D. II and III only


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Polynomial OperationsTHIS TEST IS SQUARE

Embedded Assessment 1Use after Activity 16

Congruent squares of length x are cut from the corners of a 10-inch-by-15-inch piece of cardboard to create a box without a lid.

1. Write an expression in terms of x for each. a. the height of the box b. the length of the box c. the width of the box

2. Write a function V(x) for the volume of the boxin terms of x. Leave your answer in factored form.

3. Express the domain of V(x) as an inequality, ininterval notation, and in set notation.

4. Sketch a graph of V(x) over the domain that youfound in Item 3. Include the scale on each axis.

5. Use a graphing calculator to find the coordinates of the maximum pointof V(x) over the domain for which you graphed it. Then interpret themeaning of the maximum point.

6. Use polynomial multiplication to rewrite V(x), the volume functionfrom Item 2, as a polynomial in standard form.

7. Consider the graph of V(x) over the set of real numbers. Describe theend behavior of the function using arrow notation.

8. Use long division or synthetic division to find the quotientx x x x

x5 3 23 4 2 6

1+ − + +

− .

xx

Unit 3 • Polynomials 265


Honeybee Task Honeybees are extremely important because they pollinate the crops that we eat. Unfortunately,

the honeybee population has been declining in recent years, and scientists are not exactly sure

why. One particular honeybee colony originally had approximately 2250 honeybees in it. The

population decreased by 23% every year.

1. Determine the number of honeybees that were in the colony after 3 years. Show yourwork.

2. Write a recursive rule that can be used to determine the number of honeybees forany given year. Explain the meaning of each constant and variable from your rule inthe context of the problem.

3. Write an explicit formula that can be used to determine the number of honeybees forany given year. Explain the meaning of each constant and variable from your formulain the context of the problem.

4. Estimate the number of years from the original population of 2250 honeybees that itwill take for the number of honeybees in the colony to reach 500. Explain yourreasoning.


Radicals and Rationals 1.

If 𝑓(𝑥) = √𝑥, which function describes 𝑔(𝑥), a function with a graph that is a translation of 𝑓(𝑥) two units to the right and three units down?

A. √𝑥 + 2 − 3 B. √𝑥 − 2 + 3

C. 2√𝑥 − 3 D. √𝑥 − 2 − 3

2. Which function has a domain of [3, ∞)?

A. 𝑓(𝑥) = √𝑥 − 3 B. 𝑓(𝑥) = √𝑥 + 3 − 3

C. 𝑓(𝑥) = √𝑥 − 3 + 8 D. 𝑓(𝑥) = √𝑥 + 3

3. Model with mathematics.The velocity of a cruise ship is equal to the square rootof the rate of fuel consumption minus 3 units.

a. Write the velocity function of a cruise ship as afunction of the fuel consumption and graph thefunction.

b. How many units of fuel are needed to get theship moving? How does the graph show this?

4. Which of the following transformations of the parent

function 𝑓(𝑥) = 1

𝑥 has a graph that is translated 6 units

left and 5 units up when compared to the graph of the parent function?

A. 𝑓(𝑥) =1

𝑥+6− 5

B. 𝑓(𝑥) =1

𝑥+6

C. 𝑓(𝑥) =1

𝑥−6+ 5

D. 𝑓(𝑥) =1

𝑥+6+ 5

5. Critique the reasoning of others.

Claire simplified 𝑓(𝑥) =3𝑥2+7𝑥−20

2𝑥2+11𝑥+12. She determined

that 𝑓(𝑥) = (3𝑥−5)(𝑥+4)

2𝑥+3 and that the domain

restriction was 𝑥 ≠ − 3

2. Is Claire correct? If not, what

did she do incorrectly?

6. What is the inverse function of 𝑓(𝑥) = (𝑥 + 2)2?

A. 𝑓−1(𝑥) = √𝑥 − 2 B. 𝑓−1(𝑥) = √𝑥 + 2

𝑓−1(𝑥) = 1

2𝑥D. 𝑓−1(𝑥) = √𝑥 − 2


Radicals and Rationals 7. Reason quantitatively.

Consider 2𝑥−5

−(𝑥+2)+

2𝑥+5

(𝑥+2).

A. Can this expression be simplified to 10

𝑥+2? If so,

how? If not, why not?

B. What restrictions exist on x, if any?

8.

Simplify (−𝑥+1

𝑥2+3𝑥−4) (

8𝑥2−4𝑥

(𝑥−3)(𝑥+7)) and identify any

restrictions on x.

A. 3𝑥2+16𝑥+16

𝑥2+3𝑥−4, 𝑥 ≠ −

2

3

B. 3𝑥2−3𝑥+4

−3𝑥−2, 𝑥 ≠ − 3

C. 3𝑥+4

3𝑥+2, 𝑥 ≠ −

2

3

D. 3𝑥+4

3𝑥+2, 𝑥 ≠ −

2

3, −4, 1

Algebra 2 Semester

(continued)

8

high school algebra ii semester - hcde.org

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