Algebra 2 Chapter 3 Test Answer Key

Characteristics Of Quadratic Functions 22 Exercises

Vocabulary and Core Concept and Check

Question 1.WRITING Explain how to determine whether a quadratic function will have a minimum value or a maximum value.Answer:

Question 2.WHICH ONE DOESNT BELONG? The graph of which function does not belong with the other three? Explain.Answer:

y = \x2 3x + 6Answer:

y = \x2 4x 1Answer:

Question 31.WRITING Two quadratic functions have graphs with vertices and . Explain why you can not use the axes of symmetry to distinguish between the two functions.Answer:

Question 32.WRITING A quadratic function is increasing to the left of x = 2 and decreasing to the right of x = 2. Will the vertex be the highest or lowest point on the graph of the parabola? Explain.Answer:

ERROR ANALYSIS In Exercises 33 and 34, describe and correct the error in analyzing the graph of y = 4×2 + 24x 7.

Question 33.

Question 34.Answer:

MODELING WITH MATHEMATICS In Exercises 35 and 36, x is the horizontal distance and y is the vertical distance . Find and interpret the coordinates of the vertex.

Question 35.The path of a basketball thrown at an angle of 45° can be modeled by y = -0.02×2 + x + 6.Answer:

Question 36.The path of a shot put released at an angle of 35° can be modeled by y = -0.01×2 + 0.7x + 6.Answer:

Question 37.ANALYZING EQUATIONS The graph of which function has the same axis of symmetry as the graph of y = x2 + 2x + 2?A. y = 2×2 + 2x + 2B. y = -3×2 6x + 2C. y = x2 2x + 2D. y = -5×2 + 10x + 23Answer:

Question 39.

y = \x2 3x + 2Answer:

Question 51.

Lesson 31 Solving Quadratic Equations

Essential Question How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

EXPLORATION 1

Matching a Quadratic Function with Its GraphWork with a partner. Match each quadratic function with its graph. Explain your reasoning. Determine the number of x-intercepts of the graph.a. f = x2 2xb. f = x2 2x + 1c. f = x2 2x + 2d. f = x2 + 2xe. f = x2 + 2x 1f. f = x2 + 2x 2

EXPLORATION 2

Solving Quadratic EquationsWork with a partner. Use the results of Exploration 1 to find the real solutions of each quadratic equation.a. x2 2x = 0b. x2 2x + 1 = 0c. x2 2x + 2 = 0d. x2 + 2x = 0e. x2 + 2x 1 = 0f. x2 + 2x 2 = 0

Communicate Your Answer

Question 3.How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?Answer:

Question 4.How many real solutions does the quadratic equation x2 + 3x + 2 = 0 have? How do you know? What are the solutions?Answer:Parabolas have a highest or a lowest point called the Vertex

Monitoring Progress

Solve the equation by graphing.Question 1.

Question 55.REASONINGWrite a quadratic function in the form f = x2 + bx + c that has zeros 8 and 11.Answer:

Question 56.NUMBER SENSEWrite a quadratic equation in standard form that has roots equidistant from 10 on the number line.Answer:

Write and solve an equation to find two consecutive odd integers whose product is 143.Answer:

Find the sum or difference.Question 76.

3i + Answer:

+ Answer:

Big Ideas Math Algebra 2 Answers Chapter 3 Quadratic Equations And Complex Numbers

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Y+x Objective Function: C = 7x 3y

What is the solution of the system? Solve by hand using a matrix. You may use your calculator to double check your answer. You must show all of your work to receive full credit. 9. 2x + 6y = 38 5x y = 15

Name: ______________________

ID: A

Solve the system by elimination. You must show all your work to get full credit. 10. x + 3y + z = 6 2x + y + 3z = 4 3x 3y 3z = 6

Solve the system by substitution. You must show all your work to get full credit. 11. 2x y + z = 4 z = 5 2x + 3y z = 10

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Lesson 21 Transformations Of Quadratic Functions

Essential Question

How do the constants a, h, and k affect the graph of the quadratic function g = a2 + k?The parent function of the quadratic family is f = x2. A transformation of the graph of the parent function is represented by the function g = a2 + k, where a 0.

EXPLORATION 1Identifying Graphs of Quadratic FunctionsWork with a partner. Match each quadratic function with its graph. Explain your reasoning. Then use a graphing calculator to verify that your answer is correct.a. g = -2b. g = 2 + 2c. g = -2 2d. g = 0.52 + 2e. g = 22f. g = -2 + 2

Communicate Your Answer

Question 2.How do the constants a, h, and k affect the graph of the quadratic function g =a2 + k?

Question 3.Write the equation of the quadratic function whose graph is shown at the right. Explain your reasoning. Then use a graphing calculator to verify that your equation is correct.

2.1 Lesson

Describe the transformation of f = x2 represented by g. Then graph each function.

Question 1.

g = -2 + 2

Question 7.Let the graph of g be a vertical shrink by a factor of \ followed by a translation 2 units up of the graph of f = x2. Write a rule for g and identify the vertex.

Question 8.Let the graph of g be a translation 4 units left followed by a horizontal shrink by a factor of \ of the graph of f = x2 + x. Write a rule for g.

Question 9.WHAT IF? In Example 5, the water hits the ground 10 feet closer to the fire truck after lowering the ladder. Write a function that models the new path of the water.

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Lesson 23 Focus Of A Parabola

Essential QuestionWhat is the focus of a parabola?EXPLORATION 1Analyzing Satellite DishesWork with a partner. Vertical rays enter a satellite dish whose cross section is a parabola. When the rays hit the parabola, they reflect at the same angle at which they entered. a. Draw the reflected rays so that they intersect the y-axis.b. What do the reflected rays have in common?c. The optimal location for the receiver of the satellite dish is at a point called the focus of the parabola. Determine the location of the focus. Explain why this makes sense in this situation.

EXPLORATION 2Analyzing SpotlightsWork with a partner. Beams of light are coming from the bulb in a spotlight, located at the focus of the parabola. When the beams hit the parabola, they reflect at the same angle at which they hit. Draw the reflected beams. What do they have in common? Would you consider this to be the optimal result? Explain.

Communicate Your Answer

What is the focus of a parabola?

Question 4.Describe some of the properties of the focus of a parabola.

2.3 Lesson

Monitoring Progress

Question 1.Use the Distance Formula to write an equation of the parabola with focus F and directrix y = 3.

Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation.

Question 2.

focus: )

Monitoring Progress

Write an equation of a parabola with vertex and focus .

Big Ideas Math Book Algebra 2 Answer Key Chapter 2 Quadratic Functions

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If I am sick, then I will miss school.I did not miss school.Therefore, I am not sick.

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This Video Shows You How To Factor A Third Degree Polynomial

X3 – 2×2 – 4x 8 You can factor this expression by grouping the first and third terms x3 – 4x – 2×2 8 This is equivalent to. That means that any rational solution will be a factor of 8 divided by a factor of 1. Step 1 Equation at the end of step 1. X-2 4 Group 2. We want to factor out the given expression. Tap for more steps.

Solving Equations With Brackets Algebra Activity Algebra Worksheet Solving Equations Algebra Activities Algebra Worksheets

X3 4×2 2x 8 x 3 4 x 2 – 2 x – 8. Factor out of. Factor out the greatest common factor from each group.

Big Ideas Math Book Algebra 2 Answer Key Chapter 3 Quadratic Equations And Complex Numbers

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Question 15.ABSTRACT REASONINGDetermine the possible integer values of a and c for which the trinomial ax2+ 8x+c is factorable using the Perfect Square Trinomial Pattern. Explain your reasoning.Answer:The term a is referred to as the leading coefficient, while c is the absolute term of f . Every quadratic equation has two values of the unknown variable,

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Quadratic Functions Cumulative Assessment

Question 1.You and your friend are throwing a football. The parabola shows the path of your friends throw, where x is the horizontal distance and y is the corresponding height . The path of your throw can be modeled by h = 16×2 + 65x + 5. Choose the correct inequality symbol to indicate whose throw travels higher. Explain your reasoning.

Question 2.The function g = \x 4 + 4 is a combination of transformations of f = | x|. Which combinations describe the transformation from the graph of f to the graph of g?A. translation 4 units right and vertical shrink by a factor of \, followed by a translation 4 units upB. translation 4 units right and 4 units up, followed by a vertical shrink by a factor of \C. vertical shrink by a factor of \ , followed by a translation 4 units up and 4 units rightD. translation 4 units right and 8 units up, followed by a vertical shrink by a factor of \

Question 3.Your school decides to sell tickets to a dance in the school cafeteria to raise money. There is no fee to use the cafeteria, but the DJ charges a fee of $750. The table shows the profits when x students attend the dance.a. What is the cost of a ticket?b. Your school expects 400 students to attend and finds another DJ who only charges $650. How much should your school charge per ticket to still make the same profit?c. Your school decides to charge the amount in part and use the less expensive DJ. How much more money will the school raise?

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Lesson 35 Solving Nonlinear Systems

Essential Question How can you solve a nonlinear system of equations?

EXPLORATION 1

Solving Nonlinear Systems of EquationsWork with a partner. Match each system with its graph. Explain your reasoning. Then solve each system using the graph.

EXPLORATION 2

Solving Nonlinear Systems of EquationsWork with a partner. Look back at the nonlinear system in Exploration 1. Suppose you want a more accurate way to solve the system than using a graphical approach.a. Show how you could use a numerical approach by creating a table. For instance, you might use a spreadsheet to solve the system.b. Show how you could use an analytical approach. For instance, you might try solving the system by substitution or elimination.

Communicate Your Answer

How can you solve a nonlinear system of equations?Answer:

Question 4.Would you prefer to use a graphical, numerical, or analytical approach to solve the given nonlinear system of equations? Explain your reasoning.Answer:

Solve the system using any method. Explain your choice of method.Question 1.

USING TOOLS In Exercises 4348, solve the equation by graphing.Question 43.x2 + 2x = \x2 + 2xAnswer:

2×2 12x 16 = 6×2 + 60x 144Answer:

= x2 + 6x 7Answer:

2×2 16x 25 = 6×2 + 48x + 95Answer:

2 3 = 38Answer:

42 = 1Answer:

Question 49.REASONINGA nonlinear system contains the equations of a constant function and a quadratic function. The system has one solution. Describe the relationship between the graphs.Answer:

EXPLORATION 1

Lesson 24 Modeling With Quadratic Functions

Essential QuestionHow can you use a quadratic function to model a real-life situation?

EXPLORATION 1Modeling with a Quadratic Function

Work with a partner. The graph shows a quadratic function of the formP = at2 + bt + cwhich approximates the yearly profits for a company, where P is the profit in year t.a. Is the value of a positive, negative, or zero? Explain.b. Write an expression in terms of a and b that represents the year t when the company made the least profit.c. The company made the same yearly profits in 2004 and 2012. Estimate the year in which the company made the least profit.d. Assume that the model is still valid today. Are the yearly profits currently increasing, decreasing, or constant? Explain.

EXPLORATION 2Modeling with a Graphing CalculatorWork with a partner. The table shows the heights h of a wrench t seconds after it has been dropped from a building under construction.a. Use a graphing calculator to create a scatter plot of the data, as shown at the right. Explain why the data appear to fit a quadratic model.b. Use the quadratic regression feature to find a quadratic model for the data.c. Graph the quadratic function on the same screen as the scatter plot to verify that it fits the data.d. When does the wrench hit the ground? Explain.

Communicate Your Answer

How can you use a quadratic function to model a real-life situation?

2.4 Lesson

Monitoring Progress

Question 1.WHAT IF? The vertex of the parabola is . What is the height of the net?

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Transformations Of Quadratic Functions 21 Exercises

Vocabulary and Core Concept Check

Question 1.COMPLETE THE SENTENCE The graph of a quadratic function is called a ________.Answer:

VOCABULARY Identify the vertex of the parabola given by f = 2 4.Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 312, describe the transformation of f = x2 represented by g. Then graph each function.

Question 3.

f = \2Answer:

In Exercises 3134, write a rule for g described by the transformations of the graph of f. Then identify the vertex.

Question 31.f = x2 vertical stretch by a factor of 4 and a reflection in the x-axis, followed by a translation 2 units upAnswer:

Question 32.f = x2 vertical shrink by a factor of \ and a reflection in the y-axis, followed by a translation 3 units rightAnswer:

Question 33.f = 8×2 6 horizontal stretch by a factor of 2 and a translation 2 units up, followed by a reflection in the y-axisAnswer:

Question 34.f = 2 + 3 horizontal shrink by a factor of \ and a translation 1 unit down, followed by a reflection in the x-axisAnswer:

USING TOOLS In Exercises 3540, match the function with its graph. Explain your reasoning.

Question 35.

g = 22 + 2Answer:

JUSTIFYING STEPS In Exercises 41 and 42, justify eachstep in writing a function g based on the transformationsof f = 2×2 + 6x.

Question 41.translation 6 units down followed by a reflection in the x-axisAnswer:

reflection in the y-axis followed by a translation 4 units rightAnswer:

Question 50.