6.3 Interpreting Vertex Form and Standard Form

Size: px
Start display at page:

Download "6.3 Interpreting Vertex Form and Standard Form"

Transcription

1 Name Class Date 6.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Eplore Identifing Quadratic Functions from Their Graphs Determine whether a function is a quadratic function b looking at its graph. If the graph of a function is a parabola, then the function is a quadratic function. If the graph of a function is not a parabola, then the function is not a quadratic function. Use a graphing calculator to graph each of the functions. Set the viewing window to show -10 to 10 on both aes. Determine whether each function is a quadratic function. A B Use a graphing calculator to graph ƒ () = + 1. Determine whether the function ƒ () = + 1 is a quadratic function. The function ƒ () = + 1 a quadratic function. C Use a graphing calculator to graph ƒ () = Houghton Mifflin Harcourt Publishing Compan D Determine whether the function ƒ () = is a quadratic function. The function ƒ () = E Use a graphing calculator to graph ƒ () = 2. a quadratic function. F Determine whether the function ƒ () = 2 is a quadratic function. The function ƒ () = 2 a quadratic function. Module Lesson 3

2 Use a graphing calculator to graph ƒ () = Determine whether the function ƒ () = is a quadratic function. The function ƒ () = a quadratic function. Use a graphing calculator to graph ƒ () = - ( - 3) Determine whether the function ƒ () = - ( - 3) is a quadratic function. The function ƒ () = - ( - 3) a quadratic function. Use a graphing calculator to graph ƒ () =. Determine whether the function ƒ () = is a quadratic function. The function ƒ () = a quadratic function. Reflect 1. How can ou determine whether a function is quadratic or not b looking at its graph? 2. Discussion How can ou tell if a function is a quadratic function b looking at the equation? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

3 Eplain 1 Identifing Quadratic Functions in Standard Form If a function is quadratic, it can be represented b an equation of the form = a 2 + b + c, where a, b, and c are real numbers and a 0. This is called the standard form of a quadratic equation. The ais of smmetr for a quadratic equation in standard form is given b the equation = -_ b. The verte of a 2a quadratic equation in standard form is given b the coordinates ( - _ b 2a, ƒ ( - _ 2a) b ). Eample 1 A = Determine whether the function represented b each equation is quadratic. If so, give the ais of smmetr and the coordinates of the verte. = Compare to = a 2 + b + c. This is not a quadratic function because a = 0. B = -4 Rewrite the function in the form = a 2 + b + c. = Compare to = a 2 + b + c. This a quadratic function. If = -4 is a quadratic function, give the ais of smmetr. If = -4 is a quadratic function, give the coordinates of the verte. Reflect 3. Eplain wh the function represented b the equation = a 2 + b + c is quadratic onl when a 0. Houghton Mifflin Harcourt Publishing Compan 4. Wh might it be easier to determine whether a function is quadratic when it is epressed in function notation? 5. How is the ais of smmetr related to standard form? Your Turn Determine whether the function represented b each equation is quadratic = = Module Lesson 3

4

5 Eplain 2 Changing from Verte Form to Standard Form It is possible to write quadratic equations in various forms. Eample 2 Rewrite a quadratic function from verte form, = a ( - h) 2 + k, to standard form, = a 2 + b + c. A = 4 ( - 6) = 4 ( ) + 3 Epand ( - 6) 2. = = Multipl. Simplif. The standard form of = 4 ( - 6) is = B = -3 ( + 2) 2-1 = -3 ( ) - 1 Epand ( + 2) 2. = - 1 Multipl. = Simplif. The standard form of = -3 ( + 2) 2-1 is =. Reflect 8. If in = a ( - h) 2 + k, a = 1, what is the simplified form of the standard form, = a 2 + b + c? Your Turn Rewrite a quadratic function from verte form, = a ( - h) 2 + k, to standard form, = a 2 + b + c. 9. = 2 ( + 5) = -3 ( - 7) Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

6 Eplain 3 Writing a Quadratic Function Given a Table of Values You can write a quadratic function from a table of values. Eample 3 Use each table to write a quadratic function in verte form, = a ( - h ) 2 + k. Then rewrite the function in standard form, = a 2 + b + c. A The minimum value of the function occurs at = -3. The verte of the parabola is (-3, 0). Substitute the values for h and k into = a ( - h) 2 + k. = a ( - (-3)) 2 + 0, or = a ( + 3) 2 Use an point from the table to find a. = a ( + 3) 2 1 = a (-2 + 3) 2 = a The verte form of the function is = 1 ( - (-3)) or = ( + 3) 2. Rewrite the function = ( + 3) 2 in standard form, = a 2 + b + c. = ( + 3) 2 = The standard form of the function is = B The minimum value of the function occurs at = -2. The verte of the parabola is is (-2, -3). Substitute the values for h and k into = a ( -h) 2 + k. = Use an point from the table to find a. a = Houghton Mifflin Harcourt Publishing Compan The verte form of the function is =. Rewrite the resulting function in standard form, = a 2 + b + c. = Reflect 11. How man points are needed to find an equation of a quadratic function? Module Lesson 3

7 Your Turn Use each table to write a quadratic function in verte form, = a ( - h) 2 + k. Then rewrite the function in standard form, = a 2 + b + c. 12. The verte of the function is (2, 5) The verte of the function is (-2, -7) P Eplain 4 Writing a Quadratic Function Given a Graph The graph of a parabola can be used to determine the corresponding function. Eample 4 Use each graph to find an equation for ƒ (t). A A house painter standing on a ladder drops a paintbrush, which falls to the ground. The paintbrush s height above the ground (in feet) is given b a function of the form ƒ (t) = a (t - h) 2 where t is the time (in seconds) after the paintbrush is dropped. The verte of the parabola is (h, k) = (0, 25). ƒ (t) = a ( - h) 2 + k ƒ (t) = a (t - 0) ƒ (t) = at Use the point (1, 9) to find a. Height (feet) (0, 25) (1, 9) Time (seconds) Houghton Mifflin Harcourt Publishing Compan ƒ (t) = at = a (1) = a The equation for the function is ƒ (t) = -16 t Module Lesson 3

8 B A rock is knocked off a cliff into the water far below. The falling rock s height above the water (in feet) is given b a function of the form ƒ (t) = a (t - h) 2 + k where t is the time (in seconds) after the rock begins to fall. Height (feet) (0, 40) (1, 24) Time (seconds) The verte of the parabola is (h, k) =. ƒ (t) = a (t - h) 2 + k 2 ƒ (t) = a ( ) t - +. ƒ (t) = Use the point to find a. Images/Corbis ƒ (t) = a t 2 + = a a = 2 + The equation for the function is ƒ (t) =. Reflect 14. Identif the domain and eplain wh it makes sense for this problem. 15. Identif the range and eplain wh it makes sense for this problem. Module Lesson 3

9 Your Turn 16. The graph of a function in the form ƒ () = a ( - h) 2 + k, is shown. Use the graph to find an equation for ƒ () (1, 1) (3, -3) A roofer accidentall drops a nail, which falls to the ground. The nail s height above the ground (in feet) is given b a function of the form ƒ (t) = a (t - h) 2 + k, where t is the time (in seconds) after the nail drops. Use the graph to find an equation for ƒ (t). Height (feet) (0, 45) (1, 29) Time (seconds) t Module Lesson 3

10 Elaborate 18. Describe the graph of a quadratic function. 19. What is the standard form of the quadratic function? 20. Can an quadratic function in verte form be written in standard form? 21. How man points are needed to write a quadratic function in verte form, given the table of values? 22. If a graph of the quadratic function is given, how do ou find the verte? 23. Essential Question Check-In What can ou do to change the verte form of a quadratic function to standard form? Evaluate: Homework and Practice Determine whether each function is a quadratic function b graphing. 1. ƒ () = ƒ () = 1 _ ƒ () = ƒ () = 2-3 Houghton Mifflin Harcourt Publishing Compan Determine whether the function represented b each equation is quadratic. 5. = = = = Which of the following functions is a quadratic function? Select all that appl. a. 2 = + 3 d = 0 b = 3-1 e. - = 4 c. 5 = For ƒ () = , give the ais of smmetr and the coordinates of the verte. = -b = -(8) = -8 = -4 2a 2(1) 2 = (-4) + 8(-4) - 14 = -30 Verte (-4,-30) Ais of Sm: = -4 Module Lesson 3

11 11. Describe the ais of smmetr of the graph of the quadratic function represented b the equation = a 2 + b + c. when b = 0. Rewrite each quadratic function from verte form, = a ( - h) 2 + k, to standard form, = a 2 + b + c. 12. = 5 ( - 2) = -2 ( + 4) = 3 ( + 1) = -4 ( - 3) Eplain the Error Tim wrote = -6 ( + 2) 2-10 in standard form as = Find his error. 17. How do ou change from verte form, ƒ () = a ( - h) 2 + k, to standard form, = a 2 + b + c? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

12 Use each table to write a quadratic function in verte form, = a ( - h) 2 + k. Then rewrite the function in standard form, = a 2 + b + c. 18. The verte of the function is (6, -8) The verte of the function is (4, 7) The verte of the function is (-2, -12) The verte of the function is (-3, 10) Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

13 H.O.T. Focus on Higher Order Thinking 22. Make a Prediction A ball was thrown off a bridge. The table relates the height of the ball above the ground in feet to the time in seconds after it was thrown. Use the data to write a quadratic model in verte form and convert it to standard form. Use the model to find the height of the ball at 1.5 seconds. Time (seconds) Height (feet) Multiple Representations A performer slips and falls into a safet net below. The function ƒ (t) = a (t - h) 2 + k, where t represents time (in seconds), gives the performer s height above the ground (in feet) as he falls. Use the graph to find an equation for ƒ (t). Height (feet) (0, 20) (1, 4) Time (seconds) Module Lesson 3

14 24. Represent Real-World Problems After a heav snowfall, Ken and Karin made an igloo. The dome of the igloo is in the shape of a parabola, and the height of the igloo in inches is given b the function ƒ () = a ( - h) 2 + k. Use the graph to find an equation for ƒ (). 48 (40, 48) Height (feet) (60, 36) Width (feet) Houghton Mifflin Harcourt Publishing Compan 25. Check for Reasonableness Tim hits a softball. The function ƒ (t) = a (t - h) 2 + k describes the height (in feet) of the softball, and t is the time (in seconds). Use the graph to find an equation for ƒ (t). Estimate how much time elapses before the ball hits the ground. Use the equation for the function and our estimate to eplain whether the equation is reasonable. Height (feet) (1.5, 40) (2, 36) Time (seconds) Module Lesson 3

15 Lesson Performance Task The table gives the height of a tennis ball t seconds after it has been hit, where the maimum height is 4 feet. Time (s) Height (ft) a. Use the data in the table to write the quadratic function ƒ (t) in verte form, where t is the time in seconds and ƒ (t) is the height of the tennis ball in feet. b. Rewrite the function found in part a in standard form. c. At what height was the ball originall hit? Eplain. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

7.2 Connecting Intercepts and Linear Factors

7.2 Connecting Intercepts and Linear Factors Name Class Date 7.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and

More information

Name Class Date. Understanding How to Graph g(x) = a(x - h ) 2 + k

Name Class Date. Understanding How to Graph g(x) = a(x - h ) 2 + k Name Class Date - Transforming Quadratic Functions Going Deeper Essential question: How can ou obtain the graph of g() = a( h ) + k from the graph of f () =? 1 F-BF..3 ENGAGE Understanding How to Graph

More information

20.2 Connecting Intercepts and Linear Factors

20.2 Connecting Intercepts and Linear Factors Name Class Date 20.2 Connecting Intercepts and Linear Factors Essential Question: How are -intercepts of a quadratic function and its linear factors related? Resource Locker Eplore Connecting Factors and

More information

Name Class Date. Deriving the Standard-Form Equation of a Parabola

Name Class Date. Deriving the Standard-Form Equation of a Parabola Name Class Date 1. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the Standard-Form Equation of

More information

10.2 Graphing Exponential Functions

10.2 Graphing Exponential Functions Name Class Date 10. Graphing Eponential Functions Essential Question: How do ou graph an eponential function of the form f () = ab? Resource Locker Eplore Eploring Graphs of Eponential Functions Eponential

More information

Quadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry

Quadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function = - -1 0 1 1 Eample 1: Make a table,

More information

9.5 Solving Nonlinear Systems

9.5 Solving Nonlinear Systems Name Class Date 9.5 Solving Nonlinear Sstems Essential Question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? Eplore Determining the Possible Number of

More information

10.1 Inverses of Simple Quadratic and Cubic Functions

10.1 Inverses of Simple Quadratic and Cubic Functions Name Class Date 10.1 Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of quadratic functions and cubic functions, and how can ou find them? Resource

More information

7.1 Connecting Intercepts and Zeros

7.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Common Core Math Standards The student is epected to: F-IF.7a Graph linear and quadratic functions and show intercepts, maima, and minima. Also A-REI.,

More information

4.2 Parabolas. Explore Deriving the Standard-Form Equation. Houghton Mifflin Harcourt Publishing Company. (x - p) 2 + y 2 = (x + p) 2

4.2 Parabolas. Explore Deriving the Standard-Form Equation. Houghton Mifflin Harcourt Publishing Company. (x - p) 2 + y 2 = (x + p) 2 COMMON CORE. d Locker d LESSON Parabolas Common Core Math Standards The student is epected to: COMMON CORE A-CED.A. Create equations in two or more variables to represent relationships between quantities;

More information

10.2 Graphing Square Root Functions

10.2 Graphing Square Root Functions Name Class Date. Graphing Square Root Functions Essential Question: How can ou use transformations of a parent square root function to graph functions of the form g () = a (-h) + k or g () = b (-h) + k?

More information

10.1 Inverses of Simple Quadratic and Cubic Functions

10.1 Inverses of Simple Quadratic and Cubic Functions COMMON CORE Locker LESSON 0. Inverses of Simple Quadratic and Cubic Functions Name Class Date 0. Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of

More information

15.2 Graphing Logarithmic

15.2 Graphing Logarithmic Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b 1 related to the graph of f () = log b? Resource Locker Eplore 1 Graphing

More information

Name Class Date. Inverse of Function. Understanding Inverses of Functions

Name Class Date. Inverse of Function. Understanding Inverses of Functions Name Class Date. Inverses of Functions Essential Question: What is an inverse function, and how do ou know it s an inverse function? A..B Graph and write the inverse of a function using notation such as

More information

13.1 Exponential Growth Functions

13.1 Exponential Growth Functions Name Class Date 1.1 Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > 1 related to the graph of f () = b? Resource Locker Eplore 1 Graphing and Analzing f

More information

15.4 Equation of a Circle

15.4 Equation of a Circle Name Class Date 1.4 Equation of a Circle Essential Question: How can ou write the equation of a circle if ou know its radius and the coordinates of its center? Eplore G.1.E Show the equation of a circle

More information

13.2 Exponential Growth Functions

13.2 Exponential Growth Functions Name Class Date. Eponential Growth Functions Essential Question: How is the graph of g () = a b - h + k where b > related to the graph of f () = b? A.5.A Determine the effects on the ke attributes on the

More information

9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson

9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points.

More information

11.1 Solving Linear Systems by Graphing

11.1 Solving Linear Systems by Graphing Name Class Date 11.1 Solving Linear Sstems b Graphing Essential Question: How can ou find the solution of a sstem of linear equations b graphing? Resource Locker Eplore Tpes of Sstems of Linear Equations

More information

13.2 Exponential Decay Functions

13.2 Exponential Decay Functions Name Class Date 13. Eponential Deca Functions Essential Question: How is the graph of g () = a b h + k where < b < 1 related to the graph of f () = b? Eplore 1 Graphing and Analzing f () = ( 1 and f ()

More information

Solve Quadratic Equations by Graphing

Solve Quadratic Equations by Graphing 0.3 Solve Quadratic Equations b Graphing Before You solved quadratic equations b factoring. Now You will solve quadratic equations b graphing. Wh? So ou can solve a problem about sports, as in Eample 6.

More information

15.2 Graphing Logarithmic

15.2 Graphing Logarithmic _ - - - - - - Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b

More information

5.3 Interpreting Rate of Change and Slope

5.3 Interpreting Rate of Change and Slope Name Class Date 5.3 Interpreting Rate of Change and Slope Essential question: How can ou relate rate of change and slope in linear relationships? Resource Locker Eplore Determining Rates of Change For

More information

Domain, Range, and End Behavior

Domain, Range, and End Behavior Locker LESSON 1.1 Domain, Range, and End Behavior Common Core Math Standards The student is epected to: F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship

More information

Name Class Date. Finding Real Roots of Polynomial Equations Extension: Graphing Factorable Polynomial Functions

Name Class Date. Finding Real Roots of Polynomial Equations Extension: Graphing Factorable Polynomial Functions Name Class Date -1 Finding Real Roots of Polnomial Equations Etension: Graphing Factorable Polnomial Functions Essential question: How do ou use zeros to graph polnomial functions? Video Tutor prep for

More information

5-4. Focus and Directrix of a Parabola. Key Concept Parabola VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

5-4. Focus and Directrix of a Parabola. Key Concept Parabola VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING 5- Focus and Directri of a Parabola TEKS FOCUS VOCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening.

More information

Name Class Date. Solving by Graphing and Algebraically

Name Class Date. Solving by Graphing and Algebraically Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem

More information

D: all real; R: y g (x) = 3 _ 2 x 2 5. g (x) = 5 x g (x) = - 4 x 2 7. g (x) = -4 x 2. Houghton Mifflin Harcourt Publishing Company.

D: all real; R: y g (x) = 3 _ 2 x 2 5. g (x) = 5 x g (x) = - 4 x 2 7. g (x) = -4 x 2. Houghton Mifflin Harcourt Publishing Company. AVOID COMMON ERRORS Watch for students who do not graph points on both sides of the verte of the parabola. Remind these students that a parabola is U-shaped and smmetric, and the can use that smmetr to

More information

Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.

Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept. Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of

More information

14.3 Constructing Exponential Functions

14.3 Constructing Exponential Functions Name Class Date 1.3 Constructing Eponential Functions Essential Question: What are discrete eponential functions and how do ou represent them? Resource Locker Eplore Understanding Discrete Eponential Functions

More information

Additional Factoring Examples:

Additional Factoring Examples: Honors Algebra -3 Solving Quadratic Equations by Graphing and Factoring Learning Targets 1. I can solve quadratic equations by graphing. I can solve quadratic equations by factoring 3. I can write a quadratic

More information

3.1 Graph Quadratic Functions

3.1 Graph Quadratic Functions 3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your

More information

Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a.

Mathematics 10 Page 1 of 7 The Quadratic Function (Vertex Form): Translations. and axis of symmetry is at x a. Mathematics 10 Page 1 of 7 Verte form of Quadratic Relations The epression a p q defines a quadratic relation called the verte form with a horizontal translation of p units and vertical translation of

More information

Chapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving Quadratic Equations by Finding the Square Root and Completing the Square

Chapter 9 Notes Alg. 1H 9-A1 (Lesson 9-3) Solving Quadratic Equations by Finding the Square Root and Completing the Square Chapter Notes Alg. H -A (Lesson -) Solving Quadratic Equations b Finding the Square Root and Completing the Square p. *Calculator Find the Square Root: take the square root of. E: Solve b finding square

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name Date Chapter 8 Maintaining Mathematical Proficienc Graph the linear equation. 1. = 5. = + 3 3. 1 = + 3. = + Evaluate the epression when =. 5. + 8. + 3 7. 3 8. 5 + 8 9. 8 10. 5 + 3 11. + + 1. 3 + +

More information

Writing Quadratic Functions in Standard Form

Writing Quadratic Functions in Standard Form Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval

More information

9-3 Linear Functions Going Deeper

9-3 Linear Functions Going Deeper Name Class Date 9-3 Linear Functions Going Deeper Essential question: How do ou graph a linear function? 1 CC.8.EE.5 EXPLORE Investigating Change video tutor The U.S. Department of Agriculture defines

More information

9.12 Quadratics Review

9.12 Quadratics Review Algebra Name _ B2g0gD6L jkwudtaaa msvopfwtowiarneq CLOLXCa.I K `Awljla `rtiugohhtfs_ QrIefsfeYrZvtetdf. 9.2 Quadratics Review ) What is the difference between the two mathematical statements below? Then

More information

ALGEBRA II-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION

ALGEBRA II-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION ALGEBRA II-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION The Quadratic Equation is written as: ; this equation has a degree of. Where a, b and c are integer coefficients (where a 0) The graph of

More information

11.1 Inverses of Simple Quadratic and Cubic Functions

11.1 Inverses of Simple Quadratic and Cubic Functions Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,

More information

Skills Practice Skills Practice for Lesson 1.1

Skills Practice Skills Practice for Lesson 1.1 Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Give an eample of each term.. quadratic function 9 0. vertical motion equation s

More information

Characteristics of Quadratic Functions

Characteristics of Quadratic Functions . Characteristics of Quadratic Functions Essential Question What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr? Parabolas and Smmetr Work with a partner. a. Complete

More information

3.1. Shape and Structure Forms of Quadratic Functions ESSENTIAL IDEAS TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS 169A

3.1. Shape and Structure Forms of Quadratic Functions ESSENTIAL IDEAS TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS 169A Shape and Structure Forms of Quadratic Functions.1 LEARNING GOALS KEY TERMS In this lesson, ou will: Match a quadratic function with its corresponding graph. Identif ke characteristics of quadratic functions

More information

Fair Game Review. Chapter 8. Graph the linear equation. Big Ideas Math Algebra Record and Practice Journal

Fair Game Review. Chapter 8. Graph the linear equation. Big Ideas Math Algebra Record and Practice Journal Name Date Chapter Graph the linear equation. Fair Game Review. =. = +. =. =. = +. = + Copright Big Ideas Learning, LLC Big Ideas Math Algebra Name Date Chapter Fair Game Review (continued) Evaluate the

More information

Using Intercept Form

Using Intercept Form 8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of

More information

Mth Quadratic functions and quadratic equations

Mth Quadratic functions and quadratic equations Mth 0 - Quadratic functions and quadratic equations Name Find the product. 1) 8a3(2a3 + 2 + 12a) 2) ( + 4)( + 6) 3) (3p - 1)(9p2 + 3p + 1) 4) (32 + 4-4)(2-3 + 3) ) (4a - 7)2 Factor completel. 6) 92-4 7)

More information

f(x) = 2x 2 + 2x - 4

f(x) = 2x 2 + 2x - 4 4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms

More information

2.3 Solving Absolute Value Inequalities

2.3 Solving Absolute Value Inequalities .3 Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value Inequalit You

More information

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS

UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Answer Ke Name: Date: UNIT #8 QUADRATIC FUNCTIONS AND THEIR ALGEBRA REVIEW QUESTIONS Part I Questions. For the quadratic function shown below, the coordinates of its verte are, (), 7 6,, 6 The verte is

More information

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

More information

6.1 Solving Quadratic Equations by Graphing Algebra 2

6.1 Solving Quadratic Equations by Graphing Algebra 2 10.1 Solving Quadratic Equations b Graphing Algebra Goal 1: Write functions in quadratic form Goal : Graph quadratic functions Goal 3: Solve quadratic equations b graphing. Quadratic Function: Eample 1:

More information

Mth 95 Module 4 Chapter 8 Spring Review - Solving quadratic equations using the quadratic formula

Mth 95 Module 4 Chapter 8 Spring Review - Solving quadratic equations using the quadratic formula Mth 95 Module 4 Chapter 8 Spring 04 Review - Solving quadratic equations using the quadratic formula Write the quadratic formula. The NUMBER of REAL and COMPLEX SOLUTIONS to a quadratic equation ( a b

More information

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

More information

Algebra 1 Unit 9 Quadratic Equations

Algebra 1 Unit 9 Quadratic Equations Algebra 1 Unit 9 Quadratic Equations Part 1 Name: Period: Date Name of Lesson Notes Tuesda 4/4 Wednesda 4/5 Thursda 4/6 Frida 4/7 Monda 4/10 Tuesda 4/11 Wednesda 4/12 Thursda 4/13 Frida 4/14 Da 1- Quadratic

More information

Lesson 4.1 Exercises, pages

Lesson 4.1 Exercises, pages Lesson 4.1 Eercises, pages 57 61 When approimating answers, round to the nearest tenth. A 4. Identify the y-intercept of the graph of each quadratic function. a) y = - 1 + 5-1 b) y = 3-14 + 5 Use mental

More information

PRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.

PRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line. MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for

More information

REVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES

REVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.

More information

2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.

2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner. 9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the -intercepts of a graph in Section., it follows that the -intercept

More information

14.2 Choosing Among Linear, Quadratic, and Exponential Models

14.2 Choosing Among Linear, Quadratic, and Exponential Models Name Class Date 14.2 Choosing Among Linear, Quadratic, and Eponential Models Essential Question: How do ou choose among, linear, quadratic, and eponential models for a given set of data? Resource Locker

More information

Unit 10 - Graphing Quadratic Functions

Unit 10 - Graphing Quadratic Functions Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif

More information

Lesson Master 9-1B. REPRESENTATIONS Objective G. Questions on SPUR Objectives. 1. Let f(x) = 1. a. What are the coordinates of the vertex?

Lesson Master 9-1B. REPRESENTATIONS Objective G. Questions on SPUR Objectives. 1. Let f(x) = 1. a. What are the coordinates of the vertex? Back to Lesson 9-9-B REPRESENTATIONS Objective G. Let f() =. a. What are the coordinates of the verte? b. Is the verte a minimum or a maimum? c. Complete the table of values below. 3 0 3 f() d. Graph the

More information

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

Essential Question How can you use a quadratic function to model a real-life situation? 3. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..E A..A A..B A..C Modeling with Quadratic Functions Essential Question How can ou use a quadratic function to model a real-life situation? Work with a partner.

More information

15.2 Graphing Logarithmic

15.2 Graphing Logarithmic Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > 0 and b 1 related to the graph of f () = log b? Resource Locker A.5.A Determine

More information

5.2 Solving Linear-Quadratic Systems

5.2 Solving Linear-Quadratic Systems Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker

More information

Algebra 2 Unit 2 Practice

Algebra 2 Unit 2 Practice Algebra Unit Practice LESSON 7-1 1. Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of

More information

5.1 Understanding Linear Functions

5.1 Understanding Linear Functions Name Class Date 5.1 Understanding Linear Functions Essential Question: What is a linear function? Resource Locker Eplore 1 Recognizing Linear Functions A race car can travel up to 210 mph. If the car could

More information

20.3 Applying the Zero Product Property to Solve Equations

20.3 Applying the Zero Product Property to Solve Equations 20.3 Applying the Zero Product Property to Solve Equations Essential Question: How can you use the Zero Product Property to solve quadratic equations in factored form? Resource Locker Explore Understanding

More information

Name: Period: SM Starter on Reading Quadratic Graph. This graph and equation represent the path of an object being thrown.

Name: Period: SM Starter on Reading Quadratic Graph. This graph and equation represent the path of an object being thrown. SM Name: Period: 7.5 Starter on Reading Quadratic Graph This graph and equation represent the path of an object being thrown. 1. What is the -ais measuring?. What is the y-ais measuring? 3. What are the

More information

6.5 Comparing Properties of Linear Functions

6.5 Comparing Properties of Linear Functions Name Class Date 6.5 Comparing Properties of Linear Functions Essential Question: How can ou compare linear functions that are represented in different was? Resource Locker Eplore Comparing Properties of

More information

Finding Complex Solutions of Quadratic Equations

Finding Complex Solutions of Quadratic Equations COMMON CORE y - 0 y - - 0 - Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the

More information

Graph and Write Equations of Parabolas

Graph and Write Equations of Parabolas TEKS 9.2 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Parabolas Before You graphed and wrote equations of parabolas that open up or down. Now You will graph and write equations of parabolas that open

More information

Study Guide and Intervention

Study Guide and Intervention 6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a

More information

Lesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas

Lesson Goals. Unit 4 Polynomial/Rational Functions Quadratic Functions (Chap 0.3) Family of Quadratic Functions. Parabolas Unit 4 Polnomial/Rational Functions Quadratic Functions (Chap 0.3) William (Bill) Finch Lesson Goals When ou have completed this lesson ou will: Graph and analze the graphs of quadratic functions. Solve

More information

Final Exam Review Part 2 #1 Page 1 / 21

Final Exam Review Part 2 #1 Page 1 / 21 Final Eam Review Part #1 Intermediate Algebra / MAT 135 Spring 017 Master ( Master Templates) Student Name/ID: v 1. Solve for, where is a real number. v v + 1 + =. Solve for, where is a real number. +

More information

Solving Linear-Quadratic Systems

Solving Linear-Quadratic Systems 36 LESSON Solving Linear-Quadratic Sstems UNDERSTAND A sstem of two or more equations can include linear and nonlinear equations. In a linear-quadratic sstem, there is one linear equation and one quadratic

More information

13.1 Understanding Piecewise-Defined Functions

13.1 Understanding Piecewise-Defined Functions Name Class Date 13.1 Understanding Piecewise-Defined Functions Essential Question: How are piecewise-defined functions different from other functions? Resource Locker Eplore Eploring Piecewise-Defined

More information

Properties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a

Properties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a 0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value

More information

For questions 5-8, solve each inequality and graph the solution set. You must show work for full credit. (2 pts each)

For questions 5-8, solve each inequality and graph the solution set. You must show work for full credit. (2 pts each) Alg Midterm Review Practice Level 1 C 1. Find the opposite and the reciprocal of 0. a. 0, 1 b. 0, 1 0 0 c. 0, 1 0 d. 0, 1 0 For questions -, insert , or = to make the sentence true. (1pt each) A. 5

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

2.1 Evaluate and Graph Polynomial

2.1 Evaluate and Graph Polynomial 2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of

More information

Algebra I Quadratics Practice Questions

Algebra I Quadratics Practice Questions 1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 From CCSD CSE S Page 1 of 6 1 5. Which is equivalent

More information

Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function 2 Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

More information

QUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta

QUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape

More information

4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation? 3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the

More information

10.3 Coordinate Proof Using Distance with Segments and Triangles

10.3 Coordinate Proof Using Distance with Segments and Triangles Name Class Date 10.3 Coordinate Proof Using Distance with Segments and Triangles Essential Question: How do ou write a coordinate proof? Resource Locker Eplore G..B...use the distance, slope,... formulas

More information

+ = + + = x = + = + = 36x

+ = + + = x = + = + = 36x Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the

More information

Algebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3

Algebra I Practice Questions ? 1. Which is equivalent to (A) (B) (C) (D) 2. Which is equivalent to 6 8? (A) 4 3 1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 Page 1 of 0 11 Practice Questions 6 1 5. Which

More information

SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,

More information

x Radical Sign: Radicand: the number beneath the radical sign

x Radical Sign: Radicand: the number beneath the radical sign Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing.

More information

Evaluate nth Roots and Use Rational Exponents. p Evaluate nth roots and study rational exponents. VOCABULARY. Index of a radical

Evaluate nth Roots and Use Rational Exponents. p Evaluate nth roots and study rational exponents. VOCABULARY. Index of a radical . Georgia Performance Standard(s) MMA2a, MMA2b, MMAd Your Notes Evaluate nth Roots and Use Rational Eponents Goal VOCABULARY nth root of a p Evaluate nth roots and stud rational eponents. Inde of a radical

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its

More information

1.1 Domain, Range, and End Behavior

1.1 Domain, Range, and End Behavior Name Class Date 1.1 Domain, Range, and End Behavior Essential Question: How can ou determine the domain, range, and end behavior of a function? Resource Locker Eplore Representing an Interval on a Number

More information

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.

Quadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots. Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to

More information

20.3 Applying the Zero Product Property to Solve Equations

20.3 Applying the Zero Product Property to Solve Equations Name Class Date 2. Applying the Zero Product Property to Solve Equations Essential Question: How can you use the Zero Product Property to solve quadratic equations in factored form? Resource Locker Explore

More information

MATH 60 Review Problems for Final Exam

MATH 60 Review Problems for Final Exam MATH 60 Review Problems for Final Eam Scientific Calculators Onl - Graphing Calculators Not Allowed NO CLASS NOTES PERMITTED Evaluate the epression for the given values. m 1) m + 3 for m = 3 2) m 2 - n2

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Shape and Structure. Forms of Quadratic Functions. Lesson 2.1 Assignment

Shape and Structure. Forms of Quadratic Functions. Lesson 2.1 Assignment Lesson.1 Assignment Name Date Shape and Structure Forms of Quadratic Functions 1. Analze the graph of the quadratic function. a. The standard form of a quadratic function is f() 5 a 1 b 1 c. What possible

More information

7.2 Properties of Graphs

7.2 Properties of Graphs 7. Properties of Graphs of Quadratic Functions GOAL Identif the characteristics of graphs of quadratic functions, and use the graphs to solve problems. LEARN ABOUT the Math Nicolina plas on her school

More information