Characteristics of Quadratic Functions

Size: px
Start display at page:

Download "Characteristics of Quadratic Functions"

Transcription

1 . 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 the table. Then use the values in the table to sketch the graph of the function f() = 1 on graph paper f() 3 5 f() b. Use the results in part (a) to identif the verte of the parabola. c. Find a vertical line on our graph paper so that when ou fold the paper, the left portion of the graph coincides with the right portion of the graph. What is the equation of this line? How does it relate to the verte? d. Show that the verte form f() = 1 ( ) is equivalent to the function given in part (a). ATTENDING TO PRECISION To be proficient in math, ou need to use clear definitions in our reasoning and discussions with others. Parabolas and Smmetr Work with a partner. Repeat Eploration 1 for the function given b f() = = 1 3 ( 3) +. Communicate Your Answer 3. What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr?. Describe the smmetr of each graph. Then use a graphing calculator to verif our answer. a. f() = ( 1) + b. f() = ( + 1) c. f() = ( 3) + 1 d. f() = 1 ( + ) e. f() = + 3 f. f() = 3( 5) + Section. Characteristics of Quadratic Functions 9

2 . Lesson What You Will Learn Core Vocabular ais of smmetr, p. 70 standard form, p. 70 minimum value, p. 7 maimum value, p. 7 intercept form, p. 73 Previous -intercept Eplore properties of parabolas. Find maimum and minimum values of quadratic functions. Graph quadratic functions using -intercepts. Solve real-life problems. Eploring Properties of Parabolas An ais of smmetr is a line that divides a parabola into mirror images and passes through the verte. Because the verte of f() = a( h) + k is (h, k), the ais of smmetr is the vertical line = h. Previousl, ou used transformations to graph quadratic functions in verte form. You can also use the ais of smmetr and the verte to graph quadratic functions written in verte form. (h, k) = h Using Smmetr to Graph Quadratic Functions Graph f() = ( + 3) +. Label the verte and ais of smmetr. SOLUTION Step 1 Identif the constants a =, h = 3, and k =. Step Plot the verte (h, k) = ( 3, ) and draw the ais of smmetr = 3. Step 3 Evaluate the function for two values of. = : f( ) = ( + 3) + = = 1: f( 1) = ( 1 + 3) + = Plot the points (, ), ( 1, ), and their reflections in the ais of smmetr. Step Draw a parabola through the plotted points. ( 3, ) = 3 Quadratic functions can also be written in standard form, f() = a + b + c, where a 0. You can derive standard form b epanding verte form. f() = a( h) + k Verte form f() = a( h + h ) + k Epand ( h). f() = a ah + ah + k f() = a + ( ah) + (ah + k) Distributive Propert Group like terms. 70 Chapter Quadratic Functions f() = a + b + c Let b = ah and let c = ah + k. This allows ou to make the following observations. a = a: So, a has the same meaning in verte form and standard form. b = ah: Solve for h to obtain h = b a. So, the ais of smmetr is = b a. c = ah + k: In verte form f() = a( h) + k, notice that f(0) = ah + k. So, c is the -intercept.

3 Core Concept Properties of the Graph of f() = a + b + c = a + b + c, a > 0 = a + b + c, a < 0 (0, c) = b a b = a (0, c) The parabola opens up when a > 0 and opens down when a < 0. The graph is narrower than the graph of f() = when a > 1 and wider when a < 1. The ais of smmetr is = b a ( and the verte is b a (, f a) b ). The -intercept is c. So, the point (0, c) is on the parabola. Graphing a Quadratic Function in Standard Form COMMON ERROR Be sure to include the negative sign when writing the epression for the -coordinate of the verte. Graph f() = Label the verte and ais of smmetr. SOLUTION Step 1 Identif the coefficients a = 3, b =, and c = 1. Because a > 0, the parabola opens up. Step Find the verte. First calculate the -coordinate. = b a = (3) = 1 Then find the -coordinate of the verte. f(1) = 3(1) (1) + 1 = So, the verte is (1, ). Plot this point. Step 3 Draw the ais of smmetr = 1. Step Identif the -intercept c, which is 1. Plot the point (0, 1) and its reflection in the ais of smmetr, (, 1). Step 5 Evaluate the function for another value of, (1, )) such as = 3. = 1 f(3) = 3(3) (3) + 1 = 10 Plot the point (3, 10) and its reflection in the ais of smmetr, ( 1, 10). Step Draw a parabola through the plotted points. Monitoring Progress Graph the function. Label the verte and ais of smmetr. 1. f() = 3( + 1). g() = ( ) h() = + 1. p() = Section. Characteristics of Quadratic Functions 71

4 STUDY TIP When a function f is written in verte form, ou can use h = b a and k = f ( a) b to state the properties shown. Finding Maimum and Minimum Values Because the verte is the highest or lowest point on a parabola, its -coordinate is the maimum value or minimum value of the function. The verte lies on the ais of smmetr, so the function is increasing on one side of the ais of smmetr and decreasing on the other side. Core Concept Minimum and Maimum Values For the quadratic function f() = a + b + c, the -coordinate of the verte is the minimum value of the function when a > 0 and the maimum value when a < 0. a > 0 a < 0 decreasing increasing minimum = b a Minimum value: f ( a) b Domain: All real numbers Range: f ( a) b increasing = b a maimum decreasing Maimum value: f ( a) b Domain: All real numbers Range: f ( a) b Decreasing to the left of = b a Increasing to the right of = b a Increasing to the left of = b a Decreasing to the right of = b a Check Minimum X= Y= Finding a Minimum or a Maimum Value Find the minimum value or maimum value of f() = 1 1. Describe the domain and range of the function, and where the function is increasing and decreasing. SOLUTION Identif the coefficients a = 1, b =, and c = 1. Because a > 0, the parabola opens up and the function has a minimum value. To find the minimum value, calculate the coordinates of the verte. = b a = ( 1 ) = f() = 1 () () 1 = 3 The minimum value is 3. So, the domain is all real numbers and the range is 3. The function is decreasing to the left of = and increasing to the right of =. Monitoring Progress 5. Find the minimum value or maimum value of (a) f() = and (b) h() = Describe the domain and range of each function, and where each function is increasing and decreasing. 7 Chapter Quadratic Functions

5 REMEMBER An -intercept of a graph is the -coordinate of a point where the graph intersects the -ais. It occurs where f() = 0. COMMON ERROR Remember that the -intercepts of the graph of f() = a( p)( q) are p and q, not p and q. Graphing Quadratic Functions Using -Intercepts When the graph of a quadratic function has at least one -intercept, the function can be written in intercept form, f() = a( p)( q), where a 0. Core Concept Properties of the Graph of f() = a( p)( q) Because f(p) = 0 and f(q) = 0, p and = q are the -intercepts of the graph of the function. The ais of smmetr is halfwa between (p, 0) and (q, 0). So, the ais of smmetr is = p + q. The parabola opens up when a > 0 and opens down when a < 0. Graphing a Quadratic Function in Intercept Form Graph f() = ( + 3)( 1). Label the -intercepts, verte, and ais of smmetr. SOLUTION Step 1 Identif the -intercepts. The -intercepts are p = 3 and q = 1, so the parabola passes through the points ( 3, 0) and (1, 0). Step Find the coordinates of the verte. = p + q = = 1 f( 1) = ( 1 + 3)( 1 1) = 8 So, the ais of smmetr is = 1 and the verte is ( 1, 8). Step 3 Draw a parabola through the verte and the points where the -intercepts occur. (p, 0) p + q (q, 0) ( 1, 8) ( 3, 0) = a( p)( q) = 1 (1, 0) Check You can check our answer b generating a table of values for f on a graphing calculator. -intercept -intercept X X=-1 Y The values show smmetr about = 1. So, the verte is ( 1, 8). Monitoring Progress Graph the function. Label the -intercepts, verte, and ais of smmetr.. f() = ( + 1)( + 5) 7. g() = 1 ( )( ) Section. Characteristics of Quadratic Functions 73

6 Solving Real-Life Problems Modeling with Mathematics (50, 5) The parabola shows the path of our first golf shot, where is the horizontal distance (in ards) and is the corresponding height (in ards). The path of our second shot can be modeled b the function f() = 0.0( 80). Which shot travels farther before hitting the ground? Which travels higher? SOLUTION (0, 0) (100, 0) 1. Understand the Problem You are given a graph and a function that represent the paths of two golf shots. You are asked to determine which shot travels farther before hitting the ground and which shot travels higher.. Make a Plan Determine how far each shot travels b interpreting the -intercepts. Determine how high each shot travels b finding the maimum value of each function. Then compare the values. 3. Solve the Problem First shot: The graph shows that the -intercepts are 0 and 100. So, the ball travels 100 ards before hitting the ground. 5 d 100 d Because the ais of smmetr is halfwa between (0, 0) and (100, 0), the ais of smmetr is = = 50. So, the verte is (50, 5) and the maimum height is 5 ards. Second shot: B rewriting the function in intercept form as f() = 0.0( 0)( 80), ou can see that p = 0 and q = 80. So, the ball travels 80 ards before hitting the ground. To find the maimum height, find the coordinates of the verte. = p + q = = 0 f(0) = 0.0(0)(0 80) = 3 The maimum height of the second shot is 3 ards. 0 = 5 Because 100 ards > 80 ards, the first shot travels farther. Because 3 ards > 5 ards, the second shot travels higher. 0 0 f 90. Look Back To check that the second shot travels higher, graph the function representing the path of the second shot and the line = 5, which represents the maimum height of the first shot. The graph rises above = 5, so the second shot travels higher. Monitoring Progress 8. WHAT IF? The graph of our third shot is a parabola through the origin that reaches a maimum height of 8 ards when = 5. Compare the distance it travels before it hits the ground with the distances of the first two shots. 7 Chapter Quadratic Functions

7 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Eplain how to determine whether a quadratic function will have a minimum value or a maimum value.. WHICH ONE DOESN T BELONG? The graph of which function does not belong with the other three? Eplain. f() = 3 + f() = 3 + f() = 3( )( + ) f() = 3( + 1) 7 Monitoring Progress and Modeling with Mathematics In Eercises 3 1, graph the function. Label the verte and ais of smmetr. (See Eample 1.) 3. f() = ( 3). h() = ( + ) 5. g() = ( + 3) + 5. = ( 7) 1 7. = ( ) + 8. g() = ( + 1) 3 9. f() = ( 1) h() = ( + ) + REASONING In Eercises 19 and 0, use the ais of smmetr to plot the reflection of each point and complete the parabola (, 3) 1 (1, ) (0, 1) = 0. = 3 ( 1, 1) (, ) ( 3, 3) 11. = 1 ( + ) = 1 ( 3) f() = 0.( 1) 1. g() = ANALYZING RELATIONSHIPS In Eercises 15 18, use the ais of smmetr to match the equation with its graph. 15. = ( 3) = ( + ) 17. = 1 ( + 1) = ( ) 1 A. C. = = 3 B. D. = 1 = In Eercises 1 30, graph the function. Label the verte and ais of smmetr. (See Eample.) 1. = = = f() = g() = 1. f() = 5 7. g() = f() = = = WRITING Two quadratic functions have graphs with vertices (, ) and (, 3). Eplain wh ou can not use the aes of smmetr to distinguish between the two functions. 3. WRITING A quadratic function is increasing to the left of = and decreasing to the right of =. Will the verte be the highest or lowest point on the graph of the parabola? Eplain. Section. Characteristics of Quadratic Functions 75

8 ERROR ANALYSIS In Eercises 33 and 3, describe and correct the error in analzing the graph of = The -coordinate of the verte is = b a = () = 3. In Eercises 39 8, find the minimum or maimum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing. (See Eample 3.) 39. = 1 0. = =. g() = The -intercept of the graph is the value of c, which is f() = g() = h() = 1. h() = MODELING WITH MATHEMATICS In Eercises 35 and 3, is the horizontal distance (in feet) and is the vertical distance (in feet). Find and interpret the coordinates of the verte. 35. The path of a basketball thrown at an angle of 5 can be modeled b = The path of a shot put released at an angle of 35 can be modeled b = = f() = PROBLEM SOLVING The path of a diver is modeled b the function f() = , where f() is the height of the diver (in meters) above the water and is the horizontal distance (in meters) from the end of the diving board. a. What is the height of the diving board? b. What is the maimum height of the diver? c. Describe where the diver is ascending and where the diver is descending ANALYZING EQUATIONS The graph of which function has the same ais of smmetr as the graph of = + +? A = + + B = 3 + C = + D = USING STRUCTURE Which function represents the widest parabola? Eplain our reasoning. A = ( + 3) B = 5 C = 0.5( 1) + 1 D = PROBLEM SOLVING The engine torque (in foot-pounds) of one model of car is given b = , where is the speed (in thousands of revolutions per minute) of the engine. a. Find the engine speed that maimizes torque. What is the maimum torque? b. Eplain what happens to the engine torque as the speed of the engine increases. MATHEMATICAL CONNECTIONS In Eercises 51 and 5, write an equation for the area of the figure. Then determine the maimum possible area of the figure w b w b 7 Chapter Quadratic Functions

9 In Eercises 53 0, graph the function. Label the -intercept(s), verte, and ais of smmetr. (See Eample.) 53. = ( + 3)( 3) 5. = ( + 1)( 3) 55. = 3( + )( + ) 5. f() = ( 5)( 1) 57. g() = ( + ) 58. = ( + 7) 59. f() = ( 3) 0. = ( 7) USING TOOLS In Eercises 1, identif the -intercepts of the function and describe where the graph is increasing and decreasing. Use a graphing calculator to verif our answer. 1. f() = 1 ( )( + ). = 3 ( + 1)( 3) 3. g() = ( )( ). h() = 5( + 5)( + 1) 5. MODELING WITH MATHEMATICS A soccer plaer kicks a ball downfield. The height of the ball increases until it reaches a maimum height of 8 ards, 0 ards awa from the plaer. A second kick is modeled b = ( ). Which kick travels farther before hitting the ground? Which kick travels higher? (See Eample 5.). MODELING WITH MATHEMATICS Although a football field appears to be flat, some are actuall shaped like a parabola so that rain runs off to both sides. The cross section of a field can be modeled b = ( 10), where and are measured in feet. What is the width of the field? What is the maimum height of the surface of the field? 8. OPEN-ENDED Write two different quadratic functions in intercept form whose graphs have the ais of smmetr = PROBLEM SOLVING An online music store sells about 000 songs each da when it charges $1 per song. For each $0.05 increase in price, about 80 fewer songs per da are sold. Use the verbal model and quadratic function to determine how much the store should charge per song to maimize dail revenue. Revenue (dollars) = Price (dollars/song) Sales (songs) R() = ( ) (000 80) 70. PROBLEM SOLVING An electronics store sells 70 digital cameras per month at a price of $30 each. For each $0 decrease in price, about 5 more cameras per month are sold. Use the verbal model and quadratic function to determine how much the store should charge per camera to maimize monthl revenue. Revenue (dollars) = Price (dollars/camera) Sales (cameras) R() = (30 0) (70 + 5) 71. DRAWING CONCLUSIONS Compare the graphs of the three quadratic functions. What do ou notice? Rewrite the functions f and g in standard form to justif our answer. f() = ( + 3)( + 1) g() = ( + ) 1 h() = USING STRUCTURE Write the quadratic function f() = + 1 in intercept form. Graph the function. Label the -intercepts, -intercept, verte, and ais of smmetr. surface of football field Not drawn to scale 73. PROBLEM SOLVING A woodland jumping mouse hops along a parabolic path given b = , where is the mouse s horizontal distance traveled (in feet) and is the corresponding height (in feet). Can the mouse jump over a fence that is 3 feet high? Justif our answer. 7. REASONING The points (, 3) and (, ) lie on the graph of a quadratic function. Determine whether ou can use these points to find the ais of smmetr. If not, eplain. If so, write the equation of the ais of smmetr. Not drawn to scale Section. Characteristics of Quadratic Functions 77

10 7. HOW DO YOU SEE IT? Consider the graph of the function f() = a( p)( q). 77. MAKING AN ARGUMENT The point (1, 5) lies on the graph of a quadratic function with ais of smmetr = 1. Your friend sas the verte could be the point (0, 5). Is our friend correct? Eplain. 78. CRITICAL THINKING Find the -intercept in terms of a, p, and q for the quadratic function f() = a( p)( q). a. What does f ( p + q ) represent in the graph? b. If a < 0, how does our answer in part (a) change? Eplain. 79. MODELING WITH MATHEMATICS A kernel of popcorn contains water that epands when the kernel is heated, causing it to pop. The equations below represent the popping volume (in cubic centimeters per gram) of popcorn with moisture content (as a percent of the popcorn s weight). Hot-air popping: = 0.71( 5.5)(.) Hot-oil popping: = 0.5( 5.35)( 1.8) 75. MODELING WITH MATHEMATICS The Gateshead Millennium Bridge spans the River Tne. The arch of the bridge can be modeled b a parabola. The arch reaches a maimum height of 50 meters at a point roughl 3 meters across the river. Graph the curve of the arch. What are the domain and range? What do the represent in this situation? 7. THOUGHT PROVOKING You have 100 feet of fencing to enclose a rectangular garden. Draw three possible designs for the garden. Of these, which has the greatest area? Make a conjecture about the dimensions of the rectangular garden with the greatest possible area. Eplain our reasoning. a. For hot-air popping, what moisture content maimizes popping volume? What is the maimum volume? b. For hot-oil popping, what moisture content maimizes popping volume? What is the maimum volume? c. Use a graphing calculator to graph both functions in the same coordinate plane. What are the domain and range of each function in this situation? Eplain. 80. ABSTRACT REASONING A function is written in intercept form with a > 0. What happens to the verte of the graph as a increases? as a approaches 0? Maintaining Mathematical Proficienc Find the distance between the two points. 81. (1, ), (3, ) 8. ( 7, 13), (10, 8) 83. ( 3, 9), ( 3, 1) 8. ( 1, ), ( 0, 3) Solve the proportion. Reviewing what ou learned in previous grades and lessons = 8. 3 = = = 0 78 Chapter Quadratic Functions

Graph Quadratic Functions in Standard Form

Graph Quadratic Functions in Standard Form TEKS 4. 2A.4.A, 2A.4.B, 2A.6.B, 2A.8.A Graph Quadratic Functions in Standard Form Before You graphed linear functions. Now You will graph quadratic functions. Wh? So ou can model sports revenue, as in

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

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

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

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

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

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

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

Essential Question How can you use a scatter plot and a line of fit to make conclusions about data?

Essential Question How can you use a scatter plot and a line of fit to make conclusions about data? . Scatter Plots and Lines of Fit Essential Question How can ou use a scatter plot and a line of fit to make conclusions about data? A scatter plot is a graph that shows the relationship between two data

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

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

Essential Question How can you solve a nonlinear system of equations? .5 Solving Nonlinear Sstems Essential Question Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph.

More information

Modeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models?

Modeling with Exponential and Logarithmic Functions 6.7. Essential Question How can you recognize polynomial, exponential, and logarithmic models? .7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match

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

TEST REVIEW QUADRATICS EQUATIONS Name: 2. Which of the following statements is true about the graph of the function?

TEST REVIEW QUADRATICS EQUATIONS Name: 2. Which of the following statements is true about the graph of the function? Chapter MATHEMATICS 00 TEST REVIEW QUADRATICS EQUATIONS Name:. Which equation does not represent a quadratic function?. Which of the following statements is true about the graph of the function? it has

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

Writing Equations in Point-Slope Form

Writing Equations in Point-Slope Form . Writing Equations in Point-Slope Form Essential Question How can ou write an equation of a line when ou are given the slope and a point on the line? Writing Equations of Lines Work with a partner. Sketch

More information

Functions. Essential Question What is a function?

Functions. Essential Question What is a function? 3. Functions COMMON CORE Learning Standard HSF-IF.A. Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs

More information

Functions. Essential Question What is a function? Work with a partner. Functions can be described in many ways.

Functions. Essential Question What is a function? Work with a partner. Functions can be described in many ways. . Functions Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs and the -coordinates are outputs. A relation

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name Date Chapter 3 Maintaining Mathematical Proficienc Plot the point in a coordinate plane. Describe the location of the point. 1. A( 3, 1). B (, ) 3. C ( 1, 0). D ( 5, ) 5. Plot the point that is on

More information

Factoring Polynomials

Factoring Polynomials 5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.D 2A.7.E Factoring Polnomials Essential Question How can ou factor a polnomial? Factoring Polnomials Work with a partner. Match each polnomial equation with

More information

Functions. Essential Question What are some of the characteristics of the graph of an exponential function? ) x e. f (x) = ( 1 3 ) x f.

Functions. Essential Question What are some of the characteristics of the graph of an exponential function? ) x e. f (x) = ( 1 3 ) x f. 7. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A Eponential Growth and Deca Functions Essential Question What are some of the characteristics of the graph of an eponential function? You can use a graphing

More information

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

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

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

Quadratic Functions and Factoring

Quadratic Functions and Factoring Quadratic Functions and Factoring Lesson. CC.9-2.F.IF.7a*.2 CC.9-2.F.IF.7a*.3 CC.9-2.A.SSE.3a*.4 CC.9-2.A.SSE.3a*.5 CC.9-2.A.REI.4b.6 CC.9-2.N.CN.2.7 CC.9-2.A.REI.4a.8 CC.9-2.N.CN.7.9 CC.9-2.A.REI.4b.

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

Essential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)

Essential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars) 5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast.

More information

SEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p.

SEE the Big Idea. Quonset Hut (p. 218) Zebra Mussels (p. 203) Ruins of Caesarea (p. 195) Basketball (p. 178) Electric Vehicles (p. Polnomial Functions.1 Graphing Polnomial Functions. Adding, Subtracting, and Multipling Polnomials.3 Dividing Polnomials. Factoring Polnomials.5 Solving Polnomial Equations. The Fundamental Theorem of

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

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

Name Date. and y = 5.

Name Date. and y = 5. Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;

More information

Path of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2.

Path of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2. - Quadratic Functions and Transformations Content Standards F.BF. Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative)

More information

Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane?

Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? 10.7 Circles in the Coordinate Plane Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? The Equation of a Circle with Center at the Origin Work

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

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation. 1. + =. = 3 3. 5 + = 10. 3 = 5. 3 = 6. 3 + = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < 3 10. 8 3r 5 r

More information

Solving Quadratic Equations

Solving Quadratic Equations 9 Solving Quadratic Equations 9. Properties of Radicals 9. Solving Quadratic Equations b Graphing 9. Solving Quadratic Equations Using Square Roots 9. Solving Quadratic Equations b Completing the Square

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

SECTION 3.1: Quadratic Functions

SECTION 3.1: Quadratic Functions SECTION 3.: Quadratic Functions Objectives Graph and Analyze Quadratic Functions in Standard and Verte Form Identify the Verte, Ais of Symmetry, and Intercepts of a Quadratic Function Find the Maimum or

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

Quadratic Graphs and Their Properties

Quadratic Graphs and Their Properties - Think About a Plan Quadratic Graphs and Their Properties Physics In a physics class demonstration, a ball is dropped from the roof of a building, feet above the ground. The height h (in feet) of the

More information

Laurie s Notes. Overview of Section 3.5

Laurie s Notes. Overview of Section 3.5 Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.

More information

Comparing Linear, Exponential, and Quadratic Functions

Comparing Linear, Exponential, and Quadratic Functions . Comparing Linear, Eponential, and Quadratic Functions How can ou compare the growth rates of linear, eponential, and quadratic functions? ACTIVITY: Comparing Speeds Work with a partner. Three cars start

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions.1 Eponential Growth and Deca Functions. The Natural Base e.3 Logarithms and Logarithmic Functions. Transformations of Eponential and Logarithmic Functions.5 Properties

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

Mini-Lecture 8.1 Solving Quadratic Equations by Completing the Square

Mini-Lecture 8.1 Solving Quadratic Equations by Completing the Square Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.

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

Skills Practice Skills Practice for Lesson 3.1

Skills Practice Skills Practice for Lesson 3.1 Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Define each term in our own words.. quadratic function. vertical motion Problem

More information

BIG IDEAS MATH. Ron Larson Laurie Boswell. Sampler

BIG IDEAS MATH. Ron Larson Laurie Boswell. Sampler BIG IDEAS MATH Ron Larson Laurie Boswell Sampler 3 Polnomial Functions 3.1 Graphing Polnomial Functions 3. Adding, Subtracting, and Multipling Polnomials 3.3 Dividing Polnomials 3. Factoring Polnomials

More information

Quadratic Equations and Complex Numbers

Quadratic Equations and Complex Numbers Quadratic Equations and Comple Numbers.1 Solving Quadratic Equations. Comple Numbers.3 Completing the Square. Using the Quadratic Formula.5 Solving Nonlinear Sstems. Quadratic Inequalities Robot-Building

More information

Name Class Date. Identify the vertex of each graph. Tell whether it is a minimum or a maximum.

Name Class Date. Identify the vertex of each graph. Tell whether it is a minimum or a maximum. Practice Quadratic Graphs and Their Properties Identify the verte of each graph. Tell whether it is a minimum or a maimum. 1. y 2. y 3. 2 4 2 4 2 2 y 4 2 2 2 4 Graph each function. 4. f () = 3 2 5. f ()

More information

) approaches e

) approaches e COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural

More information

3.2. Properties of Graphs of Quadratic Relations. LEARN ABOUT the Math. Reasoning from a table of values and a graph of a quadratic model

3.2. Properties of Graphs of Quadratic Relations. LEARN ABOUT the Math. Reasoning from a table of values and a graph of a quadratic model 3. Properties of Graphs of Quadratic Relations YOU WILL NEED grid paper ruler graphing calculator GOAL Describe the ke features of the graphs of quadratic relations, and use the graphs to solve problems.

More information

Laurie s Notes. Overview of Section 2.4

Laurie s Notes. Overview of Section 2.4 Overview of Section 2. Introduction The goal in this lesson is for students to create quadratic equations to represent the relationship between two quantities (HSA-CED.A.2). From the contet given, students

More information

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

More information

Essential Question How can you determine the number of solutions of a linear system?

Essential Question How can you determine the number of solutions of a linear system? .1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.3.A A.3.B Solving Linear Sstems Using Substitution Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent

More information

MATH 115: Review for Chapter 3

MATH 115: Review for Chapter 3 MATH : Review for Chapter Can ou use the Zero-Product Propert to solve quadratic equations b factoring? () Solve each equation b factoring. 6 7 8 + = + ( ) = 8 7p ( p ) p ( p) = = c = c = + Can ou solve

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

Completing the Square

Completing the Square 3.5 Completing the Square Essential Question How can you complete the square for a quadratic epression? Using Algera Tiles to Complete the Square Work with a partner. Use algera tiles to complete the square

More information

Linear Functions. Essential Question How can you determine whether a function is linear or nonlinear?

Linear Functions. Essential Question How can you determine whether a function is linear or nonlinear? . Linear Functions Essential Question How can ou determine whether a function is linear or nonlinear? Finding Patterns for Similar Figures Work with a partner. Cop and complete each table for the sequence

More information

Rational Exponents and Radical Functions

Rational Exponents and Radical Functions .1..... Rational Eponents and Radical Functions nth Roots and Rational Eponents Properties of Rational Eponents and Radicals Graphing Radical Functions Solving Radical Equations and Inequalities Performing

More information

Inverse of a Function

Inverse of a Function . Inverse o a Function Essential Question How can ou sketch the graph o the inverse o a unction? Graphing Functions and Their Inverses CONSTRUCTING VIABLE ARGUMENTS To be proicient in math, ou need to

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

Solving Quadratic Equations by Graphing 9.1. ACTIVITY: Solving a Quadratic Equation by Graphing. How can you use a graph to solve a quadratic

Solving Quadratic Equations by Graphing 9.1. ACTIVITY: Solving a Quadratic Equation by Graphing. How can you use a graph to solve a quadratic 9. Solving Quadratic Equations b Graphing equation in one variable? How can ou use a graph to solve a quadratic Earlier in the book, ou learned that the -intercept of the graph of = a + b variables is

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

Quadratics in Vertex Form Unit 1

Quadratics in Vertex Form Unit 1 1 U n i t 1 11C Date: Name: Tentative TEST date Quadratics in Verte Form Unit 1 Reflect previous TEST mark, Overall mark now. Looking back, what can ou improve upon? Learning Goals/Success Criteria Use

More information

Mathematics 2201 Midterm Exam Review

Mathematics 2201 Midterm Exam Review Mathematics 0 Midterm Eam Review Chapter : Radicals Chapter : Quadratic Functions Chapter 7: Quadratic Equations. Evaluate: 8 (A) (B) (C) (D). Epress as an entire radical. (A) (B) (C) (D). What is 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

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

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

Solving Polynomial Equations 3.5. Essential Question How can you determine whether a polynomial equation has a repeated solution?

Solving Polynomial Equations 3.5. Essential Question How can you determine whether a polynomial equation has a repeated solution? 3. Solving Polynomial Equations Essential Question Essential Question How can you determine whether a polynomial equation has a repeated solution? Cubic Equations and Repeated Solutions USING TOOLS STRATEGICALLY

More information

TRANSFORMATIONS OF f(x) = x Example 1

TRANSFORMATIONS OF f(x) = x Example 1 TRANSFORMATIONS OF f() = 2 2.1.1 2.1.2 Students investigate the general equation for a famil of quadratic functions, discovering was to shift and change the graphs. Additionall, the learn how to graph

More information

Solving Linear Systems

Solving Linear Systems 1.4 Solving Linear Sstems Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent when it has at least one solution. A linear sstem is inconsistent

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

Essential Question How can you determine whether a polynomial equation has imaginary solutions? 2 B. 4 D. 4 F.

Essential Question How can you determine whether a polynomial equation has imaginary solutions? 2 B. 4 D. 4 F. 5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS 2A.7.A The Fundamental Theorem of Algebra Essential Question How can ou determine whether a polnomial equation has imaginar solutions? Cubic Equations and Imaginar

More information

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology. Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph

More information

CHAPTER 2. Polynomial Functions

CHAPTER 2. Polynomial Functions CHAPTER Polynomial Functions.1 Graphing Polynomial Functions...9. Dividing Polynomials...5. Factoring Polynomials...1. Solving Polynomial Equations...7.5 The Fundamental Theorem of Algebra...5. Transformations

More information

Solving Quadratic Equations: Algebraically and Graphically Read 3.1 / Examples 1 4

Solving Quadratic Equations: Algebraically and Graphically Read 3.1 / Examples 1 4 CC Algebra II HW #14 Name Period Row Date Solving Quadratic Equations: Algebraically and Graphically Read 3.1 / Examples 1 4 Section 3.1 In Exercises 3 12, solve the equation by graphing. (See Example

More information

9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON

9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON CONDENSED LESSON 9.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations solve

More information

Mathematics 2201 Midterm Exam Review

Mathematics 2201 Midterm Exam Review Mathematics 0 Midterm Eam Review Chapter : Radicals Chapter 6: Quadratic Functions Chapter 7: Quadratic Equations. Evaluate: 6 8 (A) (B) (C) (D). Epress as an entire radical. (A) (B) (C) (D). What is the

More information

How can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines

How can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines .7 Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the

More information

Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest. Minimum value the least. Parabola the set of points in a

Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest. Minimum value the least. Parabola the set of points in a - Attributes and Transformations of Quadratic Functions TEKS FCUS VCABULARY TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction

More information

136 Maths Quest 10 for Victoria

136 Maths Quest 10 for Victoria Quadratic graphs 5 Barr is a basketball plaer. He passes the ball to a team mate. When the ball is thrown, the path traced b the ball is a parabola. Barr s throw follows the quadratic equation =.5 +. +.5.

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

Essential Question How can you cube a binomial? Work with a partner. Find each product. Show your steps. = (x + 1) Multiply second power.

Essential Question How can you cube a binomial? Work with a partner. Find each product. Show your steps. = (x + 1) Multiply second power. 4.2 Adding, Subtracting, and Multiplying Polynomials COMMON CORE Learning Standards HSA-APR.A.1 HSA-APR.C.4 HSA-APR.C.5 Essential Question How can you cube a binomial? Cubing Binomials Work with a partner.

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

How can you construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot

How can you construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot 9. Scatter Plots How can ou construct and interpret a scatter plot? ACTIVITY: Constructing a Scatter Plot Work with a partner. The weights (in ounces) and circumferences C (in inches) of several sports

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

Quadratic Functions ESSENTIAL QUESTIONS EMBEDDED ASSESSMENTS

Quadratic Functions ESSENTIAL QUESTIONS EMBEDDED ASSESSMENTS Quadratic Functions 5 01 College Board. All rights reserved. Unit Overview In this unit ou will stud a variet of was to solve quadratic functions and sstems of equations and appl our learning to analzing

More information

4.4 Scatter Plots and Lines of Fit 4.5 Analyzing Lines of Fit 4.6 Arithmetic Sequences 4.7 Piecewise Functions

4.4 Scatter Plots and Lines of Fit 4.5 Analyzing Lines of Fit 4.6 Arithmetic Sequences 4.7 Piecewise Functions Writing Linear Functions. Writing Equations in Slope-Intercept Form. Writing Equations in Point-Slope Form.3 Writing Equations of Parallel and Perpendicular Lines. Scatter Plots and Lines of Fit.5 Analzing

More information

Chapter 11 Quadratic Functions

Chapter 11 Quadratic Functions Chapter 11 Quadratic Functions Mathematical Overview The relationship among parabolas, quadratic functions, and quadratic equations is investigated through activities that eplore both the geometric and

More information

Polynomial vs. Non-Polynomial Functions Even vs. Odd Functions; End Behavior Read 4.1 Examples 1-3

Polynomial vs. Non-Polynomial Functions Even vs. Odd Functions; End Behavior Read 4.1 Examples 1-3 HW # Name Period Row Date Polynomial vs. Non-Polynomial Functions Even vs. Odd Functions; End Behavior Read.1 Eamples 1- Section.1. Which One Doesn't Belong? Which function does not belong with the other

More information

2 nd Semester Final Exam Review Block Date

2 nd Semester Final Exam Review Block Date Algebra 1B Name nd Semester Final Eam Review Block Date Calculator NOT Allowed Graph each function. Identif the verte and ais of smmetr. 1 (10-1) 1. (10-1). 3 (10-) 3. 4 7 (10-) 4. 3 6 4 (10-1) 5. Predict

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations 5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving

More information

6.3 Interpreting Vertex Form and Standard Form

6.3 Interpreting Vertex Form and Standard Form 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

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

c) domain {x R, x 3}, range {y R}

c) domain {x R, x 3}, range {y R} Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..

More information

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction

UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction Prerequisite Skills This lesson requires the use of the following skills: knowing the standard form of quadratic functions using graphing technolog to model quadratic functions Introduction The tourism

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

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