4.1 Circles. Explore Deriving the Standard-Form Equation
|
|
- Vincent Bailey
- 5 years ago
- Views:
Transcription
1 COMMON CORE r Locker LESSON Circles.1 Name Class Date.1 Circles Common Core Math Standards The student is epected to: COMMON CORE A-CED.A.3 Represent constraints b equations or inequalities,... and interpret solutions as viable or nonviable options in a modeling contet. Also A-CED.A., G-GPE.A.1, G-GPE.B. Mathematical Practices COMMON CORE MP.7 Using Structure Language Objective Work with a partner to match graphs of circles to their equations in standard form. ENGAGE Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? Possible answer: The standard form for the equation of a circle is ( - h) + ( - k) = r, which tells ou that the center is (h, k) and the radius is r. PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo and the generall circular nature of radio-signal reception strength. Then preview the Lesson Performance Task. Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? Eplore Deriving the Standard-Form Equation of a Circle Recall that a circle is the set of points in a plane that are a fied distance, called the radius, from a given point, called the center. A B C The coordinate plane shows a circle with center C(h, k) and radius r. P(, ) is an arbitrar point on the circle but is not directl above or below or to the left or right of C. A(, k) is a point with the same -coordinate as P and the same -coordinate as C. Eplain wh CAP is a right triangle. Since point A has the same -coordinate as point P, segment PA is a vertical segment. Since point A has the same -coordinate as point C, segment CA is a horizontal segment. This means that segments PA and CA are perpendicular, which means that CAP is a right angle and CAP is a right triangle. Identif the lengths of the sides of CAP. Remember that point P is arbitrar, so ou cannot rel upon the diagram to know whether the -coordinate of P is greater than or less than h or whether the -coordinate of P is greater than or less than k, so ou must use absolute value for the lengths of the legs of CAP. Also, remember that the length of the hpotenuse of CAP is just the radius of the circle. - h The length of segment AC is. - k The length of segment AP is. The length of segment CP is r. Appl the Pthagorean Theorem to CAP to obtain an equation of the circle. ( - h ) + ( - k ) = r r C (h, k) A (, k) Resource Locker P (, ) Module 159 Lesson 1 Name Class Date.1 Circles Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? A-CED.A.3 Represent constraints b equation or inequalities,... and interpret solutions as viable or nonviable options in a modeling contet. Also A-CED.A., G-GPE.A.1, G-GPE.B. Eplore Deriving the Standard-Form Equation of a Circle Recall that a circle is the set of points in a plane that are a fied distance, called the radius, from a given point, called the center. The coordinate plane shows a circle with center C(h, k) and radius r. P(, ) is an arbitrar point on the circle but is not directl above or below or to the left or right of C. A(, k) is a point with the same -coordinate as P and the same -coordinate as C. Eplain wh CAP is a right triangle. Since point A has the same -coordinate as point P, segment PA is a vertical segment. Since point A has the same -coordinate as point C, segment CA is a horizontal segment. This means that segments PA and CA are perpendicular, which means that CAP is a right angle and CAP is a right triangle. Identif the lengths of the sides of CAP. Remember that point P is arbitrar, so ou cannot rel upon the diagram to know whether the -coordinate of P is greater than or less than h or whether the -coordinate of P is greater than or less than k, so ou must use absolute value for the lengths of the legs of CAP. Also, remember that the length of the hpotenuse of CAP is just the radius of the circle. The length of segment AC is. The length of segment AP is. The length of segment CP is. Appl the Pthagorean Theorem to CAP to obtain an equation of the circle. = ( - ) + ( - ) - h - k r h k r Resource P (, ) C (h, k) A (, k) Module 159 Lesson 1 HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 159 Lesson.1
2 Reflect 1. Discussion Wh isn t absolute value used in the equation of the circle? Since squaring removes an negative signs just as absolute value does, there s no need to take absolute value before squaring.. Discussion Wh does the equation of the circle also appl to the cases in which P has the same -coordinate as C or the same -coordinate as C so that CAP doesn t eist? If P has the same -coordinate as C, then P s -coordinate must be either k + r or k - r. So, ( - h) + ( - k) = (h - h) + ( (k ± r) - k) = 0 + (±r) = r, and the equation of the circle is still satisfied. Similarl, if P has the same -coordinate as C, then P s -coordinate must be either h + r or h - r. So, ( - h) + ( - k) = ((h ± r) - h) + (k - k) = (±r) + 0 = r, and the equation of the circle is still satisfied. EXPLORE Deriving the Standard-Form Equation of a Circle INTEGRATE TECHNOLOGY Students have the option of completing the Eplore activit either in the book or online. Eplain 1 Writing the Equation of a Circle The standard-form equation of a circle with center C(h, k) and radius r is ( - h) + ( - k) = r. If ou solve this equation for r, ou obtain the equation r = ( -h) + ( - k), which gives ou a means for finding the radius of a circle when the center and a point P(, ) on the circle are known. Eample 1 Write the equation of the circle. The circle with center C(-3, ) and radius r = Substitute -3 for h, for k, and for r into the general equation and simplif. ( - (-3) ) + ( - ) = ( + 3) + ( - ) = 16 The circle with center C(, -3) and containing the point P(, 5) Step 1 Find the radius. r = CP = ( - ()) 5 = ( 6 ) + ( 8 ) = = _ 100 = 10 Step Write the equation of the circle. ( - () ) + ( - (-3) ) ( + ) + ( + 3) = + ( - (-3)) = Module 160 Lesson 1 Wh does point A have coordinates (, k)? It is below P (, ), which has -coordinate, and on the same horizontal line as C (h, k), which has -coordinate k. Wh is it necessar to use absolute value signs when representing the length of the legs of the right triangle? Since P could be an point on the circle, absolute value signs are used to make sure the length of each leg is a positive number. EXPLAIN 1 Writing the Equation of a Circle AVOID COMMON ERRORS Some students ma forget to square the radius when writing the equation. Others ma take its square root. Help students to avoid making these errors b having them write r above the place in the equation where the need to write the square of the radius. PROFESSIONAL DEVELOPMENT Math Background The equation of a circle is based on the fact that all of the points on the circle are a fied distance from a given point. This distance can be found using the Pthagorean Theorem. In the derivation, the fied distance, the radius, is represented b the hpotenuse of a right triangle that has one verte at the center of the circle, and one on the circle itself. Appling the Pthagorean Theorem produces the equation of the circle, ( - h) + ( - k) = r. Note that taking the square root of each side of this equation produces the equation ( - h) + ( - k) = r. This equation shows that the radius is the distance between the two points. Do ou need to be given the coordinates of the center to write an equation of a circle? If not, what other information could ou use to find the center? Eplain. No. If ou know the endpoints of a diameter of the circle, ou can find the coordinates of the center using the Midpoint Formula. Circles 160
3 INTEGRATE MATHEMATICAL Focus on Math Connections MP.1 Discuss with students how the graph of the equation ( - h) + ( - k) = r is a transformation of the graph of + = r. Have students describe the transformation and compare the two graphs. EXPLAIN Rewriting an Equation of a Circle to Graph the Circle How do ou know what number to add to make perfect square trinomials when converting to standard form? Take half of the coefficient of the term, and square it. Then do the same with the coefficient of the term. Once the equation is in standard form, how do ou find the diameter of the circle? The diameter is times the square root of the number that represents r. Your Turn Write the equation of the circle. 3. The circle with center C(1, ) and radius r = ( - 1) + ( - () ) =. The circle with center C(-, 5) and containing the point P(-, -1) Eplain Rewriting an Equation of a Circle to Graph the Circle Epanding the standard-form equation ( - h) + ( - k) = r results in a general second-degree equation in two variables having the form + + c + d + e = 0. In order to graph such an equation or an even more general equation of the form a + a + c + d + e = 0. ou must complete the square on both and to put the equation in standard form and identif the circle s center and radius. Eample ( - 1) + ( + ) = Because points C and P have the same -coordinate, the radius of the circle is just the absolute value of the difference of their -coordinates, so r = 5 - (-1) = 6 = 6. ( - (-) ) + ( - 5) = 6 ( + ) + ( - 5) = 36 Graph the circle after writing the equation in standard form = 0 Write the equation = 0 Prepare to complete the square on and. Complete both squares. ( ) + ( ) = Factor and simplif. ( - 5) + ( + 3) = The center of the circle is C(5, -3), and the radius is r = _ =. Graph the circle. ( ) + ( ) = Module 161 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Instruct each student in each pair to write an equation of a circle in standard form. Have them graph the circles, keeping their graphs hidden from their partners. Have them also convert their equations to the general form a + b + c + d + e = 0 b epanding and combining like terms. Instruct students to echange the general forms of their equations, write each other s equations in standard form, and draw the graph. Have students compare their work. 161 Lesson.1
4 B = 0 Write the equation = 0 Factor from the terms ( + ) + ( - ) + 11 = 0 and the terms. Prepare to complete ( + + ) + ( - + ) = ( ) + ( ) the square on and. Complete both ( ) + ( - + ) = ( 1 ) + ( ) squares. Factor and simplif. ( + 1 ) + ( - ) = Divide both sides b. ( + 1 ) + ( - ) = The center is C (, ), and the radius is r = _ =. Graph the circle _ 3_ 9 9_ AVOID COMMON ERRORS When adding the numbers to the constant term to maintain the equalit, students ma forget to multipl the number added to complete each square b the leading coefficient that was factored from the variable terms. Help them to avoid this error b encouraging them to circle the leading coefficients that have been factored out in the process of completing the square. INTEGRATE MATHEMATICAL Focus on Critical Thinking MP.3 Have students compare how the completing-the-square process is used to write quadratic functions in verte form with how it is used to write a circle in standard form. Have them describe both the similarities and the differences. Your Turn Graph the circle after writing the equation in standard form = = 0 ( + ) + ( + 6) = ( + + ) + ( ) = ( + ) + ( + 3) = 9 The center is C(-, -3), and the radius is r = 9 = = = 0 9 ( - 6) + 9 ( - 8) = ( ) + 9 ( ) = (9) + 9 (16) ( - 3) + 9 ( - ) = 16 16_ ( - 3) + ( - ) = 9 - The center is C (3, ), and the radius is r = _ 16 _ 9 = 3. Module 16 Lesson DIFFERENTIATE INSTRUCTION Kinesthetic Eperience Prepare a length of string with a marker attached at one end. Displa a coordinate plane. With a tack, tape, or some other means, attach the non-marker end of the string to an arbitrar point on the plane and draw a circle. Ask students to eplain how the could use a point on the circle and the center to find the length of the string. Help them to see that the can construct a right triangle whose hpotenuse is the length of the string, and appl the Pthagorean Theorem to determine the length of the string. Circles 16
5 EXPLAIN 3 Solving a Real-World Problem Involving a Circle If ou know the equation of a circle, how can ou determine whether a given point lies inside, outside, or on the circle? You can substitute the coordinates of the point for and in the equation of the circle, and see whether the value of ( - h) + ( - k) is less than, greater than, or equal to the value of r. If it is less than r, the point lies inside the circle; if greater than, the point lies outside the circle; and if equal to, the point lies on the circle. INTEGRATE MATHEMATICAL Focus on Reasoning MP. Ask students to eplain wh points that are inside the circle satisf the inequalit ( - h) + ( - k) < r. Students should recognize that a point inside the circle would lie on a circle whose radius would be shorter than r, and whose equation would be ( - h) + ( - k) = r 1, with r 1 < r. Thus, r 1 < r and ( - h) + ( - k) < r. Image Credits: Jose Luis Pelaez/Corbis Eplain 3 Solving a Real-World Problem Involving a Circle A circle in a coordinate plane divides the plane into two regions: points inside the circle and points outside the circle. Points inside the circle satisf the inequalit ( - h) + ( - k) < r, while points outside the circle satisf the inequalit ( - h) + ( - k) > r. Eample 3 Write an inequalit representing the given situation, and draw a circle to solve the problem. The table lists the locations of the homes of five friends along with the locations of their favorite pizza restaurant and the school the attend. The friends are deciding where to have a pizza part based on the fact that the restaurant offers free deliver to locations within a 3-mile radius of the restaurant. At which homes should the friends hold their pizza part to get free deliver? Place Location Alonzo s home A(3, ) Barbara s home B(, ) Constance s home C(-, 3) Dion s home D(0, -1) Eli s home E(1, ) Pizza restaurant (-1, 1) School (1, -) Write the equation of the circle with center (-1, 1) and radius 3. ( - (-1)) + ( - 1) = 3, or ( + 1) + ( - 1) = 9 The inequalit ( + 1) + ( - 1) < 9 represents the situation. Plot the points from the table and graph the circle. The points inside the circle satisf the inequalit. So, the friends should hold their pizza part at either Constance s home or Dion s home to get free deliver. B C A Restaurant - 0 D - School E In order for a student to ride the bus to school, the student must live more than miles from the school. Which of the five friends are eligible to ride the bus? Write the equation of the circle with center ( 1, - ) and radius. ( - 1 ) + ( - ( - )) = ( - 1 ) + ( + ) = The inequalit ( - 1 ) + ( + ) > represents the situation. Use the coordinate grid in Part A to graph the circle. The points outside the circle satisf the inequalit. So, Alonzo, Barbara, and Constance are eligible to ride the bus. Module 163 Lesson 1 LANGUAGE SUPPORT Communicate Math Have students work in pairs. Provide each pair of students with different graphs of circles and, on separate note cards or sheets of paper, the equations for those graphs. The first student chooses a graph and decides which equation goes with it, then eplains wh the are a match. The second student repeats the procedure using another graph and equation. 163 Lesson.1
6 Reflect 7. For Part B, how do ou know that point E isn t outside the circle? The coordinates of point E are (1, ). Substituting 1 for and for in ( - 1) + ( + ) gives (1-1) + ( + ) = 0 + (-) =, so the coordinates of E satisf the equation of the circle, which means that E is on the circle and not outside it. Your Turn Write an inequalit representing the given situation, and draw a circle to solve the problem. 8. Sasha delivers newspapers to subscribers that live within a -block radius of her house. Sasha s house is located at point (0, -1). Points A, B, C, D, and E represent the houses of some of the subscribers to the newspaper. To which houses does Sasha deliver newspapers? ( - 0) + ( - (-1)) = Elaborate + ( + 1) = 16 The inequalit + ( + 1) < 16 represents the situation. The points inside the circle satisf the inequalit + ( + 1) <16. So, Sasha delivers to the houses located at points B, D, and E. C A B - 0 Sasha - D E 9. Describe the process for deriving the equation of a circle given the coordinates of its center and its radius. First, choose an arbitrar point P on the circle. Net, find a third point A that forms a right triangle with points C and P. Then, use the coordinates of the three points to find the lengths of segments CA and PA. (The length of segment CP is the circle s radius.) Finall, use the Pthagorean Theorem to write an equation of the circle. ELABORATE How is the equation of a circle related to the equation a + b = c from the Pthagorean Theorem? The radius is c, and the lengths of the legs of the right triangle that has the radius as its hpotenuse are a and b. INTEGRATE MATHEMATICAL Focus on Reasoning MP. Discuss with students wh, in the equation a + b + c + d + e = 0, a and b must be equal for the equation to be that of a circle. Focus their attention on the steps involved in converting the equation to the standard form of a circle, and on the roles of a and b in the conversion. 10. What must ou do with the equation a + a + c + d + e = 0 in order to graph it? Complete the square on and to write the equation in standard form. From the standard-form equation ou can then identif the circle s center and radius, which ou can then use to graph the circle. 11. What do the inequalities ( - h) + ( - k) < r and ( - h) + ( - k) > r represent? The inequalit ( - h) + ( - k) < r represents points inside the circle with equation ( - h) + ( - k) = r, and the inequalit ( - h) + ( - k) > r represents points outside the circle. 1. Essential Question Check-In What information must ou know or determine in order to write an equation of a circle in standard form? You must know the center of the circle and its radius to write an equation of the circle in standard form. If onl the center and a point on the circle are known, ou can determine the radius from those two points. SUMMARIZE THE LESSON How can ou write the equation of a circle? You can use the coordinates of the center for h and k, and the radius for r, in the equation ( - h) + ( - k) = r. Module 16 Lesson 1 Circles 16
4.1 Circles. Deriving the Standard-Form Equation of a Circle. Explore
Name Class Date 4.1 Circles ssential Question: What is the standard form for the equation of a circle, and what does the standard form tell ou about the circle? plore Deriving the Standard-Form quation
More information4.2 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 information10.1 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 information15.4 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 informationDomain, 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 information11.1 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 informationThe standard form of the equation of a circle is based on the distance formula. The distance formula, in turn, is based on the Pythagorean Theorem.
Unit, Lesson. Deriving the Equation of a Circle The graph of an equation in and is the set of all points (, ) in a coordinate plane that satisf the equation. Some equations have graphs with precise geometric
More informationLaurie 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 information15.2 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 information6.2 Multiplying Polynomials
Locker LESSON 6. Multiplying Polynomials PAGE 7 BEGINS HERE Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? Common
More informationFinding 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 informationEssential 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 informationEssential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically
6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one
More informationAdding and Subtracting Rational Expressions
COMMON CORE Locker LESSON 9.1 Adding and Subtracting Rational Epressions Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions?
More informationEssential Question: What is a complex number, and how can you add, subtract, and multiply complex numbers? Explore Exploring Operations Involving
Locker LESSON 3. Complex Numbers Name Class Date 3. Complex Numbers Common Core Math Standards The student is expected to: N-CN. Use the relation i = 1 and the commutative, associative, and distributive
More information7.2 Multiplying Polynomials
Locker LESSON 7. Multiplying Polynomials Teas Math Standards The student is epected to: A.7.B Add, subtract, and multiply polynomials. Mathematical Processes A.1.E Create and use representations to organize,
More information13.2 Exponential Decay Functions
6 6 - - Locker LESSON. Eponential Deca Functions Common Core Math Standards The student is epected to: F.BF. Identif the effect on the graph of replacing f() b f() + k, kf(), f(k), and f( + k) for specific
More information13.3 Exponential Decay Functions
6 6 - - Locker LESSON. Eponential Deca Functions Teas Math Standards The student is epected to: A.5.B Formulate eponential and logarithmic equations that model real-world situations, including eponential
More information2.3. Solving Absolute Value Inequalities. Inequalities ENGAGE. 2.3 Solving Absolute Value
Resource Locker LESSO N 2.3 Solving Absolute Value Inequalities Name Class Date 2.3 Solving Absolute Value Inequalities Texas Math Standards The student is expected to: A2.6.F Solve absolute value linear
More informationAre You Ready? Find Area in the Coordinate Plane
SKILL 38 Are You Read? Find Area in the Coordinate Plane Teaching Skill 38 Objective Find the areas of figures in the coordinate plane. Review with students the definition of area. Ask: Is the definition
More information7.1 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 information20.2 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 information11.3 Solving Radical Equations
Locker LESSON 11. Solving Radical Equations Common Core Math Standards The student is expected to: A-REI. Solve simple rational and radical equations in one variable, and give examples showing how extraneous
More informationUNIT 6 MODELING GEOMETRY Lesson 1: Deriving Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: appling the Pthagorean Theorem representing horizontal and vertical distances in a coordinate plane simplifing square roots writing
More informationf(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 information8.2 Graphing More Complicated Rational Functions
1 Locker LESSON 8. Graphing More Complicated Rational Functions PAGE 33 Name Class Date 8. Graphing More Complicated Rational Functions Essential Question: What features of the graph of a rational function
More information10.2 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 informationExplore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x.
1_ 8 6 8 Locker LESSON 13. The Base e Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes of the graphs of ƒ () = b and ƒ () = log b () where b is, 1, and e
More information2.1 The Rectangular Coordinate System
. The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table
More informationConic Section: Circles
Conic Section: Circles Circle, Center, Radius A circle is defined as the set of all points that are the same distance awa from a specific point called the center of the circle. Note that the circle consists
More informationCourse 15 Numbers and Their Properties
Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.
More information2.4 Investigating Symmetry
Locker LESSON 2.4 Investigating Symmetry Texas Math Standards The student is expected to: G.3.D Identify and distinguish between reflectional and rotational symmetry in a plane figure. Mathematical Processes
More informationName 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 information7.2 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 information3.1 Solving Quadratic Equations by Taking Square Roots
COMMON CORE -8-16 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary
More information12.2 Simplifying Radical Expressions
x n a a m 1 1 1 1 Locker LESSON 1. Simplifying Radical Expressions Texas Math Standards The student is expected to: A.7.G Rewrite radical expressions that contain variables to equivalent forms. Mathematical
More informationName 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 informationMORE TRIGONOMETRY
MORE TRIGONOMETRY 5.1.1 5.1.3 We net introduce two more trigonometric ratios: sine and cosine. Both of them are used with acute angles of right triangles, just as the tangent ratio is. Using the diagram
More informationIB MATH STUDIES.
IB MATH STUDIES We are so ecited that you have decided to embark upon an eciting journey through IB Math Studies. Make no mistake, the road ahead will be challenging and perhaps overwhelming at times.
More information6.3 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 information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationLESSON #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 informationName 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 information17.1 Understanding Polynomial Expressions
COMMON CORE 4 a b Locker x LESSON Common Core Math Standards The student is expected to: COMMON CORE A-SSE.A.a Interpret parts of an expression, such as terms, factors, and coefficients. Also A-SSE.A.b,
More informationThe Distance Formula & The Midpoint Formula
The & The Professor Tim Busken Mathematics Department Januar 14, 2015 Theorem ( : 1 dimension) If a and b are real numbers, then the distance between them on a number line is a b. a b : 2 dimensions Consider
More informationFinding Complex Solutions of Quadratic Equations
y - y - - - x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes
More informationUsing 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 informationLESSON 4.3 GRAPHING INEQUALITIES
LESSON.3 GRAPHING INEQUALITIES LESSON.3 GRAPHING INEQUALITIES 9 OVERVIEW Here s what ou ll learn in this lesson: Linear Inequalities a. Ordered pairs as solutions of linear inequalities b. Graphing linear
More informationCommon Core State Standards for Activity 14. Lesson Postal Service Lesson 14-1 Polynomials PLAN TEACH
Postal Service Lesson 1-1 Polynomials Learning Targets: Write a third-degree equation that represents a real-world situation. Graph a portion of this equation and evaluate the meaning of a relative maimum.
More informationEssential Question: How can you compare linear functions that are represented in different ways? Explore Comparing Properties of Linear Functions
Locker LESSON 6.5 Comparing Properties of Linear Functions Common Core Math Standards The student is epected to: F-IF.9 Compare properties of two functions each represented in a different wa (algebraicall,
More information4 Linear Functions 45
4 Linear Functions 45 4 Linear Functions Essential questions 1. If a function f() has a constant rate of change, what does the graph of f() look like? 2. What does the slope of a line describe? 3. What
More informationLESSON #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 informationmentoringminds.com MATH LEVEL 6 Student Edition Sample Page Unit 33 Introduction Use the coordinate grid to answer questions 1 9.
Student Edition Sample Page Name Standard 6.11(A) Readiness Unit 33 Introduction Use the coordinate grid to answer questions 1 9. A 6 F 5 L E 4 3 I 1 B K 6 5 4 3 1 1 3 4 5 6 1 H C D 3 G 4 5 J 6 1 Which
More informationEssential 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 informationLinear Equation Theory - 2
Algebra Module A46 Linear Equation Theor - Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED June., 009 Linear Equation Theor - Statement of Prerequisite
More informationSummer Math Packet (revised 2017)
Summer Math Packet (revised 07) In preparation for Honors Math III, we have prepared a packet of concepts that students should know how to do as these concepts have been taught in previous math classes.
More informationFunctions. 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 informationIn order to take a closer look at what I m talking about, grab a sheet of graph paper and graph: y = x 2 We ll come back to that graph in a minute.
Module 7: Conics Lesson Notes Part : Parabolas Parabola- The parabola is the net conic section we ll eamine. We talked about parabolas a little bit in our section on quadratics. Here, we eamine them more
More informationFunctions. 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 informationSolutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1
Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Part : Multiple Choice Questions. Here ou work out the problems and then select the answer that matches our answer. No partial credit is given
More informationConic Sections CHAPTER OUTLINE. The Circle Ellipses and Hyperbolas Second-Degree Inequalities and Nonlinear Systems FIGURE 1
088_0_p676-7 /7/0 :5 PM Page 676 (FPG International / Telegraph Colour Librar) Conic Sections CHAPTER OUTLINE. The Circle. Ellipses and Hperbolas.3 Second-Degree Inequalities and Nonlinear Sstems O ne
More information5.2 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 information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationTrigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric
Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric
More information4.1 Identifying and Graphing Sequences
Name Class Date 4.1 Identifing and Graphing Sequences Essential Question: What is a sequence and how are sequences and functions related? Resource Locker Eplore Understanding Sequences A go-kart racing
More informationUNCORRECTED 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 informationAlgebra 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 informationACTIVITY: Using a Table to Plot Points
.5 Graphing Linear Equations in Standard Form equation a + b = c? How can ou describe the graph of the ACTIVITY: Using a Table to Plot Points Work with a partner. You sold a total of $6 worth of tickets
More informationTable of Contents. Unit 6: Modeling Geometry. Answer Key...AK-1. Introduction... v
These materials ma not be reproduced for an purpose. The reproduction of an part for an entire school or school sstem is strictl prohibited. No part of this publication ma be transmitted, stored, or recorded
More informationUse the coordinate plane provided to answer each question. y
Warm Up Use the coordinate plane provided to answer each question. 1. Plot points A (, ) and B (, ).. Is the distance between points A and B considered a horizontal distance, a vertical distance, or a
More informationApplications. 12 The Shapes of Algebra. 1. a. Write an equation that relates the coordinates x and y for points on the circle.
Applications 1. a. Write an equation that relates the coordinates and for points on the circle. 1 8 (, ) 1 8 O 8 1 8 1 (13, 0) b. Find the missing coordinates for each of these points on the circle. If
More informationAlgebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.
Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and
More information11.1 Double Riemann Sums and Double Integrals over Rectangles
Chapter 11 Multiple Integrals 11.1 ouble Riemann Sums and ouble Integrals over Rectangles Motivating Questions In this section, we strive to understand the ideas generated b the following important questions:
More information4.1 Understanding Polynomial Expressions
4 a b Locker x LESSON 4. Understanding Polynomial Expressions Common Core Math Standards The student is expected to: A-SSE.a Interpret parts of an expression, such as terms, factors, and coefficients.
More information10.1 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 information2.3 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 information16.2 Solving Exponential Equations
Locker LESSON 16.2 Solving Exponential Equations Texas Math Standards The student is expected to: A2.5.D Solve exponential equations of the form y = ab x where a is a nonzero real number and b is greater
More informationReteaching (continued)
Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression
More informationName Date Class Period. pencil straightedge graph paper How can you relate slope, y-intercept, and an equation?
Name Date Class Period Activit 8.5 Investigating Slope-Intercept Form MATERIALS QUESTION pencil straightedge graph paper How can ou relate slope, -intercept, and an equation? You can find the slope and
More informationSystems of Linear Inequalities
. Sstems of Linear Inequalities sstem of linear inequalities? How can ou sketch the graph of a ACTIVITY: Graphing Linear Inequalities Work with a partner. Match the linear inequalit with its graph. + Inequalit
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More information8.2 Finding Complex Solutions of Polynomial Equations
Locker LESSON 8. Finding Complex Solutions of Polynomial Equations Texas Math Standards The student is expected to: A.7.D Determine the linear factors of a polynomial function of degree three and of degree
More informationMethods for Advanced Mathematics (C3) Coursework Numerical Methods
Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on
More information15.2 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 informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationMaintaining 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 informationMath Intermediate Algebra
Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and
More information9-1. 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 information3.1. 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 information13.2 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 informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More information10.2 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 information2.3 Solving Absolute Value Inequalities
Name Class Date.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
More informationMHF 4U Unit 1 Polynomial Functions Outline
MHF 4U Unit 1 Polnomial Functions Outline Da Lesson Title Specific Epectations 1 Average Rate of Change and Secants D1., 1.6, both D1.1A s - Instantaneous Rate of Change and Tangents D1.6, 1.4, 1.7, 1.5,
More informationSummary and Vocabulary
Chapter 2 Chapter 2 Summar and Vocabular The functions studied in this chapter are all based on direct and inverse variation. When k and n >, formulas of the form = k n define direct-variation functions,
More informationStudy 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 information7.3 Triangle Inequalities
Name lass Date 7.3 Triangle Inequalities Essential Question: How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? Eplore G.5.D Verify the Triangle
More informationMath 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals
Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (
More informationFair Game Review. Chapter 10
Name Date Chapter 0 Evaluate the expression. Fair Game Review. 9 +. + 6. 8 +. 9 00. ( 9 ) 6. 6 ( + ) 7. 6 6 8. 9 6 x 9. The number of visits to a website can be modeled b = +, where is hundreds of visits
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More information