Ready To Go On? Skills Intervention 10-1 Introduction to Conic Sections

Transcription

1 Find this vocabular word in Lesson 10-1 and the Multilingual Glossar. Graphing Parabolas and Hperbolas on a Calculator A is a single curve, whereas a has two congruent branches. Identif and describe each conic section. A. 3 Read To Go On? Skills Intervention 10-1 Introduction to Conic Sections Solve for so that the epression can be graphed using a graphing calculator. 3 3 Subtract from each side. Take the square root of both sides. Graph the equations and to see the complete graph. The graph is a with verte (3, ) that opens to the. Vocabular conic section B Solve for so that the epression can be graphed using a graphing calculator Begin to isolate the -term Write the right side of the equation over the common denominator. Take the of each term. 16 Multipl each side b to solve for. Graph the equations 3 and 16 to see the complete graph. Copright b Holt, Rinehart and Winston. 16 Holt Algebra

2 Read To Go On? Problem Solving Intervention 10-1 Introduction to Conic Sections A circle, one of the four tpes of conic sections, is defined b its center and its radius. The no charge deliver area of a floral shop etends to the locations (1, 10) and (7, ). Write an equation for the no charge deliver area of the floral shop if a line between the locations represents the diameter of the deliver area. Understand the Problem 1. What is the shape of the deliver area?. What are the points to which the deliver area etends? 3. What does the line connecting these points represent?. What are ou being asked to find? Make a Plan 5. What is the midpoint formula for the segment with endpoints ( 1, 1 ) and (, )? 6. The midpoint formula will give ou the coordinates of what part of the circle? 7. What formula should ou use to find the radius of the circle? 8. Complete the formula for coordinates ( 1, 1 ) and (, ). d 9. What is the equation of a circle with center (h, k) and radius r? Solve 10. Find the midpoint of the coordinates (1, 10) and (7, ). 11. What is the center of the circle? 1. Use the center and the endpoint (1, 10) to find the radius using the distance formula, r (1 ) (10 6). What is the radius? 13. Using the center and the radius, write the equation of the circle that represents the no charge deliver area. Look Back 1. Use the center of the circle and the endpoint (7, ) to find the radius. 15. Is the radius the same as the radius in Eercise 1 using endpoint (7, )? Copright b Holt, Rinehart and Winston. 163 Holt Algebra

3 Read To Go On? Skills Intervention 10- Circles Find these vocabular words in Lesson 10- and the Multilingual Glossar. Vocabular circle tangent Writing the Equation of a Circle The of a circle is a fied distance connecting a point on a circle to the center of the circle. Since all points on a the center of the circle, ou can use the of a circle. Write the equation of each circle. A. center (5, 3) and radius r 6 Write the equation of a circle with center (h, k). ( ) ( ) r Substitute the given points into the equation of a circle. ( 5) ( ) are the same distance from to find the equation Let h 5, k and r 6. ( ) ( 3) Write the standard equation. B. center (1, 3) and containing point (, ) Use the distance formula to find the r ( 1 ) ( 1 ) of the circle. r (3) Substitute given values. r ( ) ( ) Simplif. r ( ) r ( h) ( ) Write the equation of a circle with center (h, k). ( ( )) ( (3)) Let h, k 3 and r. ( ) ( 3) Standard equation. Copright b Holt, Rinehart and Winston. 16 Holt Algebra

4 Read To Go On? Skills Intervention 10-3 Ellipses Find these vocabular words in Lesson 10-3 and the Multilingual Glossar. Vocabular ellipse foci (of an ellipse) major ais vertices of an ellipse minor ais co-vertices of an ellipse Using Standard Form to Write an Equation for an Ellipse Write the equation of an ellipse with foci (, 0) and vertices (6, 0). Step 1 Choose the appropriate form of equation. Is the horizontal or vertical ais longer? The appropriate form is: 1 a Step Identif the values of a and c. The verte gives the value of a, therefore a. The focus gives the value of c, therefore c. Step 3 Use the relationship c a b to find b. Step Write the equation b b b Graphing Ellipses Graph the ellipse 9( 1) ( 1) 36. Step 1 Write the equation in standard form. 9( 1) ( 1) 36 9( 1) ( 1) ( 1) ( 1) 1 9 Step Identif the values of h, k, a, and b. h is the -coordinate of the verte, h. k is the -coordinate of the verte, k. a 3 and b The major ais is vertical since 3. Step 3 The vertices are: (1, 1 3), or (1, ) and (1, ). The co-vertices are: (1, 1), or (, 1) and (, 1). Step Graph the ellipse. Copright b Holt, Rinehart and Winston. 165 Holt Algebra

5 Read To Go On? Problem Solving Intervention 10-3 Ellipses An elliptical arch under a bridge is constructed so that it is 60 feet wide and has a maimum height of 5 feet. Write an equation for a cross section of the bridge. Understand the Problem 1. What are ou being asked to do?. What is the width of the bridge? What is its maimum height? 5 ft 60 ft Make a Plan 3. What is the general form for the equation of an ellipse? 1. What is the length of the major ais? 5. The endpoints of the major ais are the of the ellipse. Solve 6. The vertices are halfwa from the center. What is half of 60? 7. What is the value of a? 8. The height of the bridge is the value of b. What is the value of b? 9. Substitute known values into the general form of the equation Look Back 10. Solve the equation for. The LCD is 56, , , , ,500 56, , Graph the function on a graphing calculator. Locate the maimum value of the ellipse:. Does this match our value for b? Copright b Holt, Rinehart and Winston. 166 Holt Algebra

6 Read To Go On? Skills Intervention 10- Hperbolas Find these vocabular words in Lesson 10- and the Multilingual Glossar. Vocabular hperbola foci (of a hperbola) branch (of a hperbola) transverse ais vertices of a hperbolas conjugate ais co-vertices of a hperbola Writing Equations of Hperbolas Write the equation of a hperbola with center (0, 0), verte (, 0), and endpoints of conjugate ais (0, 3). Step 1 Draw a rough sketch of the given information. Is the hperbola horizontal or vertical? The appropriate form of the equation is: 1 a Step Identif the values of a and b. The verte gives the value of a, therefore a. The conjugate ais gives the value of b, therefore b. Step 3 Write the equation. 1 9 Graphing Hperbolas ( 1 ) Graph the hperbola ( 1 ) Step 1 The equation is in standard form so the transverse ais is. Step Identif the values of h, k, a, and b. h is the -coordinate of the verte, h. k is the -coordinate of the verte, k. a and b c a b c 16 c c Step 3 The vertices are: (1, 1 ), or (1, ) and (1, ). The co-vertices are: (1 3, 1), or (, 1) and (, 1). The equation of the asmptotes are: k a ( h) b ( 1) 3 Step Draw a bo using the vertices and co-vertices. Draw the asmptotes through the corners of the bo. Step 5 Draw the hperbola using the vertices and asmptotes Copright b Holt, Rinehart and Winston. 167 Holt Algebra

7 Read To Go On? Skills Intervention 10-5 Parabolas Find these vocabular words in Lesson 10-5 and the Multilingual Glossar. Vocabular focus (of a parabola) directri Graphing Parabolas Find the verte, value of p, ais of smmetr, focus, and directri for the 1 parabola 16. Then graph. A parabola with a ais of smmetr opens upward or downward. A parabola with a ais of smmetr ma open to the left or right. 1 Step 1 The equation is written in the standard form. The verte is (, 0). p Step Solve for p b setting 1 p p 1 16 p p Step 3 The graph has a the graph opens to the. ais of smmetr and since p is negative, Step The focus is (p, 0). Substituting the value for p, the focus is (, 0). Step 5 The directri is a line. p ( ) Step 6 Sketch the graph. Using the Distance Formula to Write the Equation of a Parabola Write the equation of a parabola with focus (0, ) and directri. ( 1 ) ( 1 ) ( ) ( ) Distance Formula ( 0) ( ( )) ( ) ( ) Substitute (0, ) for 1 and 1 and (, ) for and. ( ) ( ) Simplif. ( ) Square both sides. Epand. Subtract and from both sides. Solve for. Copright b Holt, Rinehart and Winston. 168 Holt Algebra

8 Read To Go On? Problem Solving Intervention 10-5 Parabolas Parabolas are used in the design of satellite dishes to reflect light waves. A cross section of a parabolic satellite dish has the equation 1, where and are measured in feet. The receiver must be placed at the focus of the parabola. How far from the verte of the satellite dish should the receiver be placed? Understand the Problem 1. What is the shape of the satellite dish?. What does the equation 1 represent? 3. Where must the receiver be placed?. What are ou being asked to find? Make a Plan 5. What standard form of the equation can ou use? 6. What will the value of p represent? Solve 7. Write the equation in standard form, solving for. 8. How do ou find the value of p? 9. What is p equal to? 10. How far from the verte of the satellite dish should the receiver be placed? Look Back 11. Graph the parabola. F (3, 0) 1. Is the location of the receiver at the focus of the parabola? 13. Does the location of the receiver make sense in relation to the satellite dish? Copright b Holt, Rinehart and Winston. 169 Holt Algebra

9 Read To Go On? Quiz 10-1 Introduction to Conic Sections 1. The deliver area of a pizza store etends to the locations (5, 1) and (7, ). Write an equation for the deliver area of the store if a line between the locations represents a diameter of the deliver area. Identif and describe each conic section. ( 3). ( ) Circles Write the equation of each circle. 6. center (, 6) and radius r 8 7. center (3, ) and containing the point (7, 6) 8. Write the equation of the line that is tangent to 100 at (6, 8) Ellipses Find the center, vertices, co-vertices, and foci of each ellipse. Then graph ( 3 ) 36( ) Copright b Holt, Rinehart and Winston. 170 Holt Algebra

10 Read To Go On? Quiz continued 11. Write the equation of the ellipse with center (5, 7), verte (8, 7), and focus (10, 7). 1. A semi-elliptical bridge over a stream that is 60 feet wide must be feet high at its highest point to accommodate boat traffic. Write an equation for a cross section of the bridge. 10- Hberbolas 13. Find the center, vertices, co-vertices, foci, and asmptotes for the hperbola Then graph. 8 center: foci: vertices: asmptotes: co-vertices: 1. Write the equation of the hperbola with vertices at (0, 3) and (0, 3) and foci at (0, 5) Parabolas Find the center, value of p, ais of smmetr, focus, and directri for the parabola. Then graph center: value of p: 10 ais: focus: 5 directri: Write the equation of the parabola with focus (, 3) and directri A cross section of a parabolic microphone has the equation, where and are measured in inches. How far from the verte of the microphone should the feedhorn be placed? Copright b Holt, Rinehart and Winston. 171 Holt Algebra

11 Read To Go On? Enrichment Conic Equations Find the equation of the conic satisfing the given conditions. 1. Focus (, 0); directri. Focus (, 3); directri 3 3. Vertices: (7, 0) and (7, 0); foci: (3, 0) and (3, 0). Foci: (, 0) and (, 0) length of major ais: 6 5. Vertices: (1, 1) and (1, 5); endpoints of minor ais: (3, ) and (1, ) 6. Asmptotes: 3 and 3 ; one verte (, 0) 7. Vertices: (1, 0) and (1, 0); foci: (, 0) and (, 0) 8. Vertices: (3, 8) and (3, ); asmptotes: 3 1 and 3 9. Parabola with verte at (3, ) and focus at (3, ) Copright b Holt, Rinehart and Winston. 17 Holt Algebra

12 10B Identifing Conic Sections in Standard Form Identif the conic section that each equation represents. Circle ( h) ( k) r Ellipse Read To Go On? Skills Intervention 10-6 Identifing Conic Sections HORIZONTAL AXIS ( h ) ( k ) 1 a b VERTICAL AXIS ( h ) ( k ) 1 b a Hperbola ( h ) ( k ) 1 a b ( k ) ( h ) 1 a b Parabola h 1 p ( k ) k 1 ( h ) p A. ( 5 ) ( 1 ) 1 6 This equation is of the same form as a with a horizontal ais. B. ( 1) 1 ( ) This equation is of the same form as a with a ais. Finding the Standard Form of the Equation for a Conic Section Find the standard form of b completing the square. Then identif the conic. Step 1 Rearrange the terms: Step Factor from the -term and from the -term. ( ) 9( 6 ) 61 Complete both squares. 9 6 The equation represents an ( ) 9( ) ( ) 9( 3) ( ) 9( 3 ) ( ) ( 3 ) 1 with center (, 3). The vertices are ( 3, 3) or (, 3) and (, 3). Copright b Holt, Rinehart and Winston. 173 Holt Algebra

13 10B Read To Go On? Skills Intervention 10-7 Solving Nonlinear Sstems Solving a Nonlinear Sstem b Graphing Solve { b graphing. The graph of the first equation is a. The graph of the second equation is a. There ma be as man as Step 1 Solve each equation for. points of intersection. Step Graph the sstem on our calculator. Use the intersect feature to find the solution set. The point of intersection is (, 0). Solving a Nonlinear Sstem b Elimination Solve { 3 1 b using the elimination method. 3 The graph of the first equation is a. The graph of the second equation is an. There ma be as man as points of intersection. Step 1 Decide on a variable to eliminate. Eliminate. Multipl the second equation b Add the equations. Divide both sides b. Solve for. Step Find the values for. Substitute and for in either equation The solution set of the sstem is {(3, ), (3, ), (3, ), and (3, )}. Copright b Holt, Rinehart and Winston. 17 Holt Algebra

14 10B Read To Go On? Problem Solving Intervention 10-7 Solving Nonlinear Sstems A tour boat travels around an island in a pattern that can be modeled b the equation 13, with the island at the origin. Suppose that a jet skier is approaching the island on a path that can be modeled b the equation 1. Is there an danger of collision? If so, at what point(s)? Understand the Problem 1. What are ou being asked to do?. What equation represents the path of the boat? 3. What equation represents the path of the jet skier? Make a Plan {. To see if the graphs intersect, solve the sstem Solve 5. The graph of the first equation is a and the graph of the second equation is a. 6. There ma be as man as points of intersection. 7. Solve the second equation for Substitute into the first equation Substitute. 13 Simplif. 13 Combine like terms. 0 Use standard form. 6 0 Divide out GCF of. ( )( ) 0 Factor. or Solve. Find the values for. 9. The tour boat and jet skier could collide at the points (3, ) or (, ). 1 1 Look Back 10. Graph the sstem on a graphing calculator. What are the points of intersection of our graph? (, ) and (, 3). Do the points match our points in Eercise 9? Copright b Holt, Rinehart and Winston. 175 Holt Algebra

15 10B Read To Go On? Quiz 10-6 Identifing Conic Sections Identif the conic section that each equation represents ( 3 ) 16 ( 6 ) 1. ( 3) 5 ( ) Write each equation in the form A B C D E F ( 8) 10. ( ) 36 ( 3 ) 1 5 Find the standard form of each equation b completing the square. Then identif the conic Copright b Holt, Rinehart and Winston. 176 Holt Algebra

16 10B Read To Go On? Quiz continued 10-7 Solving Nonlinear Sstems Solve each sstem of equations b graphing. 15. { { { Solve each sstem of equations b using the substitution or elimination method. 18. { { { A team of jets are giving an air show performance. During the performance, the lead jet moves in a path that can be modeled b the equation 16. The other jet is in a formation along the equation. At what point(s) are the jets in danger of colliding? Find n so that the sstem { n eactl solutions. has 8 Copright b Holt, Rinehart and Winston. 177 Holt Algebra

17 10B Read To Go On? Enrichment Nonlinear Sstems of Inequalities If two or more inequalities are considered at the same time, a sstem of inequalities is formed. To find the solution set of the sstem, locate the intersection of the graphs. For eample: { 1 The graph of 1 is a parabola with verte at (1, 0). The points above or in the interior of the parabola satisf the condition. The graph of is an ellipse. It is drawn with a dashed line. To satisf the inequalit, a point must lie outside of the ellipse. Graph each sstem of nonlinear inequalities. 1. { 3. { { { 5 Copright b Holt, Rinehart and Winston. 178 Holt Algebra