# Lesson 9.1 Using the Distance Formula

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1 Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0) g. (a, 8) and (a, ) h. (c, d) and (c, d) i., and,. Find the possible values of or. a. The distance between the points (, ) and (, ) is 0 units. b. The distance between the points (, ) and (, ) is units.. For each set of vertices, identif the shortest side of ABC and find the perimeter of the triangle. Round our answers to the nearest hundredth. a. A(0, 0), B(, ), C(, ) b. A(, ), B(, ), C(, 7). Find an equation of the locus of points that satisfies the given condition. a. The points that are units from (, ) b. The points that are equidistant from (0, 0) and (, ) c. The points that are twice as far from (, 0) as the are from (0, 0). The perpendicular bisector of a segment is the locus of all points that are equidistant from the endpoints of the segment. Consider ABC with vertices A(, 0), B(, ), and C(, 0). a. Draw ABC on a coordinate plane. b. Find equations of the perpendicular bisectors of the three sides. c. Use our equations from b to find the point where the three perpendicular bisectors coincide. Call this point D. d. Find the distance from D to each verte of the triangle. What do ou notice? CHAPTER Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press

2 Lesson. Circles and Ellipses. Find the center and radius of each circle. a. b. ( ) 00 c. ( 0.) ( 0.) 0. d. e. cos t f. 0 cos t 8 sin t 0 sin t. Find the center, horizontal scale factor, and vertical scale factor for each ellipse. a. b. ( c. ) ( ) d. 7cost e. cost f. cos t sin t sin t sin t. Sketch each ellipse in Eercise. Give the eact coordinates of the endpoints of the major and minor aes, and the foci.. Write parametric equations for the graph of each equation. Identif each graph as a circle or an ellipse. a. b. ( ) ( ) c. d. e. 7 f Write an equation in standard form for each graph. a. b. c.. The entrance to a tunnel over a one-wa road is half an ellipse with height ft and width ft. a. Sketch the tunnel entrance on a coordinate plane. Place the center of the ellipse at the origin. Label the endpoints of the major and minor aes of the ellipse that appear in our sketch with their coordinates. b. Write the equation in standard form for the complete ellipse. c. Will a truck that is ft wide and 0 ft high clear the tunnel? Eplain our reasoning. Discovering Advanced Algebra More Practice Your Skills CHAPTER 7 00 Ke Curriculum Press

3 Lesson. Parabolas. For each parabola described, use the information given to find the location of the missing feature. It ma help to draw a sketch. a. If the verte is (0, 0) and the focus is (, 0), where is the directri? b. If the focus is (0, 7) and the directri is, where is the verte? c. If the verte is (, 0) and the directri is., where is the focus? d. If the focus is (, ) and the directri is, where is the verte? e. If the focus is (, ) and the verte is (, ), where is the directri?. Find the verte of each parabola and state whether the parabola opens upward, downward, to the right, or to the left. Also give the equation of the ais of smmetr. a. b. c. d. ( ) e. ( ) f.. Write parametric equations for each parabola. a. b. c. ( ) d. e.. (.) f. ( ). Write an equation in standard form for each parabola. Then write parametric equations for the parabola. a. b. c.. Solve each problem b finding the coordinates of the verte of a parabola. a. What are the dimensions of a rectangular field of maimum area that can be enclosed with 7 ft of fencing? What is the area of the field? b. The height of a projectile shot straight upward with an initial velocit of 0 m/s from the top of a 0 m tall building is given b the function h.t 0t 0. How long does it take the projectile to reach its maimum height? What is the maimum height? Round our answers to the nearest tenth. 8 CHAPTER Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press

4 Lesson. Hperbolas. Write an equation in standard form for each graph. a. b. c. d.. Write parametric equations for each graph in Eercise.. Sketch each hperbola on our paper. Include the asmptotes and the foci. Write the equations of the asmptotes, and give the eact coordinates of the vertices and the foci. a. b. c. d. co e. s t tan t f. ( ) ( ) tan t co s t. Identif each path described as an ellipse or a hperbola. Then write the equation in standard form for each path. a. A point moves in a plane so that the difference of its distances from the points (0, ) and (0, ) is alwas 8 units. b. A point moves in a plane so that the sum of its distances from the points (, 0) and (, 0) is alwas 0 units. c. A point moves in a plane so that the difference of its distances from the points (, ) and (, ) is alwas units. Discovering Advanced Algebra More Practice Your Skills CHAPTER 00 Ke Curriculum Press

5 Lesson. The General Quadratic. Complete the square for each epression. Then write the epression in factored form. a. 8 b. c. d. e.. f..7. Identif the graph of each equation as a circle, an ellipse, a parabola, or a hperbola. Then rewrite each equation in the general quadratic form, A B C D E F 0. (Include all coefficients.) a. ( 7) ( ) b. ( ) ( ) c. ( ) ( ) d. ( ) ( ) e. 0.( ).( ) f. ( ) ( ). Convert each equation to the standard form of a conic section. a. 0 b. 0 c d. 0 0 e f Name the shape described b each equation in Eercise. Give the verte of each parabola and the center of each circle, ellipse, and hperbola.. Solve each equation for b using the quadratic formula. a. 0 b. 0 0 c. 0 d Solve each sstem of equations algebraicall, using the substitution method or the elimination method. ( ) a. b. c CHAPTER Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press

6 Lesson. Introduction to Rational Functions. Write an equation and graph each transformation of the parent function f(). a. Translate the graph left units. b. Translate the graph up units and left unit. c. Translate the graph left units and down units. d. Verticall stretch the graph b a scale factor of. e. Horizontall stretch the graph b a scale factor of and translate it down units. f. Reflect the graph across the -ais.. Write equations for the asmptotes of each hperbola. a. b. c. d. e. f. g. h. i.. Solve. a. b. c. d. 8 e. f Write a rational equation that can be used to solve each problem, using as the variable. Then use our equation to solve the problem. a. A baseball plaer got hits in his first at-bats this season. How man consecutive hits must he get to bring his batting average up to.80? b. How much water must be added to 0 ml of a % alcohol solution to dilute it to a % alcohol solution? Discovering Advanced Algebra More Practice Your Skills CHAPTER 00 Ke Curriculum Press

7 Lesson.7 Graphs of Rational Functions. Rewrite each rational epression in factored form. a. b. c. 7 d. e. 8 f.. Rewrite each epression in rational form (as the quotient of two polnomials). a. b. c. 7 d. e. 7 f. 0. Find all vertical and horizontal asmptotes of the graph of each rational function. a. f() b. f() c. f() d. f() ( ) e. f() f. f() 8. Find all vertical and slant asmptotes of the graph of each rational function. a. f() b. f() c. f() d. f() e. f() f. f(). Give the coordinates of all holes in the graph of each rational function. a. f() b. f() c. f() d. f() e. f() 0 f. f() CHAPTER Discovering Advanced Algebra More Practice Your Skills 00 Ke Curriculum Press

8 Lesson.8 Operations with Rational Epressions. Factor each epression completel and reduce common factors. a. b. c. 0 0 d. e. 0 f. 0. Find the least common denominator for each pair of rational epressions. a., ( ) ( ) ( ) b. ( ), c., d. 8 7, 8. Add, subtract, multipl, or divide as indicated. Reduce an common factors. a. ( ) ( ) ( ) b. ( ) ( 7) ( ) c. 0 d. e. f. g. h.. Rewrite each fraction as a single rational epression. a. b. c. Discovering Advanced Algebra More Practice Your Skills CHAPTER 00 Ke Curriculum Press

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