Algebra Unit 9 (Chapter 9) 0. Spiral Review Worksheet 0. Find verte, line of symmetry, focus and directri of a parabola. (Section 9.) Worksheet 5. Find the center and radius of a circle. (Section 9.3) Worksheet 6 3. Find the vertices and foci of an ellipse. (Section 9.) Page 637 3 5 odd, 35 Worksheet 3-3. Find the vertices, foci and asymptotes for a hyperbola. (Section 9.5) Page 65 3 0, 5 7, 7 3 Worksheet 9 5. Classify conics, translate, and shade conics. Worksheet 5A 7 Worksheet 5B -0 Review Worksheet - 0 Worksheet Eample Find the line of symmetry and verte for y = ( ) + 3 Answer: ) This is a parabola in descriptive form y = a( h) + k ) Since a is a positive number it opens up. 3) The line of symmetry is found in the ( ), but you change the sign. y = ( ) + 3 line of symmetry = The line of symmetry is the fold line of the parabola. y The parabola is symmetric or balanced about the line of symmetry ) The verte is (h, k) y = ( ) + 3 y = a( h) + k So the verte is (, 3) (, 3) Algebra Unit 9 - - y
Problems: For each of the following, c) sketch the parabola. y = ( + 3). y = ( ) + 6 3. y = ( + 8). y = + 5 5. y = 3 6. y = ( ) Eample Find the line of symmetry and verte for y = + 6 + 8 Answer: ) This is a parabola in standard form y = a + b + c ) a is the number. Since it is a positive number it opens up 3) The line of symmetry is found by the formula = The values are a =, b = 6, c = 8 b a y So for this parabola we have = 6 or = 3 ) The verte is found by ( b a, substituting b into the original equation) a So for this problem the verte is ( 3, and plug 3 into + 6 + 8) ( 3) + 6( 3) + 8 So the verte coordinates are ( 3, ) y ( 3, ) Algebra Unit 9 - -
Problems: For each of the following, c) sketch the parabola 7. y = + 8 + 3 8. y = 6 + 5 9. y = + 6 + 3 0. y = 3 + 8 +. y = 8 + 0. y = + 3 3. y = 5 0. y = + 6 5. y = 8 For each of the following, d) find the directri 6. y = 8 7. = y 8. y = 6 9. = y 0. For the parabola y = d) find the directri. Use the answers to #0 and translate to the parabola y = + d) find the directri. Use the answers to #0 and translate to the parabola y = ( 5) d) find the directri Algebra Unit 9-3 -
3. For the parabola y = d) find the directri. Use the answers to #3 and translate to the parabola y = d) find the directri 5. Use the answers to #3 and translate to the parabola y = d) find the directri 6. Which of the following is a parabola? 3 ( + 3) a) y = b) y = c) y = d) y = 3 e) y = Worksheet Give the center and radius of each circle. Sketch problems,, and 3.. + y = 9. ( ) + (y ) = 36 3. ( 5) + y = 5. + y = 8 5. + (y + 5) = 9 6. ( + ) + (y 3) = 7 7. + (y ) = 0 8. ( + ) + (y + ) = 6 9. + y 6 = 0 0. + y 6 = 0. + y = 8y. + y + y = 0 3. + y + 0 y + 0 = 0. + y + 6y = 0 5. Write the equation of the circle that would result if the circle + y = is translated 3 units down and units right. Graph problems 6 and 7. 6. + y > 9 7. + (y ) < 5 Algebra Unit 9 - -
8. Given: ( ) + y = 3 Is the origin inside, outside or on the figure? Identify the following as a circle, a parabola, or neither. 9. 5 + y = 7 0. 6 + 6y = 0. y = 6. y = + 3 3. y = + 3. y + = 0 5. Find the verte and line of symmetry for y = + 0 + 5 6. Find the verte, focus and directri for y = + Worksheet 3 In problems 3, identify the vertices and foci of the following ellipses. Sketch each.. 3. ( ) (y + ) + =. 5 6 + y + 3 + = 3 5. Sketch y + y + = 0 8 3 y 5. Given: + = 5 Is the origin inside, outside, or on the figure? Put the following ellipses into standard form. 6. + 9y 90y + 5 = 0 7. 8 + y 0y + 5 = 0 8. 6 + 5y + 3 50y = 59 Identify the following as a parabola, a circle or an ellipse. 9. + y = 9 0. + 6y = 6. 6 + 6y = 6. 6 + + 9y 8y 0 = 0 3. + 8y + y + 6y = 3 Worksheet Find the asymptotes of the hyperbolas in problems 3.. y = 3. y =. y y + Graph: 5. Graph: 5 3 3 6. For problem #5, is the origin inside or outside the graph? Write in standard form: 3. y = 5 6 Algebra Unit 9-5 -
7. + 8 y = 0 8. 6 y 96 y + = 0 9. y 9 5 + 0y 7 = 0 Worksheet 5a State whether the following is a parabola, a circle, an ellipse or a hyperbola.. = 8y. + y = 8 3. 3 + 3y = 8. 9 y = 5. 3 + y + 8y = 8 6. 3 9 = 3y 7. y = + 3 + 8. y + 0 6y = 5 9. (y ) + ( + ) = 0. 3( + ) + (y ) = 9. Describe the translation of the graph of y = to the graph of y = ( + 0) (Did it move down 0, up 0, right 0 or left 0?). Describe the translation of the graph of y = 3 to the graph of y = 3 5 (Did it move down 5, up 5, right 5 or left 5?) 3. True or False: The equation ( + 3) (y ) + = is equivalent to the 5 36 equation + 3 y + = 5 6. True or False: The equation y + = is equivalent to the equation 9 y + = 9 5. True or False: The equation y + = 3 is equivalent to the equation + y = 3 6. Given: + py + 0y 6 = 0 Determine the shape of the following: a. If p = b. If p = c. If p = 7. Given: + y 0 + 0y + = 0 Is the origin inside, outside, or on the figure? Algebra Unit 9-6 -
Worksheet 5B Given the equation + y =. If the graph is shifted down units, which equation describes the new graph? ( ) y + + = (b) + ( y + ) = (c) ( ) + y = (d) + ( y ) = (e) ( + ) + ( y + ) = y. Given + =. If the equation is shifted left 5 units, which equation describes the 8 new graph? ( + 5) y + = (b) 8 ( 5) y + = (c) 8 ( y + 5) + = 8 (d) ( y 5) + = (e) 8 ( + 5) y = 8 3. If the given function describes the new graph? y = is shifted up 3 units and left units, which equation ( ) ( y 3) = (b) ( + ) ( y + 3) = (c) ( + ) ( y 3) = (d) ( ) ( y + 3) = (e) ( 3) ( y + ) =. If the given function the new function? y = ( + 3) + is shifted down 5 units, which equation describes y = ( + 3) + 9 (b) y = ( + 3) (c) y = + + ( 8) (d) y = ( ) + (e) y = + + 5( 3) Algebra Unit 9-7 -
5. If the graph of the equation equation describes the new graph? ( ) y 3 + + = is shifted to the right 3 units, which ( ) y 3 + + = (b) ( ) + y = 3 (c) ( + ) + y = 3 (d) ( y ) ( + ) + 3 = 3 (e) ( ) + y = 3 y 6. Given + =. If the function is shifted units to the left, write an 9 equation that describes the new function? y 7. Given =. If the function is shifted units down and units right, 9 write an equation that describes the new function? 8. Given 3y + =. If the function is shifted 8 units to the right and 3 units up, write an equation that describes the new function? 9. The function y y + y = 6 in which way? + + + 6 = 3 may be formed by shifting the function 0. The function 9 y 8 7 y + = in which way? 9 + = may be formed by shifting the function Algebra Unit 9-8 -
Review. Describe the translation of the graph of y = to the graph of y = ( 7). Describe the translation of the graph of y = to the graph of y = 7 3. How does the 8 in y = 8 ( + ) + 3 change the graph of y = ( + ) + 3 (move left, move right, move up, move down, make wider, make narrow, change direction from opening up to opening down, change direction from opening down to opening up). Find the slope of the asymptotes of y = 9 9 In problems 5 9 determine if the equation represents a parabola, a circle, an ellipse or a hyperbola. Find the requested parts and sketch. Parabola Circle Ellipse Hyperbola line of symmetry center vertices vertices verte radius foci foci focus asymptotes directri 5. + (y ) = 9 6. y = ( + 5) + 3 7. ( + ) y y + = 8. = 6 5 6 36 9. + y 0 + 6y 5 = 0 0. Find the vertices 5( + 3) + (y ) = 00. Find the vertices (y 3) =. Find the line of symmetry and the verte y = + + 3 For problems 3 5, identify the following conics and put into standard form. 3. + y + 0 y 08 = 0. 9 6y + 5 + 3y 79 = 0 5. + y + 6 6y + 9 = 0 6. Find the verte of y = + 3 7. Graph: + (y + ) > 9 8. Is the origin inside or outside the graph for problem #7? y 5 9. Graph: + = 3 0. Is the origin inside, outside or on the graph for problem #9? Algebra Unit 9-9 -