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MATH 102: College Algebra TEST 3 REVIEW Name: ______________________________ Date:______________ Score:____________ QUESTION (Show all work) 1) Find the equation of the circle of radius 7 with its center at the origin. 2) Find the center and radius of the circle 4 x 2 + 4 y 2 − 32x − 40y + 148 = 0 3) Find the vertex of the parabola y = 2 x 2 + x − 2 4) Find the vertex of the parabola y = 10 − 2 x − x 2 5) Find the coordinates of the vertex for the function y = x 2 + 4 x + 4 ANSWER 6) 7) Which equation describes an ellipse? [A] − 4 x 2 + 11x − 8 y 2 + 5 y = 6 [B] 6 x 2 + 11x + 6 y 2 − 4 y + 5 = 0 [C] − 2y 2 + 11y + 2 x 2 + 6 x − 4 = 0 [D] − 4 y 2 + 11x + 5 y = −4 Find an equation of the parabola in standard form with vertex ( -3, 3), axis of symmetry y = 3 , and passing through the point (3,-1) 8) Graph the parabola: y = −(x − 2)2 + 2 9) Sketch the graph of 10) Graph: x 2 + 4 x + y 2 + 6 y = 0 (x + 5)2 + (y + 4)2 = 9 MATH 102 TOPIC 15 PRACTICE TEST Page 2 11) 12) 13) 14) Find the center, vertices, and foci for the ellipse: x2 y 2 + =1 36 81 Obtain an equation for an ellipse with major axis of length 8 and foci at (7,1) and (1,1) . Graph: (x − 4 )2 9 + (y − 3)2 4 =1 Graph and show the asymptotes: x2 y 2 − =1 25 4 15) Simplify the equation and graph: − x 2 + 9 y 2 − 4 x + 18y − 31 = 0 16) Determine the vertices, asymptotes, and foci of the hyperbola defined by (x + 3)2 − 4(y + 3) 2 25 = 1. MATH 102 TOPIC 15 PRACTICE TEST Page 3 17) −2 −2 4 Evaluate: − 4 4 −2 0 −4 −2 18) ⎧ x − 4 y = 11 Use Cramer’s rule to solve: ⎨ ⎩5 x + y = 6 19) ⎧ x − 3y + 4 z = 3 ⎪ Write the augmented matrix for the system ⎨ 2x − 5y + 6z = 6 ⎪−3x + 3y + 4z = 6 ⎩ Then perform elementary row operations to complete the first column of the solution. 20) Write the partial fraction decomposition of MATH 102 TOPIC 15 PRACTICE TEST −3 (x + 3)(x + 4) Page 4 21) Solve the system: 22) ⎧ 2 x + y + 4z = −4 ⎪ y + 4z = 2 ⎨ ⎪4 x + 3 y + 12z = 4 ⎩ ⎧ x + 2 y − 3z = 3 ⎪ Use Cramer’s rule to solve: ⎨3 x − 2y − z = −23 ⎪2 x − y − 3z = −16 ⎩ ANSWERS 1. x 2 + y 2 = 49 12. (x − 4 )2 16 2. center (4, 5); radius = 2 13. (y − 1)2 + =1 7 y 10 −10 10 x −10 MATH 102 TOPIC 15 PRACTICE TEST Page 5 3. ⎛ − 1 , − 17 ⎞ ⎝ 4 8⎠ 14. 10 y –10 10 x –10 4. 15. ( -1, 11) 10 y –10 10 x –10 (y + 1)2 − (x + 2)2 4 5. 16. (-2, 0) 36 =1 vertices: (− 2,−3 ), (− 4,−3 ) ⎛ − 6 − 29 ⎞ ⎛ − 6 + 29 ⎞ foci: ⎜ ,−3 ⎟ ,⎜ ,−3 ⎟ ⎜ ⎟⎜ ⎟ 2 2 ⎝ ⎠⎝ ⎠ 6. 7. [A] Note the coefficients of the squared terms. (y − 3) 2 = 8 3 (x + 3 ) 17. 18. 112 ⎛ 5 , − 7⎞ ⎝ 3 3⎠ 19. 8. ⎡1 − 3 4 3⎤ ⎥ ⎢ ⎢2 − 5 6 6 ⎥ ⎥ ⎢ ⎢⎣− 3 3 4 6 ⎦⎥ y 5 –5 5 x –5 9. Circle with center (-5, -4), radius 3 20. 10. Circle with center (-2, -3), radius 11. Center : (0, 0), Vertices : (0,−9),(0,9) , Foci : 21. 13 (0,−3 5 )(, 0,3 5 ) MATH 102 TOPIC 15 PRACTICE TEST Page 6 22. −3 3 + x +3 x +4 inconsistent (-4, 5, 1) ⎡1 − 3 4 3⎤ ⎥ ⎢ ⎢0 1 − 2 0 ⎥ ⎥ ⎢ ⎢⎣0 − 6 16 15 ⎥⎦