Pre-Calculus Math 40s Standards Test - Conics ANSWERS

Pre-Calculus Math 40s Standards Test - Conics ANSWERS -...

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Unformatted text preview: Conics Copyright © 2006, Barry Mabillard. 0 Conics Standards Test - ANSWERS www.math40s.com www.math40s.com 1. Sketch the graph and write the equation of a circle with a centre at (3, -2) and is tangent to the x – axis. Start with 2 2 ( x - h) + ( y - k ) = r 2 The phrase “tangent to the x-axis” means the circle touches the x-axis. The centre is (3, -2), and the radius is 2. Plug these into the standard form equation. ( x - 3 ) + ( y - (-2 ) ) = 22 2 2 ( x - 3 ) + ( y +2 ) = 4 2 2 2. Identify the conic represented by the equation x 2 − 6 x = y + 3 Since the y2 term is missing, the conic is a parabola. 3. State the coordinates of the vertex in the relation y 2 = 2 x + 4 y 2 = 2x + 4 y 2 = 2 ( x + 2) The vertex is located at (-2, 0) 4. Change the following to standard form, then sketch: 4 y 2 + 40 y − 4 x 2 + 32 x + 20 = 0 4y 2 + 40y - 4x 2 + 32x + 20 = 0 4 ( y 2 +10y )- 4( x 4 ( y +10y + 25 ) - 4 ( x 2 2 2 ) = -20 - 8x +16 ) = -20+100 - 64 - 8x 4 ( y +5 ) - 4 ( x - 4 ) = 16 2 2 4 ( y +5 ) 4 ( x - 4 ) 16 = 16 16 16 2 ( y +5 ) 4 2 - 2 ( x - 4) 4 2 =1 This is a vertical hyperbola since y comes first Centre = (4, -5), a = 2, b = 2 Conics Standards Test - ANSWERS 1 www.math40s.com 5. a) Given the parabola below, find the equation if the vertex is at (4, -2) The vertex is located at (4, -2). These are the h & k values. Standard form for horizontal parabolas → x - h= a ( y - k ) 0 - 4 = a ( 0 - (-2 ) ) A point on the graph is located at (0, 0) This is a value you can use for x & y. 0 - 4 = a ( 0+ 2 ) 2 2 2 -4= a ( 4 ) a= -1 Plug everything in and solve for a. The equation is : x - 4 = - ( y +2 ) b) Verify the intercepts algebraically x – intercept: Set y = 0, then solve for x. 2 x - 4 = - ( 0+ 2 ) y – intercepts: Set x = 0, then solve for y. 0 - 4= - ( y + 2 ) -4= - ( y + 2 ) x - 4 = -4 x =0 4= ( y + 2 ) 4= 2 2 First evaluate for + 2 +2 = y + 2 y =0 2 (y + 2) ±2= y + 2 2 Then evaluate for - 2 -2 = y + 2 y = -4 6. A conic is represented by x 2 − 4 x + 9 y 2 − 18 y = 23 a) Sketch the conic x 2 - 4x +9y 2 - 18y = 23 (x (x 2 2 ) +9 ( y - 4x + 4 ) +9 ( y - 4x 2 2 - 2y ) = 23 - 2y +1) = 23+ 4+9 ( x - 2 ) +9 ( y - 1) = 36 2 2 ( x - 2 ) + 9 ( y - 1) = 36 2 2 36 ( x - 2) 36 36 2 + ( y - 1) 4 36 2 Centre = (2, 1) a = 6, b = 2 =1 b) State the domain & range Domain is { x |-4 ≤ x ≤ 8} Range is {y |-1 ≤ y ≤ 3} Conics Standards Test - ANSWERS 2 www.math40s.com 2 7. Identify the conic with the equation 3 x 2 − y 2 − 7 x + 2 = 0 Since the product of A & C is negative, the conic is a hyperbola. 8. The equation of a circle is given by the equation 2 x 2 + 2 y 2 − 8 x + 4 y − 22 = 0 a) Sketch the conic 2x 2 + 2y 2 - 8x + 4y - 22 = 0 2x 2 - 8x + 2y 2 + 4y = 22 2 ( x 2 - 4x ) + 2 ( y + 2y ) = 22 2 ( x - 4x + 4 ) + 2 ( y + 2y +1) = 22+8+ 2 2 2 2 2 ( x - 2 ) + 2 ( y +1) = 32 2 2 2 ( x - 2) 2 ( y +1) 32 + = 2 2 2 2 ( x - 2) 2 2 + ( y +1) = 16 2 b) State the radius 2 2 The form is ( x - h ) + ( y - k ) = r 2 . Taking the square root of the right side gives 4 units. 9. Determine the equation of the ellipse shown below The a – value is 3, and the b – value is 2. The centre is (0, 0) Plug into the equation 2 2 ( x - h) + ( y - k ) = 1 a2 b2 (x -0) 32 2 (y - 0) + 22 2 =1 x2 y2 + =1 9 4 Conics Standards Test - ANSWERS 3 www.math40s.com 10. The equation of a conic is ( x − 2) 9 2 ( y + 2) − 4 2 =1 a) Identify this conic section The minus between the terms indicates this is a hyperbola b) Sketch a clearly labeled graph Centre = (2, -2) a = 3, b = 2 c) State the domain & range Domain is { x | x ≤ -1, x ≥ 5} Range is {y |y ∈ R} 11. Find the coordinates of the vertices and sketch the equation: 9x2 + 4y2 + 40y + 64 = 0 9x 2 +4y 2 +40y +64 = 0 9x 2 +4(y 2 +10y)=-64 9x 2 +4(y 2 +10y + 25)=-64+100 9x 2 4(y +5)2 36 = + 36 36 36 2 2 x (y +5) + =1 4 9 Conics Standards Test - ANSWERS 4 www.math40s.com 12. Find the equation of the ellipse sketched below: ( x - h) 2 (y - k ) + 2 =1 a2 b2 Centre = (2, -1) ; a = 2, b = 3 Start with: ( x - 2 ) + ( y +1) = 1 2 2 ( 2) (3) 2 2 ( x - 2 ) + ( y +1) = 1 2 2 4 13. Sketch the equation and state the domain and range for 9 ( x − 4) 2 4 − y2 =1 9 Centre = (4, 0) ; a = 2, b = 3 Domain is { x | x ≤ 2, x ≥ 6} Range is {y |y ∈ R} y 2 x2 − =1 14. Sketch the equation and state the axis of symmetry: 16 25 This is a vertical hyperbola since y comes first. Since the centre is at the origin, the x-axis is the axis of symmetry. Conics Standards Test - ANSWERS 5 www.math40s.com 15. The equation of a conic is 25 x 2 - 9 y 2 -100 x + 72 y - 269 = 0 a) Write the equation in standard form b) Sketch the graph 25x 2 -9y 2 - 100x +72y - 269 = 0 25x 2 - 100x -9y 2 +72y = 269 25 ( x 2 - 4x ( ) -9 ( y 2 - 8y )( ) ) = 269 2 2 25 x - 4x + 4 - 9 y - 8y + 16 = 269 + 100 - 144 25 ( x - 2 ) -9 ( y - 4 ) = 225 2 2 25 ( x - 2 ) 9 ( y - 4 ) 225 = 225 225 225 2 ( x - 2) 9 2 - 2 (y - 4) 25 2 =1 Centre = (2, 4) a = 3, b = 5 16. An ellipse has the following vertices: A(0, 9), B(2, 5), C(0, 1), and D(-2, 5) Draw the ellipse and determine the equation. The a – value is 2, and the b – value is 4. The centre is (0, 5) Plug into the equation 2 2 ( x - h) + ( y - k ) = 1 a2 b2 ( x -0) 22 2 (y - 5 ) + 42 2 =1 x 2 (y - 5 ) + =1 4 16 2 Conics Standards Test - ANSWERS 6 www.math40s.com 17. For the conic 2 y 2 − 2 x − 4 y − 6 = 0 a) Find the intercepts of the conic section y – intercepts: Set x = 0, then solve for y. 2y 2 - 2 ( 0 ) - 4y - 6 = 0 x – intercept: Set y = 0, then solve for x. 2 2 ( 0 ) - 2x - 4 ( 0 ) - 6 = 0 2y 2 - 4y - 6 = 0 2 ( y 2 - 2y - 3 ) = 0 -2x = 6 y 2 - 2y - 3 = 0 x = -3 ( y - 3 )( y +1) = 0 y = -1, 3 b) Sketch a clearly labeled graph 2y 2 - 2x - 4y - 6 = 0 2y 2 - 4y = 2x +6 2 ( y 2 - 2y ) = 2x +6 2 ( y 2 - 2y +1) = 2x +6+ 2 2 ( y - 1) = 2x +8 2 2 ( y - 1) = 2 ( x + 4 ) 2 ( y - 1) 2 = ( x +4 ) 18. The conic x 2 + y 2 = 1 is translated 2 units to the right a) Write the equation of the new conic and sketch it A translation 2 units right can be accomplished by replacing x with x – 2. ( x - 2) 2 +y 2 = 1 b) State the domain of the new conic { x |1 ≤ x ≤ 3} c) State the range of the new conic {y|-1 ≤ y ≤ 1} Conics Standards Test - ANSWERS 7 www.math40s.com ...
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