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Graphing Translated Conic Sections

Learn how to graph translated conic sections including circles, ellipses, hyperbolas, and parabolas. Find equations, centers, foci, vertices, asymptotes, and lines of symmetry.

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Graphing Translated Conic Sections

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  1. Compare the given equation to the standard form of an equation of a circle. You can see that the graph is a circle with center at (h, k) = (2, – 3)and radius r= = 3. 9 EXAMPLE 1 Graph the equation of a translated circle Graph(x – 2)2 + (y + 3) 2 = 9. SOLUTION STEP 1

  2. EXAMPLE 1 Graph the equation of a translated circle STEP 2 Plot the center. Then plot several points that are each 3units from the center: (2 + 3, – 3) = (5, – 3) (2 – 3, – 3) = (– 1, – 3) (2, – 3 + 3) = (2, 0) (2, – 3 – 3) = (2, – 6) STEP 3 Draw a circle through the points.

  3. (y – 3)2 (x + 1)2 4 9 EXAMPLE 2 Graph the equation of a translated hyperbola Graph – = 1 SOLUTION STEP 1 Compare the given equation to the standard forms of equations of hyperbolas. The equation’s form tells you that the graph is a hyperbola with a vertical transverse axis. The center is at (h, k) = (– 1, 3). Because a2 = 4 and b2 = 9, you know that a = 2 and b = 3.

  4. Plot the center, vertices, and foci. The vertices lie a = 2 units above and below the center, at (21, 5)and (21, 1). Because c2 = a2 + b2 = 13, the foci lie c = 13 3.6 units above and below the center, at (– 1, 6.6)and (– 1, – 0.6). EXAMPLE 2 Graph the equation of a translated hyperbola STEP 2

  5. EXAMPLE 2 Graph the equation of a translated hyperbola STEP 3 Draw the hyperbola. Draw a rectangle centered at (21, 3)that is 2a = 4 units high and 2b = 6 units wide. Draw the asymptotes through the opposite corners of the rectangle. Then draw the hyperbola passing through the vertices and approaching the asymptotes.

  6. EXAMPLE 3 Write an equation of a translated parabola Write an equation of the parabola whose vertex is at (– 2, 3) and whose focus is at (– 4, 3). SOLUTION STEP 1 Determine the form of the equation. Begin by making a rough sketch of the parabola. Because the focus is to the left of the vertex, the parabola opens to the left, and its equation has the form (y – k)2 = 4p(x – h) where p < 0.

  7. EXAMPLE 3 Write an equation of a translated parabola STEP 2 Identify h and k. The vertex is at (– 2, 3), so h = –2 and k = 3. STEP 3 Find p. The vertex (– 2, 3)and focus (4, 3)both lie on the line y = 3, so the distance between them is | p | = | – 4 – (– 2) | = 2, and thus p = +2. Because p < 0, it follows that p = – 2, so 4p = – 8.

  8. EXAMPLE 3 Write an equation of a translated parabola ANSWER The standard form of the equation is (y – 3)2 = – 8(x + 2).

  9. (x – h)2 (y – k)2 + = 1 a2 b2 EXAMPLE 4 Write an equation of a translated ellipse Write an equation of the ellipse with foci at (1, 2) and (7, 2) and co-vertices at (4, 0) and (4, 4). SOLUTION STEP 1 Determine the form of the equation. First sketch the ellipse. The foci lie on the major axis, so the axis is horizontal. The equation has this form:

  10. ) ( 1 + 7 2 + 2 (h, k) = , 2 2 EXAMPLE 4 Write an equation of a translated ellipse STEP 2 Identify hand kby finding the center, which is halfway between the foci (or the co-vertices) = (4, 2) STEP 3 Find b, the distance between a co-vertex and the center (4, 2), and c, the distance between a focus and the center. Choose the co-vertex (4, 4) and the focus (1, 2): b= | 4 – 2 | = 2and c= | 1 – 4 | = 3.

  11. Find a. For an ellipse, a2 = b2 + c2 = 22 + 32 = 13,soa = 13 ANSWER The standard form of the equation is + = 1 (x – 4)2 (y – 2)2 13 4 EXAMPLE 4 Write an equation of a translated ellipse STEP 4

  12. EXAMPLE 5 Identify symmetries of conic sections Identify the line(s) of symmetry for each conic section in Examples 1 – 4. SOLUTION For the hyperbola in Example 2 x = – 1 and y = 3 are lines of symmetry For the circle in Example 1, any line through the center (2, – 3) is a line of symmetry.

  13. EXAMPLE 5 Identify symmetries of conic sections For the ellipse in Example 4, x = 4 and y = 2 are lines of symmetry. For the parabola in Example 3, y = 3 is a line of symmetry.

  14. Note thatA = 4, B = 0,andC = 1, so the value of the discriminant is: B2 – 4AC = 02 – 4(4)(1) = – 16 BecauseB2– 4AC < 0andA = C, the conic is an ellipse. To graph the ellipse, first complete the square in x. EXAMPLE 6 Classify a conic Classify the conic given by4x2 + y2 – 8x – 8 = 0. Then graph the equation. SOLUTION 4x2 + y2 – 8x – 8 = 0 (4x2 – 8x) + y2 = 8 4(x2 – 2x) + y2 = 8 4(x2– 2x + ? ) + y2 = 8 + 4( ? )

  15. (x – 1)2 y2 12 + = 1 3 From the equation, you can see that(h, k) = (1, 0), a = 12 = 2 3 ,andb = 3. Use these facts to draw the ellipse. EXAMPLE 6 Classify a conic 4(x2– 2x + 1) + y2 = 8 + 4(1) 4(x – 1)2 + y2 = 12

  16. STEP 1 Solve each equation for y. y = + 7x – 3 EXAMPLE 1 Solve a linear-quadratic system by graphing Solve the system using a graphing calculator. y2 – 7x + 3 = 0 Equation 1 2x – y = 3 Equation 2 SOLUTION y2–7x + 3 = 0 2x – y = 3 y2 = 7x – 3 – y = – 2x + 3 y = 2x – 3 Equation 1 Equation 2

  17. STEP 2 – Graph the equations y = y = and y = 2x – 3 Use the calculator’s intersect feature to find the coordinates of the intersection points. The graphs of and y = 2x – 3 intersect at (0.75, 21.5). The graphs of andy = 2x – 3intersect at(4, 5). y = 7x – 3, 7x – 3, 7x – 3, y = – 7x – 3, EXAMPLE 1 Solve a linear-quadratic system by graphing

  18. ANSWER The solutions are (0.75, – 1.5) and (4, 5). Check the solutions by substituting the coordinates of the points into each of the original equations. EXAMPLE 1 Solve a linear-quadratic system by graphing

  19. EXAMPLE 2 Solve a linear-quadratic system by substitution Solve the system using substitution. x2 + y2 = 10 Equation 1 y = – 3x + 10 Equation 2 SOLUTION Substitute –3x + 10for yin Equation 1 and solve for x. Equation 1 x2 + y2 = 10 x2 + (– 3x + 10)2 = 10 Substitute for y. x2 + 9x2 – 60x + 100 = 10 Expand the power. 10x2 – 60x + 90 = 0 Combine like terms. x2 – 6x + 9 = 0 Divide each side by 10. (x – 3)2 = 0 Perfect square trinomial x = 3 Zero product property

  20. To find the y-coordinate of the solution, substitute x = 3in Equation 2. ANSWER The solution is (3, 1). EXAMPLE 2 Solve a linear-quadratic system by substitution y = – 3(3) + 10 = 1 CHECKYou can check the solution by graphing the equations in the system. You can see from the graph shown that the line and the circle intersect only at the point (3, 1).

  21. Add the equations to eliminate the y2- term and obtain a quadratic equation in x. x2 – y2– 16 = 0 EXAMPLE 3 Solve a quadratic system by elimination Solve the system by elimination. 9x2 + y2 – 90x + 216 = 0 Equation 1 x2 – y2 – 16 = 0 Equation 2 SOLUTION 9x2+ y2 – 90x + 216 = 0 10x2 – 90x + 200 = 0 Add. x2 – 9x + 20 = 0 Divide each side by 10. (x – 4)(x – 5) = 0 Factor x = 4 orx = 5 Zero product property

  22. ANSWER The solutions are(4, 0), (5, 3),and(5, 23),as shown. EXAMPLE 3 Solve a quadratic system by elimination Whenx = 4, y = 0. Whenx = 5, y = ±3.

  23. A ship uses LORAN (long-distance radio navigation) to find its position.Radio signals from stations A and B locate the ship on the blue hyperbola, and signals from stations B and C locate the ship on the red hyperbola. The equations of the hyperbolas are given below. Find the ship’s position if it is east of the y - axis. EXAMPLE 4 Solve a real-life quadratic system Navigation

  24. STEP 1 Add the equations to eliminate the x2 - and y2 - terms. – x2 + y2 – 8y + 8 = 0 EXAMPLE 4 Solve a real-life quadratic system x2 – y2 – 16x + 32 = 0 Equation 1 – x2 + y2 – 8y + 8 = 0 Equation 2 SOLUTION x2 – y2 – 16x + 32 = 0 – 16x – 8y + 40 = 0 Add. y = – 2x + 5 Solve for y.

  25. STEP 2 Substitute – 2x + 5for yin Equation 1 and solve for x. x = – 1 orx = 73 EXAMPLE 4 Solve a real-life quadratic system x2 – y2 – 16x + 32 = 0 Equation 1 x2– (2x + 5)2– 16x + 32 = 0 Substitute for y. 3x2 – 4x – 7 = 0 Simplify. (x + 1)(3x – 7) = 0 Factor. Zero product property

  26. STEP 3 ANSWER Substitute forxiny= – 2x + 5to find the solutions (–1, 7)and , ( ). Because the ship is east of the y - axis, it is at , ( ). 73 13 73 13 EXAMPLE 4 Solve a real-life quadratic system

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