GRAPHS OF EQUATIONS in .NET framework

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GRAPHS OF EQUATIONS
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Fig. 3-8 3.8 Draw the graphs of:
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(a) If
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y2 x2 = 1; 9 4
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y2 x2 = 1. 9 4
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y2 x2 y2 x2 x2 = 1, then = + 1 1. So 1 and, therefore, |x| 3. Hence, there are no points (x, y) on the 9 4 9 4 9 graph within the in nite strip 3 < x < 3. See Fig. 3-11(b) for a sketch of the graph, which is a hyperbola.
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(b) Switching x and y in part (a), we obtain the hyperbola in Fig. 3-12.
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3.9 Draw the graphs of the following equations: (a) 3y x = 6 (d) y = 4 (b) 3y + x = 6 (e) y = x 2 1 (c) x = 1 1 (f ) y = + 1 x
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[CHAP. 3
Fig. 3-9
Fig. 3-10
(g) y = x (j) (h) y = x (i) y2 = x 2
GC Check your answers to parts (a) (i) on a graphing calculator.
3.10 On a single diagram, draw the graphs of: (a) y = x 2 (b) y = 2x 2 (c) y = 3x 2 (d) y = 1 2 x 2 (e) y = 1 2 x 3
( f ) GC Check your answers on a graphing calculator. 3.11 (a) Draw the graph of y = (x 1)2 . (Include all points with x = 2, 1, 0, 1, 2, 3, 4.) How is this graph related to the graph of y = x 2 GC Check on a graphing calculator.
CHAP. 3]
GRAPHS OF EQUATIONS
Fig. 3-11
Fig. 3-12
1 . GC Check on a graphing calculator. x 1 1 . GC Check on a graphing calculator. x+1
(b) Draw the graph of y =
(c) Draw the graph of y = (x + 1)2 . How is this graph related to that of y = x 2 GC Check on a graphing calculator. (d) Draw the graph of y =
3.12 Sketch the graphs of the following equations. GC Check your answers on a graphing calculator. x2 y2 + =1 (b) 4x 2 + y2 = 4 (c) x 2 y2 = 1 (a) 4 9 2 2 (y 2) (x 1) (d) y = x 3 + =1 (f ) xy = 2 (e) 4 9 [Hint: Parts (c) and (f ) are hyperbolas. Obtain part (e) from part (a).] 3.13 Find an equation whose graph consists of all points P(x, y) whose distance from the point F(0, p) is equal to its distance PQ from the horizontal line y = p (p is a xed positive number). (See Fig. 3-13.)
GRAPHS OF EQUATIONS
[CHAP. 3
Fig. 3-13
3.14 Find the standard equations of the circles satisfying the given conditions: (a) center (4, 3), radius 1; (b) center ( 1, 5), radius 2; (c) center (0, 2), radius 4; (d) center (3, 3), radius 3 2; (e) center (4, 1) and passing through (2, 3); (f ) center (1, 2) and passing through the origin. 3.15 Identify the graphs of the following equations: (a) x 2 + y2 12x + 20y + 15 = 0 (c) x 2 + y2 + 3x 2y + 4 = 0 (e) x 2 + y2 + 2x 2y + 2 = 0 (b) x 2 + y2 + 30y + 29 = 0 (d) 2x 2 + 2y2 x = 0 (f ) x 2 + y2 + 6x + 4y = 36
3.16 (a) Problem 3.3 suggests that the graph of the equation x 2 + y2 + Dx + Ey + F = 0 is either a circle, a point, or the null set. Prove this. (b) Find a condition on the numbers D, E, F which is equivalent to the graph s being a circle. [Hint: Complete the squares.] 3.17 Find the standard equation of a circle passing through the following points. (a) (3, 8), (9, 6), and (13, 2); (b) (5, 5), (9, 1), and (0, 10). [Hint: Write the equation in the nonstandard form x 2 + y2 + Dx + Ey + F = 0 and then substitute the values of x and y given by the three points. Solve the three resulting equations for D, E, and F.] 3.18 For what value(s) of k does the circle (x k)2 + (y 2k)2 = 10 pass through the point (1, 1) 3.19 Find the standard equations of the circles of radius 3 that are tangent to both lines x = 4 and y = 6. 3.20 What are the coordinates of the center(s) of the circle(s) of radius 5 that pass through the points ( 1, 7) and ( 2, 6) 3.21 Find the standard equation of the circle through ( 2, 1) and tangent to the line 3x 2y = 6 at the point (4, 3). 3.22 Find the standard equation of the circle through (1, 4) and (3, 2) and having its center on the line y = 2x 1.
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