Download PDF by A. C. Burdette (Auth.): An Introduction to Analytic Geometry and Calculus

By A. C. Burdette (Auth.)

ISBN-10: 0121422526

ISBN-13: 9780121422523

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Discuss and graph x2 + 6x - 4y + 13 = 0. We first note that this equation is of the form (3-8) and therefore represents a parabola. Next we reduce it to form (3-7) by writing Ay - 13 = x2 + 6x, and then, completing the square of the right member, obtain 4y - 4 = x2 + 6x + 9, or finally 4(y - 1) = (x + 3) 2 . From this form of the equation, making use of Theorem 3-5, we conclude that the parabola extends upward from its vertex ( — 3, 1), symmetric to its axis x = — 3. We combine this information with the points X y -2 5/4 0 13/4 2 29/4 and obtain the graph in Fig.

This is often the simplest method of describing the extent of a curve. 3-5. Graphing Equations If we combine the general remarks of Sec. 2-2 with the special results of Sees. 3-2, 3-3, 3-4, we have a fairly sound working basis for drawing the graph of many equations. In some cases, certain parts of the general discussion may be omitted because the labor of carrying them out is so tedious, or difficult, as to make them impractical. These results are not an end in them­ selves but rather an aid to the overall problem of drawing a graph.

A curve is symmetric with respect to the j-axis if and only if the substitution of — x for x in its equation yields an equivalent! equation. Similarly, by referring to Fig. 3-3 we may state : THEOREM 3-2. A curve is symmetric with respect to the x-axis if and only if the substitution of — y for y in its equation yields an equivalent equation. THEOREM 3-3. A curve is symmetric with respect to the origin if and only if the simultaneous substitution of — x for x and — y for y in its equation yields an equivalent equation.

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An Introduction to Analytic Geometry and Calculus by A. C. Burdette (Auth.)


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