MATH 180 FALL 1995 10:00 SECTION FONG

ASSIGNMENT #7

1.10, #36. A possible function is y = −4 sin 2x.

1.11, #1. The minimal degree is at least one more than the number of “bends” in the curve. The sign of the leading coefficient is given by the behavior of the curve as x → ±∞.

Graph  i ii iii iv v
Degree  3 4 4 5 5
Leading Sign − + − − +

1.11, #9. The rational function y = (x^{2} + 2x + 1)/(x^{2} − 4) behaves like x^{2}/x^{2} = 1 as x → ±∞, so the line y = 1 is a horizontal asymptote. Vertical asymptotes arise when the denominator x^{2} − 4 = 0, i.e., when x = ±2. To see whether y → ∞ or y → −∞, we consider the sign of y as x gets close to −2 from the left and as x gets close to −2 from the right. This analysis has to be repeated for the line x = 2. The graph has x-intercept at x = −1 and y-intercept at y = −1/4. Comment: A common mistake was drawing a graph symmetric with respect to the y-axis with x-intercept at x = 0 rather than at x = −1. This may have been suggested by the graph on the calculator, but pictures at low resolutions can mislead. Be careful!

1.11, #13. (1, 1) lies on ax^{2} +bx+c, so that a+b+c = 1. The axis of symmetry x = −b/2a passes through (1, 1), so that −b = 2a. The y-intercept is 6, so that c = 6. This pins down the function: y = 5x^{2} − 10x + 6.

1.11, #17. The cubic (x+3)x(x−4) has roots at the given points −3, 0, 4. Since y → −∞ as x → ∞, the leading coefficient of y is negative. So y = −(x+3)x(x−4) = −x^{3}+x^{2}+12x does the trick. The graph is increasing on (−1.5, 2) and decreasing on (−∞, −1.5) and (2, ∞) (the points −1.5 and 2 are guesses from the picture).

1.11, #23. (0, 0) lies on the graph, so the numerator is a quadratic with a root at x = 0, as for example x^{2}. The graph has no vertical asymptotes, so the denominator is a quadratic with no real roots, as for example x^{2} + 1. Since the line y = 2 is a horizontal asymptote, the ratio of the leading coefficients of numerator and denominator is equal to 2. Moreover, the graph lies between the lines y = 0 and y = 2. The function y = 2x^{2}/(x^{2} + 1) satisfies all of the preceding requirements and indeed its graph has the indicated form.