Precalculus review worksheet sec. 9.1 to 9.3

Interval of upward concavity: (, 1) (2, 4) (7, 8) c. We can now answer the final two questions. f 0 f 0 on an interval, then f is increasing on that interval.

2 2 y=f () Solution: First we draw picture 1 for this graph. coordinate(s) of the point(s) of inflection. Intervals over which f is concave upward or concave downward, d. Intervals over which f is increasing/decreasing, b. Below is the graph of the derivative f () of a continuous function f(). For picture 1 we will include the discontinuities. (Remember 3, -3 are not in the domain of f). This means that this function has no critical numbers, since its derivative is never 0 and it is differentiable at all in its domain. Picture = 0 f () = (2 9) (2) ( 2 9) ( 2 9) 2 = 0 = 2 9 ( 2 9) 2 The last equation has no solution, since it is equivalent to 2 = 9 which has no solution. To find the horizontal asymptotes we evaluate the following. We can find the vertical asymptotes by taking the it as approaches our discontinuities 3 and = 2 9 = 3 and = 3 are vertical asymptotes. Notice that f( ) = f(), so f is an odd function. Sketch the graph of Solution: Here f() = 2 9 y = 2 9 is a rational function, so its domain consists of all so that the denominator is nonzero, i.e. f 0 f is concave downward f is concave upward 1 (1,f(1)) is an inflection point f (0) = 6(0) 6 0Р”Р€ E. Picture 2 f () = 0 The critical number of f is = 1.

Picture = f () = 0 3( 2 2 3) = 0 3( + 1)( 3) = 0 The critical numbers of f are = 1 and = 3. We see there are no horizontal asymptotes. For horizontal asymptotes we check the its at Р’В±. To find the asymptotes, we first notice that there are no vertical asymptotes. We skip the -intercept since we would have to solve the cubic equation = 0. Here f() = is a polynomial, so its domain is (, ). Sketch the graph of y = Eamples Solution: A. Sketch: Use all of the previous information to sketch y = f().Р”Сћ 1. Concavity and Inflection Points: Draw picture 2 for f. Increase/Decrease and Etrema: Draw picture 1 for f. An oblique asymptote is a an asymptote which is neither a vertical nor a horizontal line. We have already found vertical and horizontal asymptotes of functions.

Also find the slant or oblique asymptotes of f. Asymptotes: Find the horizontal and vertical asymptotes of f. Also determine whether f is periodic and what the period P is. Symmetries: Determine whether f is an even function (f( ) = f()), and odd function (f( ) = f()) or neither.

$precalculus review worksheet sec. 9.1 to 9.3$

Skip finding -intercepts if f() is very complicated. Intercepts: Find the y-intercept (f(0)) and any -intercepts. 1 Notes on Curve Sketching The following checklist is a guide to sketching the curve y = f().

Precalculus review worksheet sec. 9.1 to 9.3