Graphs of Rational Functions. 386 Chapter 7 Linear Models and Graphs of Nonlinear Models Equation of ellipse ab
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1 Chapter 7 Linear Models and Graphs of Nonlinear Models. Equation of ellipse or feet 7..9 ab.9 ab a b A ab 9 ab 9 a a a a 9 a a 9 a a a b a b b a 9. The four tpes of conics are circles, parabolas, ellipses, and hperbolas.. An ellipse is the set of all points, such that the sum of the distances between, and two distinct fied points is a constant. a b or b a. An ellipse is a circle if the coefficients of the second degree terms are equal.. The central rectangle of a hperbola can be used to sketch its asmptotes because the asmptotes are the etended diagonals of the central rectangle. 7. is the top half of the hperbola. 9 Section 7. Graphs of Rational Functions. (a) (b) (c) Domain:....,,...
2 Section 7. Graphs of Rational Functions 7. (a) (b) (c) Domain: ,,. (a) (b) (c) Domain: 9,,, 7. f 9. f Domain: Domain:,, Vertical asmptote: Horizontal asmptote: since the degree of the,, Vertical asmptote: Horizontal asmptote: since the degree of the the leading coefficients are.. t gt t 9 Domain: t 9 t,, Vertical asmptote: t Horizontal asmptote: since the degree of the the leading coefficient of the numerator is and the leading coefficient of the denominator is.. Domain:,, Vertical asmptote: Horizontal asmptote: since the degree of the the leading coefficient of the numerator is and the leading coefficient of the denominator is.
3 Chapter 7 Linear Models and Graphs of Nonlinear Models. gt tt tt t t t,,, Vertical asmptotes: t, t Horizontal asmptote: since the degree of the 7. Domain:, no real solution Vertical asmptote: none Horizontal asmptote: since the degree of the the leading coefficient of the numerator is and the leading coefficient of the denominator is. 9. Domain:,,, Vertical asmptotes:, Horizontal asmptote: since the degree of the the leading coefficient of the numerator is and the leading coefficient of the denominator is.. gz z gz z z z Domain: z z z,, Vertical asmptote: z Horizontal asmptote: since the degree of the the leading coefficients are.. g Domain:,, Vertical asmptote: Horizontal asmptote: none since the degree of the numerator is greater than the degree of the denominator.. f matches with graph (d). Vertical asmptote: Horizontal asmptote: 7. f matches with graph (b). Vertical asmptote: Horizontal asmptote: 9. (d). (a)
4 Section 7. Graphs of Rational Functions 9. g - intercept: g undefined, none -intercept: none, numerator is never zero. Vertical asmptote: Horizontal asmptote: since the degree of the. g -intercept: g -intercept: none, numerator is never zero. Vertical asmptote: Horizontal asmptote: since the degree of the 7. f -intercept: f -intercept: none, numerator is never zero. Vertical asmptote: Horizontal asmptote: since the degree of the 9. g -intercept: -intercept: none, numerator is never zero Vertical asmptote: Horizontal asmptote: since the degree of the g. - intercept: undefined, none -intercept: Vertical asmptote: ; none Horizontal asmptote: since the degree of the
5 9 Chapter 7 Linear Models and Graphs of Nonlinear Models. hu -intercept: u u u - intercept: h undefined, none u u u u u u u uu u, none, since h is undefined. Vertical asmptote: u u uu u Horizontal asmptote: since the degrees are equal and the leading coefficient of the numerator is and the leading coefficient of the denominator is. u. -intercept: -intercept: undefined, none. Vertical asmptote: Horizontal asmptote: since the degree of the numerator is equal to the degree of the denominator and the leading coefficient of the numerator is and the leading coefficient of the denominator is intercept: -intercept: Vertical asmptote: none, has no real solutions. Horizontal asmptote: since the degree of the numerator is equal to the degree of the denominator and the leading coefficient of the numerator is and the leading coefficient of the denominator is intercept: -intercept: none, numerator is never zero. Vertical asmptote: none, no real solution Horizontal asmptote: since the degree of the
6 Section 7. Graphs of Rational Functions 9. gt t - intercept: g undefined, none -intercept: t t t t t Vertical asmptote: t Horizontal asmptote:. - intercept: -intercept: Vertical asmptote: Horizontal asmptote:, since the degree of the. f - intercept: -intercept: Vertical asmptotes: Vertical asmptote: none Horizontal asmptote: since the degree of the the leading coefficient of the numerator is and the leading coefficient of the denominator is.
7 9 Chapter 7 Linear Models and Graphs of Nonlinear Models 7. f intercept: f - intercept: - Vertical asmptotes: undefined at hole in graph Horizontal asmptote: since the degrees are equal and the leading coefficients are. f gives a hole in graph at 9. f Domain:,, Vertical asmptote: Horizontal asmptote: Kestrokes: X,T, Y GRAPH. h Domain:,, Vertical asmptote: Horizontal asmptote: Kestrokes: Y X,T, X,T, GRAPH 7. f t t Domain: t 7, Vertical asmptote: none Horizontal asmptote: Kestrokes: Y X,T, GRAPH
8 Section 7. Graphs of Rational Functions 9. Domain:,, Vertical asmptote: Horizontal asmptote: Kestrokes: Y X,T, X,T, 9 GRAPH 9 7. Domain:,,, Vertical asmptotes:,,, Horizontal asmptote: since the degree of the Kestrokes: Y X,T, X,T, GRAPH or Y X,T, X,T, X,T, GRAPH 7 9. Reduce g to lowest terms. g Kestrokes: Y X,T, X,T, GRAPH There is no vertical asmptote because the fraction is not reduced to lowest terms. 7. (a) (b) Average cost C C C Cost Number of units..,,., < $ $.7 (c) Kestrokes: Y. X,T, X,T, GRAPH Horizontal asmptote C $. since the degree of the numerator is equal to the degree of the denominator and the leading coefficient of the numerator is. and the leading coefficient of the denominator is. As the number of units produced increases, the average cost is approimatel $.. 7. (a) C is the horizontal asmptote, since the degree of the numerator is less than the degree of the denominator. The meaning in the contet of the problem is that the chemical is eliminated from the bod. (b) Kestrokes: Y X,T, X,T, Maimum occurs when t.. GRAPH.
9 9 Chapter 7 Linear Models and Graphs of Nonlinear Models 7. (a) answers will var. (c) Domain: > or, (d) Minimum perimeter: units units Kestrokes: X,T, Y X,T, GRAPH (b) A P l w P l w P (c)..79. Domain:. -intercept: Horizontal asmptote:.79. since the degrees are equal.. Vertical asmptote: 7.9 (the ecluded value of the domain) (d) Kestrokes: Y..79 X,T,. X,T, GRAPH Plot,.,,.9,,.,, 9.,, 7.,, 7.7 in STAT then enter,,,,,, in and enter.,.9,., 9., 7., 7.7, in L. STAT PLOT ON GRAPH L.. The model appears to be accurate for the restricted domain. CONTINUED
10 Review Eercises for Chapter 7 9. CONTINUED (e) The models are not accurate for the ears before 99 and after 99. Use the quadratic model to estimate the value of the shipment in 99, because the rational function evaluated at is negative.. An asmptote of a graph is a line to which the graph becomes arbitraril close as or increases without bound.. No, not when the domain is all reals. For eample, f has no vertical asmptote. Review Eercises for Chapter 7. P varies directl as the cube of t. P kt. z varies inversel as the square of s. z k s. k k k 7. T krs k.9 9, k k T rs 9. >.
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