Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food sup- plies. Under such conditions the population follows a logistic growth model: \( P ( t ) = \frac { d } { 1 + k e ^ { - c t } } \) where \( c , d \) , and \( k \) are positive constants. For a certain fish population in a small pond \( d = 1200 , k = 11 , c = 0.2 \) and \( t \) is measured in years. The fish were introduced into the pond at time \( t = 0 \) . (a) How many fish were originally put in the pond? (b) Find the population after \( 10,20 \) , and \( 30 \) years. (c) Evaluate \( P ( t ) \) for large values of \( t \) . What value does the population approach as \( t \rightarrow \infty \) ? Does the graph shown confirm your calculations? 

Given the function with the equation \( k(x)=\frac{-6}{x+2}-2 \) : \( 7.1 \quad \) Write the equations of the asymptotes of \( k \).

Given the function with the equation \( k(x)=\frac{-6}{x+2}-2 \) : 7.1 Write the equations of the asymptotes of \( k \). 7.2 Give the range of \( k \). 7.3 Give the domain of \( k \). 7.4 Drite the equation of the axis of symmetry, with a positive gradient. 7.5 If \( k \) is reflected over the \( x \)-axis and shifted upwards by 2 units, 7.6 give the equation of the new function, \( j(x) \), after the transformations. 7.7 In which quadrant(s) would the arms of \( k \) not be if it was sketched?

(a) Express the distance \( d \) from \( P \) to the origin as a function of \( x \). \( d(x)=\sqrt{\mathrm{x}^{4}-3 \mathrm{x}^{2}+4} \) (Simplify your answer. Type an exact answer, using radicals as needed.) (b) What is \( d \) if \( x=0 \) ? If \( x=0 \), the distance \( d \) is (Type an integer or decimal rounded to two decimal places as needed.)

(a) Express the distance \( d \) from \( P \) to the origin as a function of \( x \). \( d(x)=\sqrt{x^{4}-3 x^{2}+4} \) (Simplify your answer. Type an exact answer, using radicals as needed.) (b) What is \( d \) if \( x=0 \) ? If \( x=0 \), the distance \( d \) is 2 . (Type an integer or decimal rounded to two decimal placels as needed.) (c) What is \( d \) if \( x=1 \) ? If \( x=1 \), the distance \( d \) is \( \square \). (Type an integer or decimal rounded to two decimal places as needed.)

If \( (-1,8) \) is a point on the graph of \( y=f(x) \), which of the following must be on the graph of \( y=f(-x) \) ? \( (8,1) \) \( (-1,-8) \) Choose the correct answer below. \( (1,8) \) \( (-8,-1) \)