Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci at \( (0,-8) \) and \( (0,8) \); vertices at \( (0,3) \) and \( (0,-3) \)

Find the maximum \( y \)-value on the graph of \( y=f(x) \). \[ f(x)=-4 x^{2}+x-7 \] The maximum \( y \)-value is \( y=\square \). (Type an integer or a fraction.)

The size of a certain insect population is given by \( \mathrm{P}(\mathrm{t})=400 e^{.03 \mathrm{t}} \), where t is measured in days. At what time will the population equal 1600 ?

Use the remainder theorem to find the remainder when \( f(x) \) is divided by \( x-1 \). Then use the factor theorem to determine whether \( x-1 \) is a factor of \( f(x) \). \( f(x)=3 x^{3}-2 x^{2}+7 x-2 \) The remainder is \( \square \).

Use the graph to evaluate the following expressions. (a) \( (\mathrm{f} \circ \mathrm{g})(1) \) (b) \( (\mathrm{g} \circ \mathrm{f})(0) \) (c) \( (\mathrm{g} \circ \mathrm{g})(0) \) (a) Evaluate \( (\mathrm{f} \circ \mathrm{g})(1) \). ( \( \mathrm{f} \circ \mathrm{g})(1)=1 \) (b) Evaluate \( (\mathrm{g} \circ \mathrm{f})(0) \). ( \( \mathrm{g} \circ \mathrm{f})(0)=0 \) (c) Evaluate \( (\mathrm{g} \circ \mathrm{g})(0) \). ( \( \mathrm{g} \circ \mathrm{g})(0)=0 \)

Use transformations of the graph of \( y=x^{5} \) to graph the function. \( f(x)=(x-8)^{5}+6 \) Which tranformations are needed to graph the function? A. The graph of \( y=x^{5} \) should be horizontally shifted to the left by 6 units and shifted vertically down by 8 . B. The graph of \( y=x^{5} \) should be horizontally shifted to the right by 6 units and shifted vertically up by 8 . The graph of \( y=x^{5} \) should be horizontally shifted the the left by 8 units and shifted vertically down by 6 . D. The graph of \( y=x^{5} \) should be horizontally shifted to the right by 8 units and shifted vertically up by 6 .