Given an equation for a hyperbola, write the asymptotes and graph the equation. \( \frac{(y+3)^{2}}{4}-\frac{(x+5)^{2}}{9}=1 \) Positive-slope asymptote: Negative-slope asymptote:

Question 3 (4 points) What is the range of the inverse function of the function \( f(x)=3-\sqrt{x+1} \) ? a) \( (-\infty,-1] \) b) \( [-1, \infty) \) c) \( [3, \infty) \) d) \( (-\infty, 3] \)

At the beginning of an experiment, a scientist has 312 grams of radioactive goo. After 225 minutes, her sample has decayed to 39 grams. What is the half-life of the goo in minutes? 75 Find a formula for \( G(t) \), the amount of goo remaining at time \( t . G(t)=312 e^{t} \) How many grams of goo will remain after 41 minutes?

The radioactive substance cesium-137 has a half-life of 30 years. The amount \( A(t) \) (in grams) of a sample of cesium-137 remaining after \( t \) years is given by the following exponential function. \[ A(t)=621\left(\frac{1}{2}\right)^{\frac{t}{30}} \] Find the initial amount in the sample and the amount remaining after 80 years. Round your answers to the nearest gram as necessary.

(a) Use the vertex formula to find the vertex. \( \begin{array}{ll}\text { (b) Find the intervals where } f \text { is increasing and where } f \text { is } \\ \text { decreasing. } \\ \begin{array}{ll}\text { 47. } f(x)=6-x^{2} & \text { 48. } f(x)=2 x^{2}-2 x+1 \\ \text { 49. } f(x)=x^{2}-6 x & \text { 50. } f(x)=-2 x^{2}+4 x+ \\ \text { 51. } f(x)=2 x^{2}-4 x+1 & \text { 52. } f(x)=-3 x^{2}+x-\end{array}\end{array} \).

What is the domain of the inverse function of the function \( f(x)=3-\sqrt{x} \). 1 ? a) \( (-\infty, 3] \) b) \( [-1, \infty) \) c) \( (-\infty,-1] \) d) \( [3, \infty) \)