Precálculo preguntas y respuestas

a. Find all zeros and vertical asymptotes of the rational function \( f(x)=\frac{x+5}{(x+11)^{2}} \). If there is more than one answer, enter all such values for each in a comma separated list. If are no solutions, enter NONE. Do not leave a blank empty. (a) The function has zero(s) at \( x= \) (numbers) (b) The function has vertical asymptote(s) at \( x= \) (c) The function's long-run behavior is that \( y \rightarrow \) (d) On a piece of paper, sketch a graph of this function without using your calculator. b. How did you determine the vertical asymptote? O. By comparing the degree of the numerator and denominator. O. By dividing the polynimial and using the remainder

Describe any transformation of the graph of \( f \) that yields the graph of \( g \). \[ f(x)=13^{x}, g(x)=13^{-x-5} \] Step 1 Rewrite the function \( g \) in terms of function \( f \). \( g(x)=f(\square \)

Use the definition of common and natural logarithms to simplify. \( e^{\ln (5.61)}+3 \) help (numbers)

? QUESTION The number of bacteria in a culture decreases according to a continuous exponential decay model. The initial population in a study is 400 bacteria, and there are 140 bacteria left after 6 minutes. (a) Let \( t \) be the time (in minutes) since the beginning of the study, and let \( y \) be the number of bacteria at time \( t \). Write a formula relating \( y \) to \( t \). Use exact expressions to fill in the missing parts of the formula. Do not use approximations. \( y=\square \)

Graph each function. Then identify the domain and range of the function. \( \begin{array}{lll}\text { 10. } y=-4 x^{2} & \text { 12. } f(x)=3 x^{2} \\ \text { 13. } f(x)=\frac{2}{3} x^{2} & \text { 14. } y(x)=1.5 x^{2} & \text { 15. } y=-\frac{1}{3} x^{2}\end{array} \)

QUESTION 4 Sketch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x-1}+1 \) Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

1. The population of a rural town can be modelled by the function \( P(x)=3 x^{2}-102 x+25000 \), where \( x \) is the number of years sin 2000. According to the model, when will the population be lowest?

QUESTION 3 The equation of a hyperbola is given by \( f(x)=\frac{3}{x-7}-4 \). Write down the equation of the new function that is formed when \( f \) is transformed as follows: \( \begin{array}{ll}3.1 & \text { Shift two units to the left } \\ 3.2 & \text { Shift } 3 \text { units up } \\ 3.3 & \text { Shift } 1 \text { unit right and } 2 \text { units down } \\ 3.4 & \text { The equation of the new hyperbola has new asymptotes at } x=-4 \text { and } y=-1\end{array} \)

4. The function \( y=-5^{x} \) is reflected in the \( x \)-axis, reflected in the \( y \)-axis, and translated 2 units vertically. Determine the transformation function. \( \begin{array}{llll}\text { a) } y=-5^{-x+2} & \text { b) } y=5^{-x}-2 & \text { c) } y=5^{-x}+2 & \text { d) } y=5^{-x+2}\end{array} \)

The manager of a hardware store sells batteries for \( \$ 5 \) a package. She wants to see how much money she will earn if she increases the price in \( 10 \notin \) increments. A model of the price change is the revenue function \( R(x)=-x^{2}+10 x+3000 \), where \( x \) is the number of \( 10 \notin \) increments and \( R(x) \) is in dollars. Explain how to determine the maximum revenue.