91Ó°ÊÓ

A Rational Function with a Slant Asymptote In Exercises \(49-62,\) (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. $$f(t)=-\frac{t^{2}+1}{t+5}$$

Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?

The pH of a solution decreases by one unit. By what factor does the hydrogen ion concentration increase?

Home Mortgage The total interest \(u\) paid on a home mortgage of \(P\) dollars at interest rate \(r\) for \(t\) years is $$u=P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right]$$ Consider a \(\$ 120,000\) home mortgage at 7\(\frac{1}{2} \%\) (a) Use a graphing utility to graph the total interest function. (b) Approximate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paying twice as much in interest charges as the size of the mortgage?

True or False? In Exercises \(61-64\) , determine whether the statement is true or false. Justify your answer. The domain of a logistic growth function cannot be the set of real numbers.

Use synthetic division to show that \(x\) is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. \(x^{3}+2 x^{2}-2 x-4=0, \quad x=\sqrt{2}\)

Finding Quadratic Functions In Exercises \(65-70\) , find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given \(x\) -intercepts. (There are many correct answers.) $$ (-1,0),(3,0) $$

In Exercises 73 and \(74,\) use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where s represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?

Average Speed A driver averaged 50 miles per hour on the round trip between two cities 100 miles apart. The average speeds for going and returning were \(x\) and \(y\) miles per hour, respectively. (a) Show that \(y=(25 x) /(x-25)\) . (b) Determine the vertical and horizontal asymptotes of the graph of the function. (c) Use a graphing utility to graph the function. (d) Complete the table. (e) Are the results in the table what you expected? Explain. (f) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain.

Finding the Zeros of a Polynomial Function, write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$