Cassie and Cliff decided that their company would specialize in designing Web logs, or "blogs." Since production of any blog requires about the same amount of programming time, each customer could be charged the same price. Cassie and Cliff also estimated that the number of customers in a year would be related to the price. by n(x) = 600 - 2x, where x is the price charged per blog. Production of each blog will require about $200 in programmer labor. To get started in business, they bought a computer for$2,000. (a.) Write rules for the following functions, simplifying where possible. i. Annual income from blog designs I(x) if the price is x dollars per blog. ii. Annual production costs for blog designs C(x) if the price is x dollars per blog. iii. Annual profit from the business P(x) if the price is x dollars per biog. iv. Profit per customer $P_{c}(x)$ if the price is x dollars per blog. (b.) What would be reasonable domains for the functions in Part a? (c.) For what blog prices would the following functions be maximized? i. Income I(x). ii. Profit P(x). iii. Profit per customer $P_{\mathrm{c}}(x)$ (d.) What price would you recommend that Cassie and Cliff charge per blog? Explain your reasoning.

A family is relocating from St. Louis, Missouri, to California. Due to an increasing inventory of houses in St. Louis, it is taking longer than before to sell a house. The wife is concerned and wants to know when it is optimal to put their house on the market. Her realtor friend informs them that the last 26 houses that sold in their neighborhood took an average time of 218 days to sell. The realtor also tells them that based on her prior experience, the population standard deviation is 72 days.

a. What assumption regarding the population is necessary for making an interval estimate for the population mean?

b. Construct the $90 \%$ confidence interval for the mean sale time for all homes in the neighborhood.

A recent survey asked 5,324 individuals: What's most important to you when choosing where to live? The responses are shown by the following frequency distribution.

$$\begin{array}{|l|c|} \hline \text { Response } & \text { Frequency } \\ \hline \text { Good jobs } & 1,969 \\ \hline \text { Affordable homes } & 799 \\ \hline \text { Top schools } & 586 \\ \hline \text { Low crime } & 1,225 \\ \hline \text { Things to do } & 745 \\ \hline \end{array}$$

a. Calculate the margin of error used in the 95% confidence level for the population proportion of those who believe that low crime is most important.

b. Calculate the margin of error used in the 95% confidence level for the population proportion of those who believe that good jobs or affordable homes are most important.

c. Explain why the margins of error in parts a and $b$ are different.

A recent survey asked 5,324 individuals: What's most important to you when choosing where to live? The responses are shown by the following frequency distribution.

$$\begin{array}{|l|c|} \hline \text { Response } & \text { Frequency } \\ \hline \text { Good jobs } & 1,969 \\ \hline \text { Affordable homes } & 799 \\ \hline \text { Top schools } & 586 \\ \hline \text { Low crime } & 1,225 \\ \hline \text { Things to do } & 745 \\ \hline \end{array}$$

a. Calculate the margin of error used in the $95 \%$ confidence level for the population proportion of those who believe that low crime is most important.

b. Calculate the margin of error used in the $95 \%$ confidence level for the population proportion of those who believe that good jobs or affordable homes are most important.

$$\overline{\text { Page } 310}$$

c. Explain why the margins of error in parts a and $\mathrm{b}$ are different.

Make a difference table for Table O. Explain why Table O cannot come from a linear function.
$\quad \quad \quad \quad \quad \quad \quad \quad \quad$ Table O

$$\begin{array}{|c|c|c|c|c|}\hline \text{Input} & \text{Output} \\ \hline 0 & 0\\ \hline 1 & 5\\ \hline 2 & 20\\ \hline 3 & 45\\ \hline 4 & 80\\ \hline \end{array}$$

At a carnival game, a wading pool contains 50 rubber ducks, and 20 ducks are marked on the bottom with a star. A player selects a duck, does not replace it, and then selects another duck. If both ducks are marked with a star, the player wins a prize. Bella decided to play only if her probability of winning is at least 20%. Should she play the game? Use probability to justify your answer.

Consider the operation $\star$ for all real numbers x and y defined by x$\star$y=xy/(x+y). We note that $(\mathbb{R},\star)$ is not a group for a variety of reasons, not the least of which is that x$\star$y might not be defined (we might divide by 0). The situation, however, is not hopeless; let's do some repairs. First, let's deal with the divide-by-zero issue by extending the real numbers to also include the "number" $\infty$. With this extension, we can have (-3)$\star$3=(-3)$\cdot$3/(-3+3)=-9/0=$\infty$. That's acceptable, but 0/0 is a worse problem, so let's simply ban 0 from the set of allowable values for $\star$. That is, we define

$$\overset{\sim}{\mathbb{R}}=\mathbb{R}-\{0\}\cup \{\infty\}$$

That is $\overset{\sim}{\mathbb{R}}$ consists of all nonzero real numbers and the additional "number" $\infty$. Give sensible meanings to x$\star\infty$, $\infty\star$x and $\infty\star\infty$ (where x is a nonzero real number) and show that $(\overset{\sim}{\mathbb{R}},\star)$ is an Abelian group.

How is the return on assets ratio computed?

When the concert promoters asked their business manager to find a function W(x) that shows combined profit per ticket from ticket and snack bar sales, she suggested two possibilities.

$$W_{1}(x)=\frac{P_{c}(x)}{n(x)}+\frac{P_{b}(x)}{n(x)} \quad \text { and } \quad W_{2}(x)=\frac{P_{c}(x)+P_{b}(b)}{n(x)}$$

(a.) Using the algebraic expressions for each function involved, you get.

$$\begin{array}{c}{W_{1}(x)=\frac{-25 x^{2}+875 x-4,750}{-25 x+750}+\frac{-175 x+5,250}{-25 x+750}} \\ {\text { and } \quad W_{2}(x)=\frac{-25 x^{2}+700 x+500}{-25 x+750}}\end{array}$$

How could you convince someone that both ways of calculating total profit per ticket are correct by: i. reasoning about the meaning of each expression in the concert business situation? ii. using what you krow about addition of arithmetic fractions like $\frac{2}{5}+\frac{4}{5}$? (b.) Which of the two expressions for total profit per ticket would be better for: i. showing how each profit source contributes to total profit per ticket? ii. calculating total profit per ticket most efficiently? (c.) Find what you believe to be the expression that is most efficient for calculating W(x). Use that expression to find the profit per ticket from both ticket and snack bar sales when the ticket prices are $10,$15, $20, and$25. (d.) Estimate the ticket price that will yield maximum profit per ticket for: i. the concert operation alone, R(x). ii. the snack bar operation alone, V(x). iii. the combination of concert and snack bar operations, W(x).

Consider next the question of how to locate maximum or minimum points on the graph of a quadratic function $f(x)=a(x-h)^{2}+k$. Recall that this point is called the vertex of the graph. To develop an answer to this question, you might analyze several specific examples like I-IV below to make a conjecture. Then use algebraic reasoning to justify your conjecture. Or you might use more general algebraic reasoning to develop a conjecture and then your ideas with the specific examples. I. $r(x)=(x-3)^{2}+5$ II. $s(x)=3(x-7)^{2}-4$ III. $t(x)=-2(x+6)^{2}+3$ IV. $v(x)=-0.5(x+5)^{2}-7$ (a.) How can you tell from the parameters a, h, and k when the graph of the function $f(x)=a(x-h)^{2}+k$ will have a minimum point? When it will have a maximum point? (b.) Consider the cases when the graph of the function has a minimum point. i. For what value of x does the expression $a(x-h)^{2}+k$ take on its smallest value? ii. What is the y-coordinate of the corresponding point on the graph? iii. What are the coordinates of the minimum point when a > 0? (c.) Now consider the cases when the function has a maximum point. i. For what value of x does the expression $a(x-h)^{2}+k$ take on its largest possible value? ii. What is the y-coordinate of the corresponding point on the graph? iii. What are the coordinates of the maximum point when a < 0?

Presented below are data on three promissory notes. Determine the missing amounts.

Suppose that $f(x)=\frac{x^{2}}{2}$ and $g(x)=\frac{12}{x}$. (a.) If $h(x)=f(x) \cdot g(x)$, what do you think will be the simplest expression that gives correct values of h(x) for all x? (b.) Test your idea in Part a by comparing tables of values for $\left(\frac{x^{2}}{2}\right)\left(\frac{12}{x}\right)$ and for your proposed simpler expression to calculate h(x). Use values of x in the interval [-10, 10]. (c.) Test your idea in Part a by graphing $y=\left(\frac{x^{2}}{2}\right)\left(\frac{12}{x}\right)$ for values of x in the interval [-10, 10] and then graphing y = h(x), using your proposed simplified expression for h(x). (d.) What similarities and what differences between multiplication of numeric and algebraic fractions are suggested by the results of your work on Parts a-c?

Sketch graphs of n(x) = 750 - 25x and $P_{c}(x)=-25 x^{2}+875 x-4,750$. Use the graphs to answer the following questions. a. What is the practical domain of n(x)? That is, what prices x will give predicted numbers of tickets that make sense in the concert situation? b. Estimate the ticket price for which profit from concert operation $P_{c}(x)$, excluding snack bar operations, is maximized. Find that profit.

One solution of a quadratic equation $a x^{2}+b x+c=0$ is 2 + 3i. What is the other solution? Explain your reasoning.