Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the \(x\) -intercepts (if any). $$ f(x)=x^{2}+3 x+2 $$

Plutonium 239 is used as fuel for some nuclear reactors, and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long would it take 10 grams of plutonium 239 to decay to 1 gram? (Round your answer to three significant digits.)

Your company's market share is undergoing steady growth. Explain why a logarithmic function is not appropriate for long-term backward extrapolation of your market share.

Internet Use The following graph shows the actual percentage of U.S. residents who used the Internet at home as a function of income (the data points) and a logistic model of these data (the curve). The logistic model is given by $$ P(x)=\frac{64.2}{1+9.6(1.06)^{-x}} \text { percent } $$ where \(x\) is an individual's income in thousands of dollars. a. According to the model, what percentage of extremely wealthy people used the Internet at home? b. For low incomes, the logistic model is approximately exponential. Which exponential model best approximates \(P(x)\) for small \(x\) ? c. According to the model, \(50 \%\) of individuals with what income used the Internet at home in \(2000 ?\) (Round the answer to the nearest \(\$ 1,000\) ).

Suppose the height of a stone thrown vertically upward is given by a quadratic function of time. What is the significance of the coordinates of the vertex, the (possible) \(x\) -intercepts, and the \(y\) -intercept?

Simplify: \(\ln \left(e^{x}\right)\)