91Ó°ÊÓ

Power functions and exponential functions both have exponents. What major algebraic difference distinguishes these two types of functions?

a. Show that the values in the table have the multiply-add property. b. Use the first and last points to find algebraically the particular equation of the natural logarithmic function that fits the points. c. Show that the equation in part b gives the other points in the table. $$\begin{array}{rr}x & y \\\\\hline 3.6 & 1 \\\14.4 & 2 \\\57.6 & 3 \\\230.4 & 4 \\\921.6 & 5\end{array}$$

What geometrical feature do quadratic function graphs have that linear, exponential, and power function graphs do not have?

Determine whether the data has the add-add, add-multiply, multiply-multiply, or constant-second-differences pattern. Identify the type of function that has the pattern. $$\begin{array}{rr} x & f(x) \\ \hline 2 & 12 \\ 4 & 48 \\ 6 & 108 \\ 8 & 192 \\ 10 & 300 \end{array}$$

Determine whether the data has the add-add, add-multiply, multiply-multiply, or constant-second-differences pattern. Identify the type of function that has the pattern. $$\begin{array}{rr} x & f(x) \\ \hline 2 & 12 \\ 4 & 48 \\ 6 & 192 \\ 8 & 768 \\ 10 & 3072 \end{array}$$

Explain why the reciprocal function \(f(x)=\frac{1}{x}\) is also a power function.

Graph the functions and identify their domains. $$f(x)=4 \log _{2}(3 x+5)$$

Suppose that \(y\) increases exponentially with \(x\) and that \(z\) is directly proportional to the square of \(x\). Sketch the graph of each type of function. In what ways are the two graphs similar to one another? What major graphical difference would allow you to tell which graph is which if they were not marked?

Suppose that \(y\) varies directly with \(x\) and that \(z\) increases linearly with \(x .\) Explain why any direct-variation function is a linear function but a linear function is not necessarily a direct-variation function.

The areas of similarly shaped objects are directly proportional to the square of a linear dimension. a. Give the formula for the area of a circle. Explain why the area varies directly with the square of the radius. b. If a grapefruit has twice the diameter of an orange, how do the areas of their rinds compare? c. When Gutzon Borglum designed the reliefs he carved into Mount Rushmore in South Dakota, he started with models \(\frac{1}{12}\) the lengths of the actual reliefs. How does the area of each model compare to the area of each of the final reliefs? Explain why a relatively small decrease in the linear dimension results in a relatively large decrease in the surface areas to be carved. d. Gulliver traveled to Brobdingnag, where people were 10 times as tall as normal people. If Gulliver had \(2 \mathrm{m}^{2}\) of skin, how much skin surface would you expect a Brobdingnagian to have had?