Chapter 2

Sketch the graph of the function. \(f(x)=x^{2}-4 x\)

Find the domain of the function. \(f(x)=5 x^{2}+2 x-1\)

The value (in \(1982-1984\) dollars) of each dollar paid by consumers in each of the years from 1991 to 2005 in the United States is represented by the following ordered pairs. (Source: U.S. Bureau of Labor Statistics) \(\begin{array}{lll}(1991,0.734) & (1992,0.713) & (1993,0.692) \\ (1994,0.675) & (1995,0.656) & (1996,0.638) \\ (1997,0.623) & (1998,0.614) & (1999,0.600) \\\ (2000,0.581) & (2001,0.565) & (2002,0.556) \\ (2003,0.544) & (2004,0.530) & (2005,0.512)\end{array}\) (a) Use a spreadsheet software program to generate a scatter plot of the data. Let \(t=1\) represent 1991 . Do the data appear to be linear? (b) Use the regression feature of a spreadsheet software program to find a linear model for the data. (c) Use the model to estimate the value (in \(1982-1984\) dollars) of 1 dollar paid by consumers in 2007 and in 2008\. Discuss the reliability of your estimates based on your scatter plot and the graph of your linear model for the data.

Find an equation of the line passing through the points. \((3,5),(3,-2)\)

The weekly cost \(C\) of producing \(x\) units in a manufacturing process is given by the function \(C(x)=50 x+495\) The number of units \(x\) produced in \(t\) hours is given by \(x(t)=30 t\) Find and interpret \((C \circ x)(t)\).

Consider the graph of \(f(x)=|x|\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is vertically stretched by a factor of 4 and reflected in the \(x\) -axis.

Sketch the graph of the function. \(f(x)=1-x^{4}\)

Find the domain of the function. \(h(t)=\frac{4}{t}\)

Find an equation of the line passing through the points. \((-1,7),(3,7)\)

Consider the graph of \(f(x)=|x|\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. . The graph of \(f\) is vertically shrunk by a factor of \(\frac{1}{3}\) and shifted two units to the left.

Sketch the graph of the function. \(f(x)=x^{4}-4 x^{2}\)

Find the domain of the function. \(s(y)=\frac{3 y}{y+5}\)

Find an equation of the line passing through the points. \((3,-2),(-8,-2)\)

Consider the graph of \(g(x)=\sqrt{x}\) Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(g\) is shifted four units to the right and three units downward.

Sketch the graph of the function. \(f(x)=\frac{1}{3}(3+|x|)\)

Find the domain of the function. \(g(y)=\sqrt[3]{y-10}\)

Annual data from three years are used to create linear models for the population and the yearly snowfall of Reno, Nevada. Which model is more likely to give better predictions for future years? Discuss the appropriateness of using only three data points in each situation.

Find an equation of the line passing through the points. \(\left(2, \frac{1}{2}\right),\left(\frac{1}{2}, \frac{5}{4}\right)\)

Consider the graph of \(g(x)=\sqrt{x}\) Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(g\) is reflected in the \(x\) -axis, shifted two units to the left, and shifted one unit upward.

Sketch the graph of the function. \(f(x)=-1(1+|x|)\)

Find the domain of the function. \(f(t)=\sqrt[3]{t+4}\)

Find an equation of the line passing through the points. \((1,1),\left(6,-\frac{2}{3}\right)\)

Consider the graph of \(g(x)=\sqrt{x}\) Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(g\) is vertically shrunk by a factor of \(\frac{1}{2}\) and shifted three units to the right.

Sketch the graph of the function. \(f(x)=\sqrt{x+3}\)

Find the domain of the function. \(f(x)=\sqrt[4]{1-x^{2}}\)

Find an equation of the line passing through the points. \((1,0.6),(-2,-0.6)\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=5-3 x\)

The number of bacteria in a certain food product is given by \(N(T)=10 T^{2}-20 T+600, \quad 1 \leq T \leq 20\) where \(T\) is the temperature of the food. When the food is removed from the refrigerator, the temperature of the food is given by \(T(t)=3 t+1\) where \(t\) is the time in hours. Find (a) the composite function \(N(T(t))\) and (b) the time when the bacteria count reaches 1500 .

Consider the graph of \(g(x)=\sqrt{x}\) Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(g\) is vertically stretched by a factor of 2, reflected in the \(x\) -axis, and shifted three units upward.

Sketch the graph of the function. \(f(x)=\sqrt{x-1}\)

Find the domain of the function. \(g(x)=\sqrt{x+1}\)

Find an equation of the line passing through the points. \((-8,0.6),(2,-2.4)\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=2 x-3\)

The number of bacteria in a certain food product is given by \(N(T)=25 T^{2}-50 T+300, \quad 2 \leq T \leq 20\) where \(T\) is the temperature of the food. When the food is removed from the refrigerator, the temperature of the food is given by \(T(t)=2 t+1\) where \(t\) is the time in hours. Find (a) the composite function \(N(T(t))\) and \((\mathrm{b})\) the time when the bacteria count reaches 750 .

Find the domain of the function. \(g(x)=\frac{1}{x}-\frac{3}{x+2}\)

A fellow student does not understand why the slope of a vertical line is undefined. Describe how you would help this student understand the concept of undefined slope.

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=1-x^{2}\)

A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outermost ripple is given by \(r(t)=0.6 t\) where \(t\) is time in seconds after the pebble strikes the water. The area of the outermost circle is given by the function \(A(r)=\pi r^{2}\) Find and interpret \((A \circ r)(t)\).

Sketch the graph of the function. \(f(x)=2[x \rrbracket\)

Find the domain of the function. \(h(x)=\frac{10}{x^{2}-2 x}\)

Another student overhears your conversation in Exercise 61 and states, 'I do not understand why a horizontal line has zero slope and how that is different from undefined or no slope." Describe how you would explain the concepts of zero slope and undefined slope and how they are different from each other.

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=x^{2}-1\)

The suggested retail price of a new hybrid car is \(p\) dollars. The dealership advertises a factory rebate of $$\$ 2000$$ and a \(10 \%\) discount. (a) Write a function \(R\) in terms of \(p\) giving the cost of the hybrid car after receiving the rebate from the factory. (b) Write a function \(S\) in terms of \(p\) giving the cost of the hybrid car after receiving the dealership discount. (c) Form the composite functions \((R \circ S)(p)\) and \((S \circ R)(p)\) and interpret each. (d) Find \((R \circ S)(20,500)\) and \((S \circ R)(20,500)\). Which yields the lower cost for the hybrid car? Explain.

The point \((3,9)\) on the graph of \(f(x)=x^{2}\) has been shifted to the point \((4,7)\) after a rigid transformation. Identify the shift and write the new function \(g\) in terms of \(f\).

Sketch the graph of the function. \(f(x)=\llbracket x-1 \rrbracket\)

Find the domain of the function. \(f(x)=\frac{\sqrt{x+1}}{x-2}\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=x^{2}-4 x+3\)

The point \((8,2)\) on the graph of \(f(x)=\sqrt[3]{x}\) has been shifted to the point \((5,0)\) after a rigid transformation. Identify the shift and write the new function \(h\) in terms of \(\bar{f}\)

Sketch the graph of the function. \(f(x)=\llbracket x+1 \rrbracket\)

Find the domain of the function. \(f(s)=\frac{\sqrt{s-1}}{s-4}\)