Physics Equipment A physics student obtains the following

Factor the following expressions completely over the real numbers. \(x^{2}-16\)

Factor the following expressions completely over the real numbers. \(x^{2}+10 x+25\)

Factor the following expressions completely over the real numbers. \(81 y^{2}-4\)

Factor the following expressions completely over the real numbers. \(3 x^{3}-15 x^{2}+18 x\)

Factor the following expressions completely over the real numbers. \(16 h^{4}-81\)

Factor the following expressions completely over the real numbers. \(x^{2}+2 x h+h^{2}\)

Factor the following expressions completely over the real numbers. \(x^{2}+3 x-4\)

Factor the following expressions completely over the real numbers. \(x^{2}-3 x+4\)

Factor the following expressions completely over the real numbers. \(2 x^{2}-11 x+5\)

Factor the following expressions completely over the real numbers. \(x^{4}+x^{2}-20\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

In Exercises 1–10, match the numerical model to the corresponding graphical model (a–j) and algebraic model (k–t). (k) \(y=x^{2}+x\) (l) \(y=40-x^{2}\) (m) y = (x + 1) (x - 1) (n) \(y=\sqrt{x-3}\) (o) y = 100 - 2x (p) y = 3x - 2 (q) y = 2x (r) \(y=x^{2}+2\) (s) y = 2x + 3 (t) \(y=\frac{x-3}{2}\)

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. (a) According to the numerical model, what has been the trend in females joining the work force since 1954? (b) In what 5-year interval did the percentage of women who were employed change the most?

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. (a) According to the numerical model, what has been the trend in males joining the work force since 1954? (b) In what 5-year interval did the percentage of men who were employed change the most?

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. Model the data graphically with two scatter plots on the same graph, one showing the percentage of women employed as a function of time and the other showing the same for men. Measure time in years since 1954.

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. Are the male percentages falling faster than the female percentages are rising, or vice versa?

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. Model the data algebraically with linear equations of the form y = mx + b. Write one equation for the women’s data and another equation for the men’s data. Use the 1954 and 1999 ordered pairs to compute the slopes.

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. If the percentages continue to follow the linear models you found in Exercise 15, what will the employment percentages for women and men be in the year 2009?

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. If the percentages continue to follow the linear models you found in Exercise 15, when will the percentages of women and men in the civilian work force be the same? What percentage will that be?

Exercises 11–18 refer to the data in Table 1.6 below, showing the percentage of the female and male populations in the United States employed in the civilian work force in selected years from 1954 to 2004. Explain why the percentages cannot continue indefinitely to follow the linear models that you wrote in Exercise 15.

Doing Arithmetic with Lists Enter the data from the Total column of Table 1.2 of Example 2 into list \(L_1\) in your calculator. Enter the data from the Female column into list \(L_2\). Check a few computations to see that the procedures in (a) and (b) cause the calculator to divide each element of \(L_2\) by the corresponding entry in \(L_1\) , multiply it by 100, and store the resulting list of percentages in \(L_3\). (a) On the home screen, enter the command: \(100 \times \mathrm{L}_{2}/ \mathrm{L}_{1} \rightarrow \mathrm{L}_{3}\). (b) Go to the top of list \(L_3\) and enter \(\mathrm{L}_{3}=100\left(\mathrm{L}_{2} / \mathrm{L}_{1}\right)\).

Comparing Cakes A bakery sells a \(9^{\prime \prime}\) by \(13^{\prime \prime}\) cake for the same price as an \(8^{\prime \prime}\) diameter round cake. If the round cake is twice the height of the rectangular cake, which option gives the most cake for the money?

Stepping Stones A garden shop sells \(12^{\prime \prime}\) by \(12^{\prime \prime}\) square stepping stones for the same price as \(13^{\prime \prime}\) round stones. If all of the stepping stones are the same thickness, which option gives the most rock for the money?

Free Fall of a Smoke Bomb At the Oshkosh, WI, air show, Jake Trouper drops a smoke bomb to signal the official beginning of the show. Ignoring air resistance, an object in free fall will fall d feet in t seconds, where d and t are related by the algebraic model \(d=16 t^{2}\). (a) How long will it take the bomb to fall 180 feet? (b) If the smoke bomb is in free fall for 12.5 seconds after it is dropped, how high was the airplane when the smoke bomb was dropped?

Physics Equipment A physics student obtains the following data involving a ball rolling down an inclined plane, where t is the elapsed time in seconds and y is the distance traveled in inches. Find an algebraic model that fits the data.

U.S. Air Travel The number of revenue passengers enplaned in the United States over the 14-year period from 1994 to 2007 is shown in Table 1.7. (a) Graph a scatter plot of the data. Let x be the number of years since 1994. (b) From 1994 to 2000 the data seem to follow a linear model. Use the 1994 and 2000 points to find an equation of the line and superimpose the line on the scatter plot. (c) According to the linear model, in what year did the number of passengers seem destined to reach 900 million? (d) What happened to disrupt the linear model?

Exercises 25–28 refer to the graph below, which shows the minimum salaries in major league baseball over a recent 18-year period and the average salaries in major league baseball over the same period. Salaries are measured in dollars and time is measured after the starting year (year 0).

Exercises 25–28 refer to the graph below, which shows the minimum salaries in major league baseball over a recent 18-year period and the average salaries in major league baseball over the same period. Salaries are measured in dollars and time is measured after the starting year (year 0). After Peter Ueberroth’s resignation as baseball commissioner in 1988 and his successor’s untimely death in 1989, the team owners broke free of previous restrictions and began an era of competitive spending on player salaries. Identify where the 1990 salaries appear in the graph and explain how you can spot them.

Exercises 25–28 refer to the graph below, which shows the minimum salaries in major league baseball over a recent 18-year period and the average salaries in major league baseball over the same period. Salaries are measured in dollars and time is measured after the starting year (year 0). The owners attempted to halt the uncontrolled spending by proposing a salary cap, which prompted a players’ strike in 1994. The strike caused the 1995 season to be shortened and left many fans angry. Identify where the 1995 salaries appear in the graph and explain how you can spot them.

Exercises 25–28 refer to the graph below, which shows the minimum salaries in major league baseball over a recent 18-year period and the average salaries in major league baseball over the same period. Salaries are measured in dollars and time is measured after the starting year (year 0). Analyze the general patterns in the graphical model and give your thoughts about what the long- term implications might be for (a) the players; (b) the team owners; (c) the baseball fans.

In Exercises 29–38, solve the equation algebraically and confirm graphically. \(v^{2}-5=8-2 v^{2}\)

In Exercises 29–38, solve the equation algebraically and confirm graphically. \((x+11)^{2}=121\)

In Exercises 29–38, solve the equation algebraically and confirm graphically. \(2 x^{2}-5 x+2=(x-3)(x-2)+3 x\)

In Exercises 29–38, solve the equation algebraically and confirm graphically. \(x^{2}-7 x-\frac{3}{4}=0\)

In Exercises 29–38, solve the equation algebraically and confirm graphically. x(2x - 5) = 12

In Exercises 29–38, solve the equation algebraically and confirm graphically. x(2x - 1) = 10

In Exercises 29–38, solve the equation algebraically and confirm graphically. x(x + 7) = 14

In Exercises 29–38, solve the equation algebraically and confirm graphically. \(x^{2}-3 x+4=2 x^{2}-7 x-8\)

In Exercises 29–38, solve the equation algebraically and confirm graphically. \(x+1-2 \sqrt{x+4}=0\)

In Exercises 29–38, solve the equation algebraically and confirm graphically. \(\sqrt{x}+x=1\)

In Exercises 39–46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. \(2 x-5=\sqrt{x+4}\)

In Exercises 39–46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. \(|3 x-2|=2 \sqrt{x+8}\)

In Exercises 39–46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. |2x - 5| = 4 - |x - 3|

In Exercises 39–46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. \(\sqrt{x+6}=6-2 \sqrt{5-x}\)

In Exercises 39–46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. \(2 x-3=x^{3}-5\)

In Exercises 39–46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. \(x+1=x^{3}-2 x-5\)

In Exercises 39-46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. (x + 1)-1 = x-1 + x

In Exercises 39-46, solve the equation graphically by converting it to an equivalent equation with 0 on the right-hand side and then finding the x-intercepts. x2 = |x|

Swan Auto Rental charges $32 per day plus $0.18 per mile for an automobile rental. (a) Elaine rented a car for one day and she drove 83 miles. How much did she pay? (b) Ramon paid $69.80 to rent a car for one day. How far did he drive?

Connecting Graphs and Equations The curves on the graph below are the graphs of the three curves given by y1 = 4x + 5 y2 = x3 + 2x2 - x + 3 y3 = -x3 - 2x2 + 5x + 2. (a) Write an equation that can be solved to find the points of intersection of the graphs of y1 and y2 . (b) Write an equation that can be solved to find the x-intercepts of the graph of y3? (c) Writing to Learn How does the graphical model reflect the fact that the answers to (a) and (b) are equivalent algebraically? (d) Confirm numerically that the x-intercepts of y3 give the same values when substituted into the expressions for y1 and y2.

Exploring Grapher Failure Let y = (x200)1/200. (a) Explain algebraically why y= x for all x \(\geq\) 0 (b) Graph the equation y = (x200)1/200 in the window [0, 1] by [0, 1]. (c) Is the graph different from the graph of y=x? (d) Can you explain why the grapher failed?

Connecting Algebra and Geometry Explain how the algebraic equation (x + b)2 = x2 + 2bx + b2 models the areas of the regions in the geometric figure shown below on the left:

Exploring Hidden Behavior Solving graphically, find all real solutions to the following equations. Watch out for hidden behavior. (a) y = 10x3 + 7.5x2 - 54.85x + 37.95 (b) y = x3 + x2 - 4.99x + 3.03

Connecting Algebra and Geometry The geometric figure shown on the right above is a large square with a small square missing. (a) Find the area of the figure. (b) What area must be added to complete the large square? (c) Explain how the algebraic formula for completing the square models the completing of the square in (b).

Proving a Theorem Prove that if n is a positive integer, then n2 + 2n is either odd or a multiple of 4. Compare your proof with those of your classmates.

Writing to Learn The graph below shows the distance from home against time for a jogger. Using information from the graph, write a paragraph describing the joggers workout.

True or False A product of real numbers is zero if and only if every factor in the product is zero. Justify your answer.

True or False An algebraic model can always be used to make accurate predictions.

In Exercises 57–60, you may use a graphing calculator to decide which algebraic model corresponds to the given graphical or numerical model. (A) y = 2x + 3 (B) y= x2+5 (C) y = 12 - 3x (D) y =4x + 3 (E) \(y=\sqrt{8-x}\)

In Exercises 57–60, you may use a graphing calculator to decide which algebraic model corresponds to the given graphical or numerical model. (A) y = 2x + 3 (B) y= x2+5 (C) y = 12 - 3x (D) y =4x + 3 (E) \(y=\sqrt{8-x}\)

In Exercises 57–60, you may use a graphing calculator to decide which algebraic model corresponds to the given graphical or numerical model. (A) y = 2x + 3 (B) y= x2+5 (C) y = 12 - 3x (D) y =4x + 3 (E) \(y=\sqrt{8-x}\)

In Exercises 57–60, you may use a graphing calculator to decide which algebraic model corresponds to the given graphical or numerical model. (A) y = 2x + 3 (B) y= x2+5 (C) y = 12 - 3x (D) y =4x + 3 (E) \(y=\sqrt{8-x}\)

Analyzing the Market Both Ahmad and LaToya watch the stock market throughout the year for stocks that make significant jumps from one month to another. When they spot one, each buys 100 shares. Ahmads rule is to sell the stock if it fails to perform well for three months in a row. LaToyas rule is to sell in December if the stock has failed to perform well since its purchase. The graph below shows the monthly performance in dollars (JanDec) of a stock that both Ahmad and LaToya have been watching. (a) Both Ahmad and LaToya bought the stock early in the year. In which month? (b) At approximately what price did they buy the stock? (c) When did Ahmad sell the stock? (d) How much did Ahmad lose on the stock? (e) Writing to Learn Explain why LaToyas strategy was better than Ahmads for this particular stock in this particular year. (f) Sketch a 12-month graph of a stocks performance that would favor Ahmads strategy over LaToyas.

You can create your own graphs with hidden behavior. Working in groups of two or three, try this exploration. (a) Graph the equation \(y=(x+2)\left(x^{2}-4 x+4\right)\) in the window [-4, 4] by [-10, 10]. (b) Confirm algebraically that this function has zeros only at x = -2 and x = 2 (c) Graph the equation \(y=(x+2)\left(x^{2}-4 x+4.01\right)\) in the window [-4, 4] by [-10, 10] (d) Confirm algebraically that this function has only one zero, at x= -2. (Use the discriminant.) (e) Graph the equation \((x+2)\left(x^{2}-4 x+3.99\right)\) in the window by [-4, 4] by [-10, 10] (f) Confirm algebraically that this function has three zeros. (Use the discriminant.)

The Proliferation of Cell Phones Table 1.8 shows the number of cellular phone subscribers in the United States and their average monthly bill in the years from 1998 to 2007. (a) Graph the scatter plots of the number of subscribers and the average local monthly bill as functions of time, letting time t= the number of years after 1990. (b) One of the scatter plots clearly suggests a linear model in the form y=mx+b. Use the points at t=8 and t=16 to find a linear model. (c) Superimpose the graph of the linear model onto the scatter plot. Does the fit appear to be good? (d) Why does a linear model seem inappropriate for the other scatter plot? Can you think of a function that might fit the data better? (e) In 1995 there were 33.8 million subscribers with an average local monthly bill of $51.00. Add these points to the scatter plots. (f) Writing to Learn The 1995 points do not seem to fit the models used to represent the 1998-2004 data. Give a possible explanation for this.

Group Activity (Continuation of Exercise 63) Discuss the economic forces suggested by the two models in Exercise 63 and speculate about the future by analyzing the graphs.