Chapter 2

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-1,6),(1,4) ; L_{2}:(3,-3),(6,-9)\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(x^{2}+y^{2}=16\)

If \(f\) is an even function, determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=f(x-2)\)

The inventor of a new game determines that the variable cost of producing the game is $$\$ 2.95$$ per unit and the fixed costs are $$\$ 8000$$. The inventor sells each game for $$\$ 8.79 .$$ Let \(x\) be the number of games sold. (a) Write the total cost \(C\) as a function of the number of games sold. (b) Write the average cost per unit \(\bar{C}=C / x\) as a function of \(x\) (c) Complete the table. $$\begin{array}{|l|l|l|l|l|}\hline x & 100 & 1000 & 10,000 & 100,000 \\\\\hline \bar{C} & & & & \\ \hline\end{array}$$ (d) Write a paragraph analyzing the data in the table. What do you observe about the average cost per unit as \(x\) gets larger?

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(0,-1),(5,9) ; L_{2}:(0,3),(4,1)\)

Find the coordinates of a second point on the graph of a function \(f\) if the given point is on the graph and the function is (a) even and (b) odd. \(\left(-\frac{3}{2}, 4\right)\)

A manufacturer determines that the variable cost for a new product is $$\$ 2.05$$ per unit and the fixed costs are $$\$ 57,000$$. The product is to be sold for $$\$ 5.89$$ per unit. Let \(x\) be the number of units sold. (a) Write the total cost \(C\) as a function of the number of units sold. (b) Write the average cost per unit \(\bar{C}=C / x\) as a function of \(x\). (c) Complete the table. $$\begin{array}{|l|l|l|l|l|}\hline x & 100 & 1000 & 10,000 & 100,000 \\ \hline \bar{C} & & & & \\\\\hline\end{array}$$ (d) Write a paragraph analyzing the data in the table. What do you observe about the average cost per unit as \(x\) gets larger?

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(3,6),(-6,0) ; L_{2}:(0,-1),\left(5, \frac{7}{3}\right)\)

For groups of 80 or more people, a charter bus company determines the rate per person (in dollars) according to the formula Rate \(=8-0.05(n-80) \quad n \geq 80\) where \(n\) is the number of people in the group. (a) Write the total revenue \(R\) for the bus company as a function of \(n\). (b) Complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline n & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\\\\hline R & & & & & & & \\ \hline\end{array}$$ (c) Write a paragraph analyzing the data in the table.

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-2,-1),(1,5) ; L_{2}:(1,3),(5,-5)\)

A stone is thrown into the middle of a calm pond, causing ripples to form in concentric circles. The radius \(r\) of the outermost ripple increases at the rate of \(0.75\) foot per second. (a) Write a function for the radius \(r\) of the circle formed by the outermost ripple in terms of time \(t\). (b) Write a function for the area \(A\) enclosed by the outermost ripple. Complete the table. $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Time, } t & 1 & 2 & 3 & 4 & 5 \\\\\hline \text { Radius, } r \text { (in feet) } & & & & & \\\\\hline \text { Area, } A \text { (in square feet) } & & & & & \\ \hline\end{array}$$ (c) Compare the ratios \(A(2) / A(1)\) and \(A(4) / A(2) .\) What do you observe? Based on your observation, predict the area when \(t=8\). Verify by checking \(t=8\) in the area function.

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(4,8),(-4,2) ; L_{2}:(3,-5),\left(-1, \frac{1}{3}\right)\)

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-1,7),(-6,4) ; L_{2}:(0,1),(5,4)\)

Find the center and radius of the circle given the endpoints of the diameter of the circle.

Determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. (a) The sales tax on a purchased item is a function of the selling price. (b) Your score on the next algebra exam is a function of the number of hours you study for the exam.

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-1,3),(2,-5) ; L_{2}:(3,0),(2,-7)\)

Find the center and radius of the circle given the endpoints of the diameter of the circle.

Determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of its initial height.

Write equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point \(\quad\) Line \(\begin{array}{ll}\underline{\phantom{xxx}}(6,2) & y-2 x=-1\end{array}\)

Find the standard form of the equation of the specified circle. Center: \((0,0) ;\) radius: 3

Write equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point \(\quad\) Line \((-5,4) \quad x+y=8\)

Find the standard form of the equation of the specified circle. Center: \((0,0) ;\) radius: 5

Write equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point \(\quad\) Line \(\begin{array}{ll}\underline{\phantom{xxx}}\left(\frac{1}{4},-\frac{2}{3}\right) & 2 x-3 y=5\end{array}\)

Find the standard form of the equation of the specified circle. Center: \((-4,1)\); radius: \(\sqrt{2}\)

Write equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point \(\quad\) Line \(\begin{array}{ll}\underline{\phantom{xxx}}\left(\frac{7}{8}, \frac{3}{4}\right) & 5 x+3 y=0\end{array}\)

Find the standard form of the equation of the specified circle. Center: \(\left(0, \frac{1}{2}\right)\); radius: \(\frac{2}{3}\)

Find the standard form of the equation of the specified circle. Center: \((-1,2)\); point on circle: \((0,0)\)

Write equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point \(\quad\) Line \((2,5)\) \(x=4\)

Find the standard form of the equation of the specified circle. Center: \((3,-2) ;\) point on circle: \((-1,1)\)

Find an equation of the line that gives the relationship between the temperature in degrees Celsius \(C\) and the temperature in degrees Fahrenheit \(F\). Remember that water freezes at \(0^{\circ}\) Celsius ( \(32^{\circ}\) Fahrenheit) and boils at \(100^{\circ}\) Celsius \(\left(212^{\circ}\right.\) Fahrenheit).

Find the standard form of the equation of the specified circle. Endpoints of a diameter: \((-3,4),(5,-2)\)

Find the standard form of the equation of the specified circle. Endpoints of a diameter: \((-4,-1),(4,1)\)

A person deposits \(P\) dollars in an account that pays simple interest. After 2 months, the balance in the account is $$\$ 813$$ and after 3 months, the balance in the account is $$\$ 819.50$$. Find an equation that gives the relationship between the balance \(A\) and the time \(t\) in months.

Write the equation of the circle in standard form. Then sketch the circle. \(x^{2}+y^{2}-6 x+4 y-3=0\)

Write the equation of the circle in standard form. Then sketch the circle. \(x^{2}+y^{2}-2 x+6 y-15=0\)

The maximum recommended slope of a wheelchair ramp is \(\frac{1}{12}\). A business is installing a wheelchair ramp that rises 34 inches over a horizontal length of 30 feet. Is the ramp steeper than recommended? (Source: Americans with Disabilities Act Handbook)

Write the equation of the circle in standard form. Then sketch the circle. \(x^{2}+y^{2}-4 x+6 y+9=0\)

A line representing daily revenues \(y\) in terms of time \(x\) in days has a slope of \(m=100\). Interpret the change in daily revenues for a one-day increase in time.

Write the equation of the circle in standard form. Then sketch the circle. \(5 x^{2}+5 y^{2}+10 x+1=0\)

A small college had 3125 students in 2005 and 3582 students in 2008 . The enrollment follows a linear growth pattern. How many students will the college have in \(2012 ?\)

Write the equation of the circle in standard form. Then sketch the circle. \(2 x^{2}+2 y^{2}-2 x-2 y-3=0\)

Your salary was \(\$ 30,200\) in 2007 and \(\$ 33,500\) in 2009 . Your salary follows a linear growth pattern. What salary will you be making in \(2012 ?\)

Write the equation of the circle in standard form. Then sketch the circle. \(4 x^{2}+4 y^{2}-4 x+2 y-1=0\)

Write the equation of the circle in standard form. Then sketch the circle. \(16 x^{2}+16 y^{2}+16 x+40 y-7=0\)

In 1998, there were 37,107 motor vehicle traffic crashes involving fatalities in the United States. In 2005 , there were 39,189 such crashes. Assume that the trend is linear. Predict the number of crashes with fatalities in 2007. (Source: National Highway Traffic Safety Administration)

Write the equation of the circle in standard form. Then sketch the circle. . \(x^{2}+y^{2}-4 x+2 y+3=0\)

In 2000, Yahoo! Inc. had revenues of \(\$ 1110.2\) million. In 2003 , their revenues were \(\$ 1625.1\) million. Assume the revenue followed a linear trend. What would the approximate revenue have been in \(2005 ?\) The actual revenue in 2005 was \(\$ 5257.7\) million. Do you think the yearly revenue followed a linear trend? Explain your reasoning. (Source: Yahoo! Inc.)

An equation of a circle is written in standard form. Indicate the coordinates of the center of the circle and determine the radius of the circle. Rewrite the equation of the circle in general form. . \((x-3)^{2}+(y+1)^{2}=25\)

Applebee's is one of the largest casual dining chains in the United States. In 2000 , Applebee's had revenues of $$\$ 690.2$$ million. In 2004 , their revenues were $$\$ 1111.6$$ million. Assume the yearly revenue followed a linear trend. What would the approximate revenue have been in 2005 ? The actual revenue in 2005 was \(\$ 1216.6\) million. From these data, is it possible that Applebee's yearly revenue followed a linear trend? Explain your reasoning. (Source: Applebee's Intemational, Inc.)

An equation of a circle is written in standard form. Indicate the coordinates of the center of the circle and determine the radius of the circle. Rewrite the equation of the circle in general form. \(\left(x-\frac{1}{2}\right)^{2}+(y-2)^{2}=7\)