Chapter 8

Graph each rational function. \(f(x)=\frac{x^{2}-5 x+4}{x-4}\)

Simplify each expression. $$ \frac{11}{9}-\frac{7}{2 w}-\frac{6}{5 w} $$

Graph each rational function. $$ f(x)=\frac{6}{(x-6)^{2}} $$

Simplify each expression. \(\frac{3 x^{2}-2 x-8}{3 x^{2}-12}\)

Solve each equation by factoring. \(x^{2}+6 x+8=0\)

Simplify each expression. $$ \frac{1}{h^{2}-9 h+20}-\frac{5}{h^{2}-10 h+25} $$

Graph each rational function. $$ f(x)=\frac{1}{(x+2)^{2}} $$

Astronomers can use the brightness of two light sources, such as stars, to compare the distances from the light sources. The intensity, or brightness, of light I is inversely proportional to the square of the distance from the light source \(d .\) If two people are viewing the same light source, and one person is three times the distance from the light source as the other person, compare the light intensities that the two people observe.

Simplify each expression. \(\frac{a^{2}-4}{6-3 a}\)

Identify the type of function represented by each equation. Then graph the equation. (lesson 8.5\()\) $$ y=2 \sqrt{x} $$

Solve each equation by factoring. \(2 q^{2}+11 q=21\)

Simplify each expression. $$ \frac{x}{x^{2}+5 x+6}-\frac{2}{x^{2}+4 x+4} $$

Graph each rational function. $$ f(x)=\frac{x^{2}+6 x+5}{x+1} $$

According to the Law of Universal Gravitation, the attractive force \(F\) in Newtons between any two bodies in the universe is directly proportional to the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and inversely proportional to the square of the distance \(d\) in meters between the bodies. That is, \(F=G \frac{m_{1} m_{2}}{d_{2}} . G\) is the universal gravitational constant. Its value is \(6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\). The distance between Earth and the Moon is about \(3.84 \times 10^{8}\) meters. The mass of the Moon is \(7.36 \times 10^{22}\) kilograms. The mass of Earth is \(5.97 \times 10^{24}\) kilograms. What is the gravitational force that the Moon and Earth exert upon each other?

Simplify each expression. \(\frac{b^{2}-4 b+3}{3-2 b-b^{2}}\)

Identify the type of function represented by each equation. Then graph the equation. (lesson 8.5\()\) $$ y=0.8 x $$

\(\begin{array}{|c|c|c|}\hline \text { Balloons } & {\text { Distance from Ground }} & {\text { Rate of Ascension }} \\ {} & {(\mathrm{m})} & {(\mathrm{m} / \mathrm{min})} \\ \hline A & {60} & {15} \\ \hline \mathrm{B} & {40} & {20} \\ \hline\end{array}\) If both balloons are launched at the same time, how long will it take for them to be the same distance from the ground?

Simplify each expression. $$ \frac{m^{2}+n^{2}}{m^{2}-n^{2}}+\frac{m}{n-m}+\frac{n}{m+n} $$

Graph each rational function. $$ f(x)=\frac{x^{2}-4 x}{x-4} $$

Simplify each expression. \(\frac{6 x^{2}-6}{14 x^{2}-28 x+14}\)

If \(y\) varies inversely as \(x\) and \(y=24\) when \(x=9,\) find \(y\) when \(x=6\)

Simplify each expression. $$ \frac{y+1}{y-1}+\frac{y+2}{y-2}+\frac{y}{y^{2}-3 y+2} $$

HISTORY For Exercises \(47-49\) , use the following information. In Maria Gaetana Agnesi's book Analytical Institutions, Agnesi discussed the characteristics of the equation \(x^{2} y=a^{2}(a-y),\) the graph of which is called the "curve of Agnesi." This equation can be expressed as \(y=\frac{a^{3}}{x^{2}+a^{2}}\) Graph \(f(x)=\frac{a^{3}}{x^{2}+a^{2}}\) if \(a=4\)

According to the Law of Universal Gravitation, the attractive force \(F\) in Newtons between any two bodies in the universe is directly proportional to the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and inversely proportional to the square of the distance \(d\) in meters between the bodies. That is, \(F=G \frac{m_{1} m_{2}}{d_{2}} . G\) is the universal gravitational constant. Its value is \(6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\). Find the gravitational force exerted on each other by two 1000-kilogram iron balls a distance of 0.1 meter apart.

Simplify each expression. \(\frac{25 a^{2} b^{3}}{6 x^{2} y} \cdot \frac{8 x y^{2}}{20 a^{3} b^{2}}\)

Solve each inequality. $$ (x+11)(x-3)>0 $$

Find the LCM of each set of polynomials. \(15 a b^{2} c, 6 a^{3}, 4 b c^{2}\)

Write \(\left(\frac{2 s}{2 s+1}-1\right) \div\left(1+\frac{2 s}{1-2 s}\right)\) in simplest form.

HISTORY For Exercises \(47-49\) , use the following information. In Maria Gaetana Agnesi's book Analytical Institutions, Agnesi discussed the characteristics of the equation \(x^{2} y=a^{2}(a-y),\) the graph of which is called the "curve of Agnesi." This equation can be expressed as \(y=\frac{a^{3}}{x^{2}+a^{2}}\) Describe the graph. What are the limitations on the domain and range?

Describe two real life quantities that vary directly with each other and two quantities that vary inversely with each other.

Simplify each expression. \(\frac{-9 c d}{8 x w} \cdot \frac{(-4 w)^{2}}{15 c}\)

Solve each inequality. $$ x^{2}-4 x \leq 0 $$

Find the LCM of each set of polynomials. \(9 x^{3}, 5 x y^{2}, 15 x^{2} y^{3}\)

What is the simplest form of \(\left(3+\frac{5}{a+2}\right) \div\left(3-\frac{10}{a+7}\right) ?\)

HISTORY For Exercises \(47-49\) , use the following information. In Maria Gaetana Agnesi's book Analytical Institutions, Agnesi discussed the characteristics of the equation \(x^{2} y=a^{2}(a-y),\) the graph of which is called the "curve of Agnesi." This equation can be expressed as \(y=\frac{a^{3}}{x^{2}+a^{2}}\) Make a conjecture about the shape of the graph of \(f(x)=\frac{a^{3}}{x^{2}+a^{2}} \mathrm{ff} a=-4\) Explain your reasoning.

Simplify each expression. \(\frac{2 x^{3} y}{z^{5}} \div\left(\frac{4 x y}{z^{3}}\right)^{2}\)

Solve each inequality. $$ 2 b^{2}-b<6 $$

Find the LCM of each set of polynomials. \(5 d-10,3 d-6\)

Simplify each expression. \(\frac{w^{2}-11 w+24}{w^{2}-18 w+80} \cdot \frac{w^{2}-15 w+50}{w^{2}-9 w+20}\)

Find each product, if possible. $$ \left[\begin{array}{rr}{3} & {-5} \\ {2} & {7}\end{array}\right] \cdot\left[\begin{array}{rrr}{5} & {1} & {-3} \\ {8} & {-4} & {9}\end{array}\right] $$

Find the LCM of each set of polynomials. \(x^{2}-y^{2}, 3 x+3 y\)

For Exercises 51 and \(52,\) use the following information. In an electrical circuit, if two resistors with resistance \(R_{1}\) and \(R_{2}\) are connected in parallel as shown, the relationship between these resistances and the resulting combination resistance \(R\) is \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). If \(R_{1}\) is \(x\) ohms and \(R_{2}\) is 4 ohms less than twice \(x\) ohms, write an expression for \(\frac{1}{R}\).

REASONING Compare and contrast the graphs of \(f(x)=\frac{(x-1)(x+5)}{x-1}\) and \(g(x)=x+5\)

Suppose \(b\) varies inversely as the square of \(a\) . If \(a\) is multiplied by \(9,\) which of the following is true for the value of \(b ?\) A. It is multiplied by \(\frac{1}{3}\) B. It is multiplied by \(\frac{1}{9}\) C. It is multiplied by \(\frac{1}{81}\) D. It is multiplied by 3

Simplify each expression. \(\frac{r^{2}+2 r-8}{r^{2}+4 r+3} \div \frac{r-2}{3 r+3}\)

Find each product, if possible. $$ \left[\begin{array}{rrr}{4} & {-1} & {6} \\ {1} & {5} & {-8}\end{array}\right] \cdot\left[\begin{array}{cc}{1} & {3} \\ {9} & {-6}\end{array}\right] $$

Find the LCM of each set of polynomials. \(a^{2}-2 a-3, a^{2}-a-6\)

For Exercises 51 and \(52,\) use the following information. In an electrical circuit, if two resistors with resistance \(R_{1}\) and \(R_{2}\) are connected in parallel as shown, the relationship between these resistances and the resulting combination resistance \(R\) is \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). A circuit with two resistors connected in parallel has an effective resistance of 25 ohms. One of the resistors has a resistance of 30 ohms. Find the resistance of the other resistor.

If \(a b=1\) and \(a\) is less than 0 which of the following statements cannot be true? F. \(b\) is negative. G. \(b\) is less than \(a\) H. As \(a\) increases, \(b\) decreases. J. As \(a\) increases, \(b\) increases.

Simplify each expression. \(\frac{\frac{5 x^{2}-5 x-30}{45-15 x}}{\frac{6+x-x^{2}}{4 x-12}}\)