Polynomial and Rational Functions

# Modeling Using Variation

### Learning Objectives

In this section, you will:

- Solve direct variation problems.
- Solve inverse variation problems.
- Solve problems involving joint variation.

A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.

### Solving Direct Variation Problems

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formulatells us her earnings,come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See (Figure).

, sales price | Interpretation | |
---|---|---|

$4,600 | A sale of a $4,600 vehicle results in $736 earnings. | |

$9,200 | A sale of a $9,200 vehicle results in $1472 earnings. | |

$18,400 | A sale of a $18,400 vehicle results in $2944 earnings. |

Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called **direct variation**. Each variable in this type of relationship **varies directly **with the other.

(Figure) represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formulais used for direct variation. The valueis a nonzero constant greater than zero and is called the **constant of variation**. In this case,andWe saw functions like this one when we discussed power functions.

### Direct Variation

Ifare related by an equation of the form

then we say that the relationship is direct variation and varies directly with, or is proportional to, thepower ofIn direct variation relationships, there is a nonzero constant ratiowhereis called the constant of variation, which help defines the relationship between the variables.

### How To

**Given a description of a direct variation problem, solve for an unknown.
**

- Identify the input,and the output,
- Determine the constant of variation. You may need to divideby the specified power ofto determine the constant of variation.
- Use the constant of variation to write an equation for the relationship.
- Substitute known values into the equation to find the unknown.

### Solving a Direct Variation Problem

The quantityvaries directly with the cube ofIfwhenfindwhenis 6.

[hidden-answer a=”fs-id1165137642960″]

The general formula for direct variation with a cube isThe constant can be found by dividingby the cube of

Now use the constant to write an equation that represents this relationship.

Substituteand solve for

#### Analysis

The graph of this equation is a simple cubic, as shown in (Figure).

**Do the graphs of all direct variation equations look like (Figure)?**

*No. Direct variation equations are power functions—they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through
*

### Try It

The quantityvaries directly with the square ofIfwhenfindwhenis 4.

[hidden-answer a=”fs-id1165137659774″]

[/hidden-answer]

### Solving Inverse Variation Problems

Water temperature in an ocean varies inversely to the water’s depth. The formulagives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F.

If we create (Figure), we observe that, as the depth increases, the water temperature decreases.

depth | Interpretation | |
---|---|---|

500 ft | At a depth of 500 ft, the water temperature is 28° F. | |

1000 ft | At a depth of 1,000 ft, the water temperature is 14° F. | |

2000 ft | At a depth of 2,000 ft, the water temperature is 7° F. |

We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be **inversely proportional** and each term **varies inversely** with the other. Inversely proportional relationships are also called **inverse variations**.

For our example, (Figure) depicts the inverse variation. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formulafor inverse variation in this case uses

### Inverse Variation

Ifandare related by an equation of the form

whereis a nonzero constant, then we say that varies inversely with thepower ofIn inversely proportional relationships, or inverse variations, there is a constant multiple

### Writing a Formula for an Inversely Proportional Relationship

A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.

[hidden-answer a=”fs-id1165137654947″]

Recall that multiplying speed by time gives distance. If we letrepresent the drive time in hours, andrepresent the velocity (speed or rate) at which the tourist drives, thenBecause the distance is fixed at 100 miles, soBecause time is a function of velocity, we can write

We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction. We say that time varies inversely with velocity.

[/hidden-answer]

### How To

**Given a description of an indirect variation problem, solve for an unknown.
**

- Identify the input,and the output,
- Determine the constant of variation. You may need to multiplyby the specified power ofto determine the constant of variation.
- Use the constant of variation to write an equation for the relationship.
- Substitute known values into the equation to find the unknown.

### Solving an Inverse Variation Problem

A quantityvaries inversely with the cube ofIfwhenfindwhenis 6.

[hidden-answer a=”fs-id1165137547532″]

The general formula for inverse variation with a cube isThe constant can be found by multiplyingby the cube of

Now we use the constant to write an equation that represents this relationship.

Substituteand solve for

#### Analysis

The graph of this equation is a rational function, as shown in (Figure).

### Try It

A quantityvaries inversely with the square ofIfwhenfindwhenis 4.

[hidden-answer a=”fs-id1165137683037″]

[/hidden-answer]

### Solving Problems Involving Joint Variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variablecost, varies jointly with the number of students,and the distance,

### Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, ifvaries directly with bothand we haveIfvaries directly withand inversely withwe haveNotice that we only use one constant in a joint variation equation.

### Solving Problems Involving Joint Variation

A quantityvaries directly with the square ofand inversely with the cube root ofIfwhen andfindwhen and

[hidden-answer a=”928767″]Begin by writing an equation to show the relationship between the variables.

Substitute[latex]y=2,\,[/latex]andto find the value of the constant

Now we can substitute the value of the constant into the equation for the relationship.

To findwhenandwe will substitute values forandinto our equation.

[/hidden-answer]

### Try It

A quantityvaries directly with the square ofand inversely withIfwhenandfindwhen

and

[hidden-answer a=”fs-id1165137755838″]

[/hidden-answer]

Access these online resources for additional instruction and practice with direct and inverse variation.

Visit this website for additional practice questions from Learningpod.

### Key Equations

Direct variation | |

Inverse variation |

### Key Concepts

- A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See (Figure).
- Two variables that are directly proportional to one another will have a constant ratio.
- A relationship where one quantity is a constant divided by another quantity is called inverse variation. See (Figure).
- Two variables that are inversely proportional to one another will have a constant multiple. See (Figure).
- In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See (Figure).

### Section Exercises

#### Verbal

What is true of the appearance of graphs that reflect a direct variation between two variables?

[hidden-answer a=”fs-id1165137759992″]

The graph will have the appearance of a power function.

[/hidden-answer]

If two variables vary inversely, what will an equation representing their relationship look like?

Is there a limit to the number of variables that can vary jointly? Explain.

[hidden-answer a=”fs-id1165134192951″]

No. Multiple variables may jointly vary.

[/hidden-answer]

#### Algebraic

For the following exercises, write an equation describing the relationship of the given variables.

varies directly asand when

varies directly as the square ofand when

[hidden-answer a=”fs-id1165134381645″]

[/hidden-answer]

varies directly as the square root ofand when

varies directly as the cube ofand when

[hidden-answer a=”fs-id1165137760761″]

[/hidden-answer]

varies directly as the cube root ofand when

varies directly as the fourth power ofand when

[hidden-answer a=”fs-id1165137667538″]

[/hidden-answer]

varies inversely asand when

varies inversely as the square ofand when

[hidden-answer a=”fs-id1165137662476″]

[/hidden-answer]

varies inversely as the cube ofand when

varies inversely as the fourth power ofand when

[hidden-answer a=”fs-id1165135524553″]

[/hidden-answer]

varies inversely as the square root ofand when

varies inversely as the cube root ofand when

[hidden-answer a=”fs-id1165137664262″]

[/hidden-answer]

varies jointly withandand whenand

varies jointly asandand whenthen

[hidden-answer a=”fs-id1165137563934″]

[/hidden-answer]

varies jointly as the square ofand the square ofand whenandthen

varies jointly asand the square root ofand whenandthen

[hidden-answer a=”fs-id1165134323767″]

[/hidden-answer]

varies jointly as the square ofthe cube ofand the square root ofWhenandthen

varies jointly asand inversely asWhenandthen

[hidden-answer a=”fs-id1165137723403″]

[/hidden-answer]

varies jointly as the square ofand the square root ofand inversely as the cube ofWhenandthen

varies jointly asandand inversely as the square root ofand the square ofWhenandthen

[hidden-answer a=”767549″][/hidden-answer]

#### Numeric

For the following exercises, use the given information to find the unknown value.

varies directly asWhenthenFindwneh

varies directly as the square ofWhenthenFindwhen

[hidden-answer a=”fs-id1165135194956″]

[/hidden-answer]

varies directly as the cube ofWhenthenFindwhen

varies directly as the square root ofWhenthenFindwhen

[hidden-answer a=”fs-id1165137571788″]

[/hidden-answer]

varies directly as the cube root ofWhenthenFindwhen

varies inversely withWhenthenFindwhen

[hidden-answer a=”fs-id1165137445714″]

[/hidden-answer]

varies inversely with the square ofWhenthenFindwhen

varies inversely with the cube ofWhenthenFindwhen

[hidden-answer a=”fs-id1165137431582″]

[/hidden-answer]

varies inversely with the square root ofWhenthenFindwhen

varies inversely with the cube root ofWhenthenFindwhen

[hidden-answer a=”fs-id1165137542372″]

[/hidden-answer]

varies jointly asWhenandthenFindwhenand

varies jointly asWhen[latex]z=1,\,[/latex]andthenFindwhen[latex]z=2,\,[/latex] and

[hidden-answer a=”fs-id1165137406882″]

[/hidden-answer]

varies jointly asand the square ofWhenandthenFindwhenand

varies jointly as the square ofand the square root ofWhenandthenFindwhenand

[hidden-answer a=”fs-id1165135543455″]

[/hidden-answer]

varies jointly asandand inversely asWhen[latex]z=2, \,[/latex]andthenFindwhenandand

varies jointly as the square ofand the cube ofand inversely as the square root ofWhen[latex]z=2,\,[/latex]andthenFindwhen[latex]z=3,\,[/latex]and

[hidden-answer a=”fs-id1165137470711″]

[/hidden-answer]

varies jointly as the square ofand ofand inversely as the square root ofand ofWhen[latex]z=3,\,[/latex]andthenFindwhen[latex]z=2,\,[/latex]and

#### Technology

For the following exercises, use a calculator to graph the equation implied by the given variation.

varies directly with the square ofand when

[hidden-answer a=”fs-id1165137832227″]

[/hidden-answer]

varies directly as the cube ofand when

varies directly as the square root ofand when

[hidden-answer a=”fs-id1165137936864″]

[/hidden-answer]

varies inversely withand when

varies inversely as the square ofand when

[hidden-answer a=”fs-id1165134248054″]

[/hidden-answer]

#### Extensions

For the following exercises, use Kepler’s Law, which states that the square of the time,required for a planet to orbit the Sun varies directly with the cube of the mean distance,that the planet is from the Sun.

Using Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating and

Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.

[hidden-answer a=”fs-id1165137871569″]

1.89 years

[/hidden-answer]

Using Earth’s distance of 150 million kilometers, find the equation relatingand

Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.

[hidden-answer a=”fs-id1165137668810″]

0.61 years

[/hidden-answer]

Using Earth’s distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.

#### Real-World Applications

For the following exercises, use the given information to answer the questions.

The distancethat an object falls varies directly with the square of the time,of the fall. If an object falls 16 feet in one se**c**ond, how long for it to fall 144 feet?

[hidden-answer a=”fs-id1165133101738″]

3 seconds

[/hidden-answer]

The velocityof a falling object varies directly to the time,of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?

The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

[hidden-answer a=”fs-id1165137728416″]

48 inches

[/hidden-answer]

The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs 50 pounds when it is 3960 miles from Earth’s center, what would it weigh it were 3970 miles from Earth’s center?

[hidden-answer a=”fs-id1165137760890″]

49.75 pounds

[/hidden-answer]

The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.

The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

[hidden-answer a=”fs-id1165135194664″]

33.33 amperes

[/hidden-answer]

The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

[hidden-answer a=”fs-id1165137641112″]

2.88 inches

[/hidden-answer]

The kinetic energyof a moving object varies jointly with its massand the square of its velocityIf an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

### Chapter Review Exercises

#### Quadratic Functions

For the following exercises, write the quadratic function in standard form. Then give the vertex and axes intercepts. Finally, graph the function.

[hidden-answer a=”fs-id1165137935589″]

For the following exercises, find the equation of the quadratic function using the given information.

The vertex isand a point on the graph is

[hidden-answer a=”fs-id1165137724885″]

[/hidden-answer]

The vertex isand a point on the graph is

For the following exercises, complete the task.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

[hidden-answer a=”fs-id1165137725075″]

300 meters by 150 meters, the longer side parallel to river.

[/hidden-answer]

An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height,in terms of horizontal distance traveled,given byFind the maximum height the object attains.

#### Power Functions and Polynomial Functions

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

[hidden-answer a=”fs-id1165137451016″]

Yes, degree = 5, leading coefficient = 4

[/hidden-answer]

[hidden-answer a=”125615″]Yes, degree = 4, leading coefficient = 1[/hidden-answer]

For the following exercises, determine end behavior of the polynomial function.

[hidden-answer a=”fs-id1165134155145″]

[/hidden-answer]

#### Graphs of Polynomial Functions

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

[hidden-answer a=”fs-id1165137432758″]

–3 with multiplicity 2,with multiplicity 1, –1 with multiplicity 3

[/hidden-answer]

[hidden-answer a=”fs-id1165137749873″]

4 with multiplicity 1

[/hidden-answer]

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

[hidden-answer a=”fs-id1165137539727″]

with multiplicity 1, 3 with multiplicity 3

[/hidden-answer]

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function

#### Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.

[hidden-answer a=”fs-id1165137423736″]

with remainder 12

[/hidden-answer]

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

[hidden-answer a=”fs-id1165135365932″]

[/hidden-answer]

[hidden-answer a=”fs-id1165137772480″]

, so factored form is

[/hidden-answer]

#### Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

[hidden-answer a=”fs-id1165137780754″]

[/hidden-answer]

[hidden-answer a=”fs-id1165137654771″]

[/hidden-answer]

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

[hidden-answer a=”fs-id1165134108392″]

0 or 2 positive, 1 negative

[/hidden-answer]

#### Rational Functions

For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.

[hidden-answer a=”fs-id1165137740757″]

Intercepts , Asymptotes

and

[/hidden-answer]

[hidden-answer a=”fs-id1165137452244″]

Intercepts (3, 0), (-3, 0), and, Asymptotes

[/hidden-answer]

For the following exercises, find the slant asymptote.

[hidden-answer a=”fs-id1165137642577″]

[/hidden-answer]

#### Inverses and Radical Functions

For the following exercises, find the inverse of the function with the domain given.

[hidden-answer a=”fs-id1165135640618″]

[/hidden-answer]

[hidden-answer a=”fs-id1165135210050″]

[/hidden-answer]

[hidden-answer a=”fs-id1165134224016″]

[/hidden-answer]

#### Modeling Using Variation

For the following exercises, find the unknown value.

varies directly as the square of If whenfindif

[hidden-answer a=”fs-id1165135319534″]

[/hidden-answer]

varies inversely as the square root of If whenfindif

varies jointly as the cube ofand asIf whenand[latex]y=6,\,[/latex]findifand

[hidden-answer a=”fs-id1165137934401″]

[/hidden-answer]

varies jointly asand the square ofand inversely as the cube ofIf when[latex]z=4,\,[/latex]and[latex]y=48,\,[/latex]findif[latex]z=5,\,[/latex]and

For the following exercises, solve the application problem.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

[hidden-answer a=”fs-id1165135698641″]148.5 pounds[/hidden-answer]

The volumeof an ideal gas varies directly with the temperatureand inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

### Chapter Test

Give the degree and leading coefficient of the following polynomial function.

Determine the end behavior of the polynomial function.

[hidden-answer a=”fs-id1165135192394″]

[/hidden-answer]

Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

[hidden-answer a=”fs-id1165135503809″]

, vertex, intercepts

[/hidden-answer]

Given information about the graph of a quadratic function, find its equation.

Vertexand point on graph

Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

[hidden-answer a=”fs-id1165137673750″]

60,000 square feet

[/hidden-answer]

Find all zeros of the following polynomial functions, noting multiplicities.

[hidden-answer a=”fs-id1165134040553″]

0 with multiplicity 4, 3 with multiplicity 2

[/hidden-answer]

Based on the graph, determine the zeros of the function and multiplicities.

Use long division to find the quotient.

[hidden-answer a=”fs-id1165137427782″]

[/hidden-answer]

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

[hidden-answer a=”fs-id1165135152117″]

. So factored form is

[/hidden-answer]

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

[hidden-answer a=”fs-id1165135188171″]

(has multiplicity 2),

[/hidden-answer]

[hidden-answer a=”fs-id1165137427321″]

(has multiplicity 3),

[/hidden-answer]

Given the following information about a polynomial function, find the function.

It has a double zero atand zeros atand. Its *y*-intercept is

It has a zero of multiplicity 3 atand another zero at. It contains the point

[hidden-answer a=”fs-id1165137595278″]

[/hidden-answer]

Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

[hidden-answer a=”fs-id1165135208461″]

Intercepts, Asymptotes.

[/hidden-answer]

Find the slant asymptote of the rational function.

[hidden-answer a=”fs-id1165135151319″]

[/hidden-answer]

Find the inverse of the function.

[hidden-answer a=”fs-id1165137920721″]

[/hidden-answer]

Find the unknown value.

varies inversely as the square ofand when[latex]y=2.\,[/latex]Findif

[hidden-answer a=”fs-id1165137722658″]

[/hidden-answer]

varies jointly withand the cube root ofIf whenand[latex]y=12,\,[/latex]findifand

Solve the following application problem.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?

[hidden-answer a=”fs-id1165137734636″]

4 seconds

[/hidden-answer]

### Glossary

- constant of variation
- the non-zero value

that helps define the relationship between variables in direct or inverse variation

- direct variation
- the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other

- inverse variation
- the relationship between two variables in which the product of the variables is a constant

- inversely proportional
- a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases

- joint variation
- a relationship where a variable varies directly or inversely with multiple variables

- varies directly
- a relationship where one quantity is a constant multiplied by the other quantity

- varies inversely
- a relationship where one quantity is a constant divided by the other quantity