Power Functions with Power Functions with Modeling 2.1 & 2.2Modeling 2.1 & 2.2

Pre-CalculusPre-Calculus

Ms. HardyMs. Hardy

IntroductionIntroduction

Compare & contrast the Compare & contrast the different power functions from different power functions from the explore worksheet after the explore worksheet after the test.the test.

What are the general shapes?What are the general shapes? What is the end behavior for What is the end behavior for

each?each? How are these related to our How are these related to our

13 basic functions?13 basic functions?

Now let’s look at applications Now let’s look at applications of the Power Functionsof the Power Functions

Power FunctionsPower FunctionsAny function that can be Any function that can be

written in the formwritten in the formf(x) = kf(x) = k•x•xaa k, a ≠ 0 k, a ≠ 0

Where “a” is the power and Where “a” is the power and “k” is the constant of “k” is the constant of variation.variation.

VariationVariation is a is a kind of modeling kind of modeling that uses power that uses power functions.functions.

Direct VariationDirect Variation as one as one variable increases the variable increases the other increases.other increases.

y = kxy varies directly as x ory is proportional to x.

K is called the constant of variation.

Inverse VariationInverse Variation – as one – as one variable increases the variable increases the other decreases.other decreases.

Joint VariationJoint Variation is where is where more than two quantities more than two quantities are involved.are involved.

z = kxyz = kxyZ varies jointly as x & y ORZ varies jointly as x & y OR

Z is jointly proportional to x Z is jointly proportional to x & y. & y.

Write each variation statement as a power function

1) The circumference of a circle 1) The circumference of a circle varies directly as the radius.varies directly as the radius.

2) The average speed (rate) is 2) The average speed (rate) is inversely proportional to the time inversely proportional to the time traveled.traveled.

Write each variation statement as a power function

3) A car is traveling on a curve that 3) A car is traveling on a curve that forms a circular arc. The force forms a circular arc. The force FF needed to keep the car from needed to keep the car from skidding is jointly proportional to skidding is jointly proportional to the weight the weight ww of the car and the of the car and the square of its speed and is inversely square of its speed and is inversely proportional to the radius proportional to the radius rr of the of the curve.curve.

Write a sentence that expresses the relationship in the formula, using the language of variation or proportion.

Solving Variation Problems

1)1) Write a general variation Write a general variation equation for the scenario using equation for the scenario using variables (remember to include variables (remember to include the variation constant the variation constant KK).).

2)2) Use the established information Use the established information in the equation to solve for in the equation to solve for KK..

3)3) Write the variation equation with Write the variation equation with the the KK value and use it to answer value and use it to answer the question. the question.

Solving Variation Solving Variation ProblemsProblems

1) 1) DRAG FORCE ON A BOATDRAG FORCE ON A BOAT: : The drag force The drag force FF on a boat is on a boat is jointly proportional to the jointly proportional to the wetted surface area wetted surface area AA on the on the hull and the square of the hull and the square of the speed speed ss of the boat. of the boat.

1)1) DRAG FORCE ON A BOAT DRAG FORCE ON A BOAT continuedcontinued: : A boat experiences A boat experiences a drag force of 220 lb when a drag force of 220 lb when traveling at 5 mph with a traveling at 5 mph with a wetted surface area of 40 ftwetted surface area of 40 ft22. .

How fast must a boat be How fast must a boat be traveling if it has 28 fttraveling if it has 28 ft22 of of wetted surface area and is wetted surface area and is experiencing a drag force of experiencing a drag force of 175 lb?175 lb?

2) 2) LOUDNESS OF SOUND LOUDNESS OF SOUND The The loudness loudness LL of a sound of a sound (measured in decibels, dB) is (measured in decibels, dB) is inversely proportional to the inversely proportional to the square of the distance square of the distance dd from from the source of the sound.the source of the sound.

A person 10 ft from a lawn-A person 10 ft from a lawn-mower experiences a sound mower experiences a sound level of 70 dB. How loud is the level of 70 dB. How loud is the lawn mower when the person lawn mower when the person is 80 ft away? (round to tenth)is 80 ft away? (round to tenth)

2) 2) POWER FROM A WINDMILL POWER FROM A WINDMILL The power The power PP that can be that can be obtained from a windmill obtained from a windmill varies directly with the cube of varies directly with the cube of the wind speed the wind speed ss..

A windmill produces 96 watts of A windmill produces 96 watts of power when the wind is power when the wind is blowing at 20 mph. How much blowing at 20 mph. How much power will this windmill power will this windmill produce if the wind speed produce if the wind speed increases to 30 mph?increases to 30 mph?

Modeling with Power Modeling with Power FunctionsFunctions

Sometimes we aren’t sure what Sometimes we aren’t sure what power the regression data is best power the regression data is best modeled by.modeled by.

We can then calculate a power We can then calculate a power regression A:PwrReg on the regression A:PwrReg on the graphing calculator. It will graphing calculator. It will determine exactly what exponent determine exactly what exponent best models the data.best models the data.

Data is given below for y as Data is given below for y as a power function of x. a power function of x. Write an equation for the Write an equation for the power function, state its power function, state its power & constant of power & constant of variation.variation.

XX 11 44 99 1616 2525

YY 22 44 66 88 1010

Modeling with Power Modeling with Power FunctionsFunctions

Kepler’s Third Law states that the Kepler’s Third Law states that the square of the period of orbit T (time square of the period of orbit T (time required for full revolution around required for full revolution around the sun) varies directly with the cube the sun) varies directly with the cube of its average distance “a” from the of its average distance “a” from the Sun.Sun.

The following table gives data for the The following table gives data for the 6 planets that were known in 6 planets that were known in Kepler’s time (1600s).Kepler’s time (1600s).

PlanetPlanet Avg Avg distance distance from Sunfrom Sun

Days of Days of OrbitOrbit

MercuryMercury 57.957.9 8888

VenusVenus 108.2108.2 225225

EarthEarth 149.6149.6 365.2365.2

MarsMars 227.9227.9 687687

JupiterJupiter 778.3778.3 4,3324,332

SaturnSaturn 14271427 10,76010,760

Using your graphing calculator – Using your graphing calculator – calculate the power function calculate the power function model for the orbital period as a model for the orbital period as a function of the average distance function of the average distance from the sun. from the sun.

Use the model to predict the orbital period for Neptune which is 4497 Gm* from the Sun.

*Gm = gigameters or millions of kilometers

For a full explanation of the For a full explanation of the previous example and another previous example and another example – see pages 194 – 196 example – see pages 194 – 196 ex 5 & 6 in your textbook….ex 5 & 6 in your textbook….

HomeworkHomework

pp 197 – 199 #s 17 – 21 odd, 49, pp 197 – 199 #s 17 – 21 odd, 49, 51, 52, 5551, 52, 55

Study for quiz next time – draw Study for quiz next time – draw graphs of power functions (know graphs of power functions (know all ten) and complete the square.all ten) and complete the square.