Power Functions with Modeling 2.1 & 2.2 Pre-Calculus Ms. Hardy

Introduction Compare & contrast the different power functions from the explore worksheet after the test. What are the general shapes? What is the end behavior for each? How are these related to our 13 basic functions?

Now let’s look at applications of the Power Functions Any function that can be written in the form f(x) = k•xa k, a ≠ 0 Where “a” is the power and “k” is the constant of variation.

Variation is a kind of modeling that uses power functions.

Direct Variation as one variable increases the other increases. y = kx y varies directly as x or y is proportional to x. K is called the constant of variation.

Inverse Variation – as one variable increases the other decreases.  

Joint Variation is where more than two quantities are involved. z = kxy Z varies jointly as x & y OR Z is jointly proportional to x & y.

Write each variation statement as a power function 1) The circumference of a circle varies directly as the radius. 2) The average speed (rate) is inversely proportional to the time traveled.

Write each variation statement as a power function 3) A car is traveling on a curve that forms a circular arc. The force F needed to keep the car from skidding is jointly proportional to the weight w of the car and the square of its speed and is inversely proportional to the radius r of the curve.

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

Solving Variation Problems Write a general variation equation for the scenario using variables (remember to include the variation constant K). Use the established information in the equation to solve for K. Write the variation equation with the K value and use it to answer the question.

Solving Variation Problems 1) DRAG FORCE ON A BOAT: The drag force F on a boat is jointly proportional to the wetted surface area A on the hull and the square of the speed s of the boat.

DRAG FORCE ON A BOAT continued: A boat experiences a drag force of 220 lb when traveling at 5 mph with a wetted surface area of 40 ft2. How fast must a boat be traveling if it has 28 ft2 of wetted surface area and is experiencing a drag force of 175 lb?

2) LOUDNESS OF SOUND The loudness L of a sound (measured in decibels, dB) is inversely proportional to the square of the distance d from the source of the sound. A person 10 ft from a lawn-mower experiences a sound level of 70 dB. How loud is the lawn mower when the person is 80 ft away? (round to tenth)

2) POWER FROM A WINDMILL The power P that can be obtained from a windmill varies directly with the cube of the wind speed s. A windmill produces 96 watts of power when the wind is blowing at 20 mph. How much power will this windmill produce if the wind speed increases to 30 mph?

Modeling with Power Functions Sometimes we aren’t sure what power the regression data is best modeled by. We can then calculate a power regression A:PwrReg on the graphing calculator. It will determine exactly what exponent best models the data.

Data is given below for y as a power function of x Data is given below for y as a power function of x. Write an equation for the power function, state its power & constant of variation. X 1 4 9 16 25 Y 2 6 8 10

Modeling with Power Functions Kepler’s Third Law states that the square of the period of orbit T (time required for full revolution around the sun) varies directly with the cube of its average distance “a” from the Sun. The following table gives data for the 6 planets that were known in Kepler’s time (1600s).

Planet Avg distance from Sun Days of Orbit Mercury 57.9 88 Venus 108.2 225 Earth 149.6 365.2 Mars 227.9 687 Jupiter 778.3 4,332 Saturn 1427 10,760

Using your graphing calculator – calculate the power function model for the orbital period as a function of the average distance 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 previous example and another example – see pages 194 – 196 ex 5 & 6 in your textbook….

Homework pp 197 – 199 #s 17 – 21 odd, 49, 51, 52, 55 Study for quiz next time – draw graphs of power functions (know all ten) and complete the square.