5.3 Modeling with Power Functions

1. 5.3 Modeling with Power Functions

2. Power Data & Linear Data Connection • If f is a power function of x, then ln(f) is a linear function of ln(x). Getting a Power Model from Data

3. Ex. 1) Volume Inside a Sphere a.)Plot the graph of ln(V) versus ln(r) to determine whether or not it is reasonable to think that the volume is related to the radius by a power function. b.) Use regression to find a formula for the line formed by the plot of ln(V) as a function of ln(r). Then, use that formula to write a formula for the volume of a sphere. Round parameters to 2 decimal places. c.) Use regression to find a formula for the volume of a sphere as a function of the radius. Round to 2 decimal places. d.) Plot both the function from b and c and the original data point on the same graph. e.) What is the radius of a sphere with a volume of 453.25? (Use the function from part c.

4. Ex. 2) Generation Time as a Power Function of Length • a.) Plot ln(T) against ln(L). Determine whether it is reasonable to model T as a power function of L. • b.) Find a formula for the regression line of Ln(T) against ln(L). • c.) Find a formula that models T as a power function of L, and plot the function along with the given data. Round to 2 decimal places. • d.) If one generation time is 70% more than another, how much greater would the length be (given as a percentage)?

5. Ex. 3) Length vs. Flying Speed • a.) Verify that it is appropriate to model the data given in the table using a power function. • b.) Using regression, find a formula that models F as a power function of L. • c.) Is the average flying speed for a Flying fish more or less than would be expected from the model? • d.) If one animal’s flying speed is 35% less than another, how much shorter is it’s length (given as a percentage)?

6. Practice: a.) Make a power model for weight versus height. b.) According to the model from part a, what percentage increase in weight can be expected if height is increased by 10%?

7. 5.4: Composition of Functions • Composing functions is a way of manipulating equality relationships in order to change the independent variable in an equation. • It is useful because it can directly link functions that were previously linked by one or more other variables. This allows us to show clearer relationships between variables.

8. 5.4: Composition of Functions • Composing Functions: Real world application • If a boy’s height depends upon his age according to h(t)=1.9t + 40, and his weight depends on his height according to w(h) = 4.4h – 15, how can we write an equation for weight that depends on age?

9. Ex.) Composition: Algebra Based

10. Ex.) Composition: Algebra Based using the same independent variable.

11. Ex.) Combining Functions: Adding, Subtracting, and Multiplying.

12. Practice: