Radical Functions
Section 3.4
Modeling continuous relationship which is
increasing and concave down
Model: Cost Function
A company manufactures a
quantity q of a product per month.
They invest $2.3 million in capital
towards technology (machinery,
computers, etc.) and a fixed cost
investment of $5.4 million. Data
is collected on the cost for varying
quantities q (in 100,000 units).
Determine a model for cost.
Cost Data
Properties of Model
Is the data increasing or decreasing?
Cost increases as production increases
Is the data concave up or concave down?
Data is concave down
Use the Ladder of Powers to determine a
model with these characteristics. Can
it be a
polynomial model?
No, polynomial functions are either increasing
concave up or decreasing
concave down
Radical Functions
Function of form
where p(x) is a polynomial, integer n>1
If n is even, what is the domain of r(x)?
If n is odd, what is the domain of r(x)?
What is the concavity of a radical power
function y = ax^{n} when 0 < n < 1?
Radical Function:
index 0<n<1
Determining Radical Model
Ladder of Powers to decide if n = ½,
n = 1/3, n = ¼ etc.
Graph and , which is
a better fit to the Cost Data?
Use Derive or a graphing calculator to fit a
radical function to the data
Solution:
CobbDouglas Production Function
Cost Model
Solving Radical Equations
Algebraic Method
Isolate the radical expression
Undo the nth root by using the inverse
function –the nth power
For which nth roots do we need to check for
extraneous roots?
If nth root is even, taking an even power can
result in extraneous roots
If nth root is odd, taking an odd power will not
result in extraneous roots
Double Radical Equation
Isolate one of the radical expressions
Take the square to remove one of the square roots
Isolate the remaining radical and square again
Check for extraneous roots
Solve using algebraic and graphic method
Double Radical Equation
Class Participation Exercise
1. Determine what
type of function
has the given
graph? Is it
polynomial or
radical? Why?
2. Solve the following radical equation:
Linearization
Use radical function to
make nonlinear polynomial
data linear
Model the linear data
Convert linear model into
polynomial model
Linearization
Use Derive or graphing calculator to
plot data and transformed data
Which of these makes the data linear?
Linearization –Transformed Data
Linearization –Data Plot
Linearization
Which transformed data is linear?
Solution: Cube root, so data is modeled by
a cubic function
Linearized data model: y = 1.3x + 0.77
Actual data model:
Best fit cubic model:
