Exponents and Radicals
Division of Radicals
Exponents and Radicals
Radicals and Rational Exponents
Radical Equations
Solving Radical Equations
Roots and Radicals
Simplifying Radical Expressions
Radical Expressions
Solving Radical Equations
Solving Radical Equations
Exponents and Radicals
Exponents and Radicals
Roots;Rational Exponents;Radical Equations
Solving and graphing radical equations
Solving Radical Equations
Radicals and Rational Exponents
Roots, Radicals, and Root Functions
Multiplication of Radicals
Solving Radical Equations
Radical Expressions and Equations
Equations Containing Radicals and Complex Numbers
Square Roots and Radicals
Solving Radical Equations in One Variable Algebraically
Polynomials and Radicals
Roots,Radicals,and Fractional Exponents
Adding, Subtracting, and Multiplying Radical Expressions
Square Formula and Powers with Radicals
Simplifying Radicals
Exponents and Radicals Practice
Solving Radical Equations
Solving Radical Equations
Solving Radical Equations
Lecture-Radical Expressions
Radical Functions
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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 = axn 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

Cobb-Douglas 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:


Use radical function to
make nonlinear polynomial
data linear

Model the linear data

Convert linear model into
polynomial model


Use Derive or graphing calculator to
plot data and transformed data

Which of these makes the data linear?

Linearization –Transformed Data

Linearization –Data Plot


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:

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