1. Transforming Polynomial Functions
What you’ll learn
To apply transformations to graphs of polynomials.
Vocabulary
Power function.
Constant of proportionality.

2. Example 1: Transforming .What is an equation
of the graph under a vertical compression
by the factor followed by a reflection across the
x-axis, a horizontal translation 3 units to the right,
and then a vertical translation 2 units up?
Answer
Step 1: Multiply by 1/2
Step 2: multiply by -1 to reflect
Step 3: Replace x with x-3 to translate horizontally.
Step 4: Add 2 to translate vertically.

3. Your turn
What is the equation of the graph of y=x^3 under a vertical stretch by the factor 2 followed by a horizontal translation 3 units to the left and then a vertical translation 4 units down?
Answer

4. Find all real zeros of each function
Problem 2: Finding Zeros of a transformed cubic function.
Answer
Make equation =0
Add 8
Multiply by -8
Apply
Your turn
Find all real zeros of each function
Answer

5. What is a quartic function with only two real zeros, x=5 and x=9?
Problem 3: Constructing a Quartic Function with Two Real Zeros.
What is a quartic function with only two real zeros,
x=5 and x=9?
Method 1: Using transformations
First, find a quartic with
Translate the basic quartic 16 units down;
9 is 7 units to the right of 2.
Translate 7 units to the right.
A quartic function with its only real zeros at 5 and 9 is

6. Method 2: Use algebraic methods.
We have
We can use the factor and Q(x).
This is any quadratic with no real zeros
Another quartic function with its only real zeros at
5 and 9 is

7. Your turn
a)What is a quartic function f(x) with only two real
zeros, x=0 and x=6?
b)Does the quartic –f(x) have the same zeros? Explain
Answer
b) yes, -f(x) is the function reflected across
the x-axis, so the zeros will stay the same.

8. Take a note: The offspring of the parent function
y=x^4 is a subfamily of all quartic polynomials. This
subfamily consists of quartics of the form
These functions also belong to another category of
polynomials, and in this category you can generate
families as unusual.
Definition: A power function is a function of the form
where a and b are non zero real numbers.
Then y varies directly with or
is proportional to the bth power
of x. a is the
constant of proportionality.
Examples:

9. Problem 4: Modeling with a power function.
Wind-Generated Power: Wind farms are a source of a renewable
energy found around the world. The power P( in kilowatts)
generated by a wind turbine varies directly as the cube of the
wind speed v(in meter per second). The picture shows the power
output of one turbine at one wind speed. To the nearest
kilowatt, how much power does this turbine generate in a
10m/s wind?
Answer
Use the formula
Get the information from picture
The turbine generated about
1172 kw of power in a 10m/s wind.

10. Classwork and Homework
TB pgs Exercises 7-43