Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
Writing Function Formulas for Polynomials
The goal of today’s lesson is to write a polynomial function formula that fits a given graph and
information. The process is very similar to what you did in the past for quadratic functions:
start with a function formula that has an unknown a value then solve for a.
Examples
Example 1: The graph shows a monomial of degree 3. Find its
function formula.
f(x) = a x3
4 = a 23
4 =a· 8
1
2 = a
Answer: f(x) = 12 x3
From the given information:
Picking the point (2, 4), substitute:
Solve for a:
Example 2: The graph shows a degree 3 polynomial. Find
the function formula.
Start with factored form. Use a squared factor for (x – 2)2
because the graph just touches the axis at x = 2.
f(x) = a (x – 2)2 (x + 1)
Picking the point (0, –4), substitute and solve:
–4 = a (0 – 2)2 (0 + 1)
–4 = a (4)
(1)
–1 = a
Answer: f(x) = –1 (x – 2)2 (x + 1)
Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
PROBLEMS:
1. Each graph shows a function of either degree 3 or degree 4. Find function formulas.
a.
b.
c.
d.
Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
2. Each graph shows a function of either degree 3 or degree 4. Find function formulas
in factored form. (Follow the method shown in Example 2.)
a.
b.
c.
d.
Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
3. Find function formulas for these polynomial graphs. All of the zeros are visible on the graph,
and there are no multiplicities higher than 2. (The only kinds of factors are (x–r)’s and (x–r)2 ’s.)
a.
b.
c.
d.
Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
Polynomial Application Problems
4. The shape of a water slide ride is given by this function:
f(x) = –0.001 x3 + 0.13 x2 – 5x + 100
The x-axis represents the ground. The ride starts on the y-axis and ends on the x-axis.
All numbers in this problem represent measurements in feet.
a. Graph the function on your calculator using settings
Xmin = –10, Xmax = 100, Ymin = –10, Ymax = 100.
Sketch the graph in the box at the right.
b. How high above the ground is the ride’s starting point?
c. At what x value is the end of the ride located?
d. You should see a “dip” (low point) and a “bump” (high point) in the ride graph. Using the
minimum and maximum commands on your calculator, find the coordinates of these two
points.
Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
5. The power, in megawatts, produced between midnight and noon by a power plant is given by
f(x) = x4 – 12x3 + 4000, where x stands the hour of the day, and the domain is 0 ≤ x ≤ 12.
a. Fill in this table of values.
time x
power f(x)
0
2
4
6
8
10
12
b. Graph the function on your calculator. Set your
window to use x values from 0 to 12, and y values
such that the whole graph can be seen on the screen
(the table from part a shows you how large the y
values could be). Sketch the graph in the box.
c. Use your calculator to determine at what time the power output the lowest. How much
power is output at that time?
d. Does f(x) have any zeros? What does this tell you about the power plant?
Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
6. The amount of energy radiated by an object is
approximately given by the formula
f(x) = 0.057 (x + 273)4,
where x stands for temperature in degrees Celsius.1
a. Using the table on your calculator, fill in this table
of values for the function.
x
f(x)
–10
0
10
20
30
40
b. A metal roof on a hot day has a radiated energy of 650,000,000 (in the units of this
problem). What is the roof’s temperature? Find out by setting up an equation and solving it
algebraically.
1
This fact is called the Stefan-Boltzmann law, and the units for the output are
formula in the study of climates and global warming.
microwatts
meters 2degreesK4
. It’s an important
Name:
Date: _______________
Algebra 2
Writing Function Formulas for Polynomials
Answers to “Writing Function Formulas for Polynomials”
For any of these answers, if you have an a value that is slightly different from mine (for example,
1
4
2 instead of 9 ), both of our answers are right.
1. a. f(x) =  14 x 3
b. f(x) = 2x4
c. f(x) =  18 x 4
d. f(x) =
1
8
x3
2. a. f(x) = 94 ( x  3) 2 ( x  2)
b. f(x) = 18 ( x  5)( x  3)( x)( x  2)
c. f(x) = 92 ( x  3) 2 ( x  1)
d. f(x) =  23 ( x  3)( x  2)( x  1) 2
3. a. f(x) = – 14 ( x  2)( x  1)( x  4)
b. f(x) =
1
500
c. f(x) = –1 (x – 1)2 (x – 3)2
d. f(x) =
2
27
( x  8)( x  4) x( x  4)( x  8)
( x  3) 2 ( x  1)( x  3)
Answers to Application Problems:
4. a.
b. 100 feet
c. Use [2nd][Trace] zero, get x ≈ 85.027.
d. minimum at (28.804, 39.939), maximum at (57.863, 52.209)
5. a.
b.
c. lowest at x = 9 a.m., power is 1813 MW.
d. f(x) has no zeros, meaning there is never a time that the plant is not producing any power.