Transcript Section 5.9 PP.ppt

EXAMPLE 1
Write a cubic function
Write the cubic function whose graph is shown.
SOLUTION
STEP 1
Use the three given
x - intercepts to write the
function in factored form.
f (x) = a (x + 4)(x – 1)(x – 3)
STEP 2
Find the value of a by substituting the coordinates of
the fourth point.
EXAMPLE 1
Write a cubic function
–6 = a (0 + 4) (0 – 1) (0 – 3)
–6 = 12a
– 1 = a
2
ANSWER
The function is f (x) = – 1 (x + 4)(x – 1)(x – 3).
2
CHECK
Check the end behavior of f. The degree of f is odd
and a < 0. So f (x) +∞ as x → –∞ and f (x) → –∞ as
x → +∞ which matches the graph.
EXAMPLE 2
Find finite differences
The first five triangular numbers are shown below. A
formula for the n the triangular number is
1
f (n) = 2 (n2 + n). Show that this function has
constant second-order differences.
EXAMPLE 2
Find finite differences
SOLUTION
Write the first several triangular numbers. Find the
first-order differences by subtracting consecutive
triangular numbers. Then find the second-order
differences by subtracting consecutive first-order
differences.
EXAMPLE 2
Find finite differences
ANSWER
Each second-order difference is 1, so the secondorder differences are constant.
GUIDED PRACTICE
for Examples 1 and 2
Write a cubic function whose graph passes through
the given points.
1.
(– 4, 0), (0, 10), (2, 0), (5, 0)
ANSWER
The function is f (x) = 14 (x + 4) (x – 2) (x – 5).
y = 0.25x3 – 0.75x2 – 4.5x +10
GUIDED PRACTICE
2.
for Examples 1 and 2
(– 1, 0), (0, – 12), (2, 0), (3, 0)
ANSWER
The function is f (x) = – 2 (x + 1) (x – 2) (x – 3).
y = – 2 x3 – 8x2 – 2x – 12
GUIDED PRACTICE
3.
for Examples 1 and 2
GEOMETRY Show that f (n) = 12 n(3n – 1), a
formula for the nth pentagonal number, has
constant second-order differences.
ANSWER
Write function values for
equally-spaced n - values.
First-order differences
Second-order differences
Each second-order difference is 3, so the secondorder differences are constant.
EXAMPLE 3
Model with finite differences
The first seven triangular pyramidal numbers are
shown below. Find a polynomial function that gives
the nth triangular pyramidal number.
SOLUTION
Begin by finding the finite differences.
EXAMPLE 3
Model with finite differences
Because the third-order differences are constant,
you know that the numbers can be represented by a
cubic function of the form f (n) = an3 + bn2 + cn + d.
By substituting the first four triangular pyramidal
numbers into the function, you obtain a system of
four linear equations in four variables.
EXAMPLE 3
Model with finite differences
a(1)3 + b(1)2 + c(1) + d = 1
a+b+c+d=1
a(2)3 + b(2)2 + c(2) + d = 4
a(3)3 + b(3)2 + c(3) + d = 10
a(4)3 + b(4)2 + c(4) + d = 20
8a + 4b + 2c + d = 4
27a + 9b + 3c + d = 10
64a + 16b + 4c + d = 20
Write the linear system as a matrix equation AX = B.
Enter the matrices A and B into a graphing
calculator, and then calculate the solution X = A– 1 B.
Model with finite differences
EXAMPLE 3
1 1
8 4
27 9
64 16
A
1
2
3
4
1
1
1
1
a
b
c
d
1
= 4
10
20
x
B
1
1
1
The solution is a = 6 ,b = 2 , c = 3 , and d = 0. So,
the nth triangular pyramidal number is given by
f (n) = 1 n3 + 1 n2 + 1 n.
2
3
6
GUIDED PRACTICE
for Example 3
4. Use finite differences to find a polynomial function
that fits the data in the table.
ANSWER
f (x) = –x3 + 5x2 + x + 1.
EXAMPLE 4
Solve a multi-step problem
Space Exploration
The table shows the typical speed y (in feet per
second) of a space shuttle x seconds after launch.
Find a polynomial model for the data. Use the model
to predict the time when the shuttle’s speed reaches
4400 feet per second, at which point its booster
rockets detach.
EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Enter the data into a graphing calculator
and make a scatter plot. The points
suggest a cubic model.
EXAMPLE 4
STEP 2
Solve a multi-step problem
Use cubic regression to obtain this
polynomial model:
y = 0.00650x3 – 0.739x2 + 49.0x – 236
EXAMPLE 4
STEP 3
Solve a multi-step problem
Check the model by graphing it and the
data in the same viewing window.
EXAMPLE 4
STEP 4
Solve a multi-step problem
Graph the model and y = 4400 in the same
viewing window. Use the intersect feature.
The booster rockets detach about 106 seconds
after launch.
GUIDED PRACTICE
for Example 4
Use a graphing calculator to find a polynomial
function that fits the data.
5.
ANSWER
y = 2.71x3 – 25.5x2 + 71.8x – 45.7
GUIDED PRACTICE
for Example 4
6.
ANSWER
y = –0.587x3 + 8.99x2 – 23.4x + 9.62