Sequences 101
What is the difference between an arithmetic and geometric sequence?
Arithmetic add/subtract the same number to each term.
Geometric multiply/divide the same number to each term.
When do you use n versus n-1 in an equation?
N when the sequence starts with the 0 term
n-1 when the sequence starts with the 1st term
State the degree and Leading Coefficient:
11x6- 5x5 + 4x2
Degree: 6
Leading Coefficient: 11
Use Desmo or graphing calculator to locate the zeros and determine if their multiplicities are even or odd
g(x) = x3 + 3x2
(0,0) : Even Multiplicity
(-3,0) : Odd Multiplicity
What are the two other terms for x-intercepts?
Zeros and Roots
Classify the following sequence
and identify the common difference and/or ratio:
f(1) = 5, f(2) = 9, f(3) = 13, f(4) = 17...
Arithmetic
Common difference: +4
The explicit formula for the sequence 3, 9, 27, 81,......
f(x) = = 3 * 3(n-1)
State the degree and Leading Coefficient
8x3 - 3x6 + 4x -5
Degree:
6
Leading Coefficient:
-3
Use Desmos to describe the end behavior of the graph
g(x) = x3 +3x2
As x goes toward negative infinity
f(x) goes towards negative infinity
As x goes towards positive infinity
f(x) goes towards positive infinity
What is an x-intercept?
What is a y-intercept?
x-intercept: point on the x-axis (horizontal)
y-intercept: point on the y-axis (vertical)
Classify the following sequence
and identify the common difference and/or ratio:
f(1) = 2, f(2) = 6, f(3) = 18, f(4) = 54...
Geometric Sequence
Common Ratio: 3
Which formula defines the sequence:
f(1) = 2, f(2) =6, f(3) = 10, f(4) = 14, f(5) = 18
A) f(1) =2, f(n) = 6 + f (n-1) for n>2
B) f(1) =2, f(n) = 4 + f (n-1) for n>2
C) f(1) =2, f(n) = 2 + f (n-1) for n>2
D) f(1) =6, f(n) = 4 + f (n-1) for n>2
B) f(1) =2, f(n) = 4 + f (n-1) for n>2
Find the degree:
f(x) = - (x-3)2(x+1)(x-4)
4
Sketch the graph of polynomials functions with the following characteristics.
An odd degree function
A positive leading coefficient
Zeros at -5, -3, 0, 2, and 4
Check Drawing
How can you find the x and y intercepts from the factored form of a polynomial?
x-intercept:
plug in 0 for y
or
set each factor = 0
y-intercept:
Plug in 0 for x and solve
Write an arithmetic and geometric sequence that both start with...
(at least 5 terms)
3, 6, ...
A: 3, 6, 9, 12, 15
G: 3, 6, 12, 24, 48
The explicit formula for the sequence 128, 32, 8...
f(x) = 128 * 1/4(n-1)
or
f(x) = 128 * 0.25(n-1)
Find the degree and determine if the leading coefficient is positive or negative:
f(x) = - (x+4)2(x+1)(x-1)2
Degree:
5
Leading Coefficient:
Negative
Sketch the graph of polynomials functions with the following characteristics.
An even degree function
Zeros at -2, 1, 3, and 5
A y-intercept at (0,4)
Check drawing
Find the x and y intercepts of the following polynomial:
f(x) = (x-5)(x+2)2
x: (5,0) and (-2,0)
y: (0,-20)
Create a series of four pictures that show an arithmetic sequence.
Check pictures
Create explicit equation for the following sequence:
f(0) = 81, f(1) = 27, f(2) = 9, f(3) = 3, f(4) = 1
f(x) = 81 * 1/3n
Find the degree and determine if the leading coefficient is positive or negative:
f(x) = x2(x+2)
Degree: 3
Leading Coefficient: Positive
Sketch a graph for the following function:
f(x) = (x-2)(x+1)2
Check sketches
The polynomial p is a function of x . The graph of p has four zeros at −4, -2/3, 0, and 9. Select all the expressions that could represent p.
A) 3x(x - 4)(x + 2/3)(x + 9)
B) -3x(x + 4)(3x +2)(x - 9)
C) 3x(x + 4)(2x - 3)(x -9)
D) -3x(x + 4)(3x +2)(x - 9)2
E) -x(x + 4)(x + 2/3)(x - 9)
D) -3x(x + 4)(3x +2)(x - 9)2
E) -x(x + 4)(x + 2/3)(x - 9)