2.2A Change in linear and exponential functions
Describe the similarities between a linear function and an arithmetic sequence.
What is the form of an arithmetic sequence? Explain each component of this form.
What is the form of a linear function? Explain each component of this form.
Write the equation of the linear function which
Has a slope of 4 and passes through the point (2, 5)
Has a slope of -3 and passes through the point (-4, -6)
Explain how to represent an arithmetic sequence a_n=a_k + d(n-k) as linear function.
Slope-intercept form
Point-slope form
For the linear function f(x) = 3+2x :
What is the initial value?
What is the constant rate of change (slope)?
What will be the representation of the function as arithmetic sequence?
What will be first term in the sequence?
What will be common difference?
Represent the arithmetic sequence defined by the following equations as linear equations.
a_n = 5 + 3n
a_n = 7 - 2n
a_n = 10 + 6(n-1)
a_n = -5 + 8(n-1)
Represent the following linear functions as arithmetic sequences in the form {a_n=a_k+d(n-k)} where k\neq0.
Represent the following sequences as linear equations in the following forms.
Slope-intercept form
Point-slope form
3, 10, 17, 24, 31, \ldots
-4, 2, 8, 14, 20, \ldots
100, 92, 84, 76, 68, \ldots
1, 6, 11, 16, 21, \ldots
Consider the arithmetic sequence 4, 7, 10, 13, \ldots and the linear function y = 3x + 1.
State the domain of the arithmetic sequence.
State the domain of the linear function.
Discuss any differences in their domains.
The linear function y = 5x - 3 is represented by the arithmetic sequence -3, 2, 7, 12, 17, \ldots.
State the domain of the arithmetic sequence.
State the domain of the linear function.
Discuss the similarities and differences in the domains of the arithmetic sequence and the linear function.
The following tables show a function for a sequence:
Determine if this function is linear.
Provide the rate of change if the function is linear.
State the linear function.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | 7 | 12 | 17 | 22 | 27 | 32 |
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | -5 | -3 | -1 | 1 | 3 | 5 |
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | 2 | 5 | 8 | 11 | 14 | 17 |
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | 1 | 3 | 9 | 27 | 81 | 243 |
The 5th term of an arithmetic sequence is 16 and the 10th term is 31. Determine the corresponding linear function.
An arithmetic sequence starts with 7 and has a common difference of -2. Determine the linear function that describes this sequence.
The linear function y =-2x+5 represents an arithmetic sequence.
Determine the common difference.
Determine the first 5 terms.
Suppose a sequence is defined by the recursive formula a_n = a_{n-1} + 5 with the initial term a_0 = 2.
Write a linear function f(x) that generates the same sequence as the terms of the arithmetic sequence.
Verify your function by listing the first five terms of both the sequence and the function's outputs for the inputs x=0,\,1,\,2,\,3,\,4.
Given a linear function f(x) = 3x - 2 and an arithmetic sequence a_n = 2 + 3n.
Explain why or why not the function and sequence are equivalent.
If they are not equivalent, modify one of them so they become equivalent.
A company's profit over the first 10 years of operation could be modeled by a function similar to an arithmetic sequence. In the first year, the company made a profit of \$2000 and each subsequent year the profit increased by \$500.
Write a linear function that models the company's profit over time.
Predict the company's profit in the 8th year.
After how many years will the company's profit exceed \$10\,000?
A population of a rare bird species in a wildlife reserve is observed over time. The bird population (p) can be modeled by a linear function p(t) = a_k + d(t - k), where t is the time in years since the observation started, and a_k is the population at year k. At the end of the first year of observation, there were 200 birds, and at the end of the fifth year, there were 120 birds.
Find the equation of the linear function that models the bird population.
Predict the bird population at the end of the tenth year.
If a conservation effort is put in place that increases the bird population by 20 birds each year, write a new linear function to model this scenario.