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Explain the relationship between linear functions and arithmetic sequences.
Arithmetic sequences are discrete points, while linear functions connect those points into a smooth line. Both involve an initial value and a constant rate of change.
Explain the relationship between exponential functions and geometric sequences.
Geometric sequences are discrete points, while exponential functions connect those points into a smooth curve. Both involve an initial value and a constant ratio.
What does a positive slope indicate?
A line that goes up from left to right. As x increases, y increases.
What does a negative slope indicate?
A line that goes down from left to right. As x increases, y decreases.
Describe exponential growth.
A function where the rate of increase is proportional to the current value, leading to rapid growth.
Describe exponential decay.
A function where the rate of decrease is proportional to the current value, leading to a rapid decline.
What is the significance of the base $b$ in an exponential function $f(x) = ab^x$?
If $b > 1$, it represents exponential growth. If $0 < b < 1$, it represents exponential decay.
How are linear functions and arithmetic sequences used to model real-world scenarios?
They model situations with constant change, such as population growth or simple interest.
How are exponential functions and geometric sequences used to model real-world scenarios?
They model exponential growth or decay, such as compound interest or radioactive decay.
What is the importance of domain when dealing with sequences and functions?
Sequences are usually defined for positive integers, while functions can have a wider range of values (like all real numbers).
What are the differences between linear and exponential functions in terms of rate of change?
Linear: Constant rate of change (constant slope). Exponential: Proportional rate of change (constant ratio).
What are the key differences between arithmetic and geometric sequences?
Arithmetic: Constant difference between terms. Geometric: Constant ratio between terms.
Compare the general forms of linear and exponential functions.
Linear: $f(x) = b + mx$ (addition). Exponential: $f(x) = ab^x$ (multiplication).
What are the differences between using addition and multiplication in linear and exponential functions?
Linear: Constant addition of slope. Exponential: Constant multiplication by the base.
Compare the parameters needed to define linear and exponential functions.
Linear: Initial value (y-intercept) and slope. Exponential: Initial value and base.
How do the domains of sequences and functions differ?
Sequences: Usually positive integers. Functions: Can have a wider range of values, like all real numbers.
Compare how linear and exponential functions are affected by transformations such as shifts and stretches.
Linear: Shifts and stretches affect slope and y-intercept directly. Exponential: Shifts and stretches affect initial value and base, impacting growth/decay rate.
What are the similarities and differences between modeling simple interest and compound interest?
Simple Interest: Modeled by linear functions (constant addition). Compound Interest: Modeled by exponential functions (constant multiplication).
Compare the long-term behavior of linear and exponential functions.
Linear: Grows (or decays) at a constant rate. Exponential: Grows (or decays) much faster in the long run.
How do you determine if a real-world situation is best modeled by a linear or an exponential function?
Linear: Look for constant addition/subtraction. Exponential: Look for constant multiplication/division (percentage increase/decrease).
What is the formula for a linear function?
$f(x) = b + mx$
What is the formula for an arithmetic sequence?
$a_n = a_0 + dn$
What is the point-slope form of a linear function?
$f(x) = y_i + m(x - x_i)$
What is the formula for an exponential function?
$f(x) = ab^x$
What is the formula for a geometric sequence?
$g_n = g_0 * r^n$
What is the formula for a geometric sequence with a known term?
$g_n = g_k * r^(n-k)$
What is the formula for an exponential function with a known point?
$f(x) = y_i * r^(x-x_i)$
Formula for arithmetic sequence with a known term?
$a_n = a_k + d(n-k)$
How do you calculate the slope ($m$) given two points $(x_1, y_1)$ and $(x_2, y_2)$?
$m = \frac{y_2 - y_1}{x_2 - x_1}$
What is the decay factor formula?
$(1 - r)$, where $r$ is the rate of decay.