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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.
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).
How do you determine the equation of a line given two points?
1. Calculate the slope using $m = (y_2 - y_1) / (x_2 - x_1)$. 2. Use the point-slope form: $y - y_1 = m(x - x_1)$. 3. Simplify to slope-intercept form if needed.
How do you determine if a function represents exponential growth or decay?
Examine the base $b$ in $f(x) = ab^x$. If $b > 1$, it's growth. If $0 < b < 1$, it's decay.
How do you find the value of an exponential function after a certain number of years?
Substitute the number of years for $t$ in the function $V(t) = a(b)^t$ and calculate the result.
How do you find the equation of an exponential function given two points?
1. Substitute the points into $f(x) = ab^x$ to get two equations. 2. Solve for $a$ in one equation. 3. Substitute the expression for $a$ into the other equation and solve for $b$. 4. Substitute the value of $b$ back into the equation for $a$.
How do you solve for $t$ in an exponential decay problem?
1. Set up the equation $V(t) = a(b)^t$. 2. Isolate the exponential term. 3. Take the logarithm of both sides. 4. Solve for $t$.
How to determine the common difference in an arithmetic sequence?
Subtract any term from its subsequent term: $d = a_{n+1} - a_n$.
How to determine the common ratio in a geometric sequence?
Divide any term by its preceding term: $r = g_{n+1} / g_n$.
How do you convert an exponential function from base $b$ to base $e$?
Use the identity $b^x = e^{x ln(b)}$, so $f(x) = ab^x$ becomes $f(x) = ae^{x ln(b)}$.
How to write the equation of a line given a point and a slope?
Use the point-slope form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the slope. Then, rearrange to slope-intercept form if needed.
How do you determine the initial value of an exponential function from a table of values?
Find the y-value when x=0. This value is the initial value $a$ in the function $f(x) = ab^x$.