Linear: Constant rate of change (constant slope). Exponential: Proportional rate of change (constant ratio).

Arithmetic: Constant difference between terms. Geometric: Constant ratio between terms.

Linear: $f(x) = b + mx$ (addition). Exponential: $f(x) = ab^x$ (multiplication).

Linear: Constant addition of slope. Exponential: Constant multiplication by the base.

Linear: Initial value (y-intercept) and slope. Exponential: Initial value and base.

Sequences: Usually positive integers. Functions: Can have a wider range of values, like all real numbers.

Linear: Shifts and stretches affect slope and y-intercept directly. Exponential: Shifts and stretches affect initial value and base, impacting growth/decay rate.

Simple Interest: Modeled by linear functions (constant addition). Compound Interest: Modeled by exponential functions (constant multiplication).

Linear: Grows (or decays) at a constant rate. Exponential: Grows (or decays) much faster in the long run.

Linear: Look for constant addition/subtraction. Exponential: Look for constant multiplication/division (percentage increase/decrease).

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.

Geometric sequences are discrete points, while exponential functions connect those points into a smooth curve. Both involve an initial value and a constant ratio.

A line that goes up from left to right. As x increases, y increases.

A line that goes down from left to right. As x increases, y decreases.

A function where the rate of increase is proportional to the current value, leading to rapid growth.

A function where the rate of decrease is proportional to the current value, leading to a rapid decline.

If $b > 1$, it represents exponential growth. If $0 < b < 1$, it represents exponential decay.

They model situations with constant change, such as population growth or simple interest.

They model exponential growth or decay, such as compound interest or radioactive decay.

Sequences are usually defined for positive integers, while functions can have a wider range of values (like all real numbers).

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.

Examine the base $b$ in $f(x) = ab^x$. If $b > 1$, it's growth. If $0 < b < 1$, it's decay.

Substitute the number of years for $t$ in the function $V(t) = a(b)^t$ and calculate the result.

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 $a$.

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 $t$.

Subtract any term from its subsequent term: $d = a_{n+1} - a_n$.

Divide any term by its preceding term: $r = g_{n+1} / g_n$.

Use the identity $b^x = e^{x ln(b)}$, so $f(x) = ab^x$ becomes $f(x) = ae^{x ln(b)}$.

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.

Find the y-value when x=0. This value is the initial value $a$ in the function $f(x) = ab^x$.