What is the formula for a linear function?

$f(x) = b + mx$

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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.

How do you determine the equation of a line given two points?

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?

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?

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 $f(x) = ab^x$ .

Define a linear function.

A function of the form $f(x) = b + mx$ , where $b$ is the y-intercept and $m$ is the slope.

What is the slope of a line?

The rate of change of a linear function; steepness and direction.

Define an arithmetic sequence.

A sequence with a constant difference between consecutive terms, defined as $a_n = a_0 + dn$ .

What is the common difference in an arithmetic sequence?

The constant value added to each term to get the next term.

Define an exponential function.

A function of the form $f(x) = ab^x$ , where $a$ is the initial value and $b$ is the base.

What is the base of an exponential function?

The factor by which the function's value changes for each unit increase in $x$ .

Define a geometric sequence.

A sequence with a constant ratio between consecutive terms, defined as $g_n = g_0 * r^n$ .

What is the common ratio in a geometric sequence?

The constant value by which each term is multiplied to get the next term.

What is the y-intercept?

The point where the line crosses the y-axis, where $x=0$ .

What does the initial value represent in an exponential function?

The starting value of the function when x=0.

All Flashcards

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.

How do you determine the equation of a line given two points?

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?

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?

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 $f(x) = ab^x$ .

Define a linear function.

A function of the form $f(x) = b + mx$ , where $b$ is the y-intercept and $m$ is the slope.

What is the slope of a line?

The rate of change of a linear function; steepness and direction.

Define an arithmetic sequence.

A sequence with a constant difference between consecutive terms, defined as $a_n = a_0 + dn$ .

What is the common difference in an arithmetic sequence?

The constant value added to each term to get the next term.

Define an exponential function.

A function of the form $f(x) = ab^x$ , where $a$ is the initial value and $b$ is the base.

What is the base of an exponential function?

The factor by which the function's value changes for each unit increase in $x$ .

Define a geometric sequence.

A sequence with a constant ratio between consecutive terms, defined as $g_n = g_0 * r^n$ .

What is the common ratio in a geometric sequence?

The constant value by which each term is multiplied to get the next term.

What is the y-intercept?

The point where the line crosses the y-axis, where $x=0$ .

What does the initial value represent in an exponential function?

The starting value of the function when x=0.