Glossary

Exponential functions

Criticality: 3

Functions where the independent variable appears in the exponent, typically in the form $f(x) = ab^x$, modeling growth or decay.

Example:

The population growth of a city over time can often be modeled by an exponential function, such as $P(t) = P_0(1+r)^t$ .

Exponential unit fraction

Criticality: 2

An exponent in the form $1/k$, which represents the *k*th root of the base.

Example:

In the expression $27^{1/3}$ , the exponent $1/3$ is an exponential unit fraction, indicating the cube root of 27.

Horizontal dilation

Criticality: 2

A transformation that stretches or compresses a graph horizontally from the y-axis, affecting its width.

Example:

The graph of $y=5^{2x}$ is a horizontal dilation (compression) of $y=5^x$ by a factor of 1/2.

Horizontal translation

Criticality: 2

A transformation that shifts a graph left or right along the x-axis without changing its shape or orientation.

Example:

Shifting the graph of $y=3^x$ two units to the right results in a horizontal translation to $y=3^{x-2}$ .

Negative exponent property

Criticality: 3

States that a base raised to a negative exponent is equivalent to the reciprocal of the base raised to the positive exponent: $b^{-n} = \frac{1}{b^n}$.

Example:

To rewrite $6^{-3}$ without a negative exponent, you apply the negative exponent property to get $\frac{1}{6^3}$ or $\frac{1}{216}$ .

Power property

Criticality: 3

When raising an exponential term to another power, you multiply the exponents: $(b^m)^n = b^{mn}$.

Example:

To simplify $(x^4)^3$ , you use the power property to obtain $x^{12}$ .

Product property

Criticality: 3

When multiplying exponential terms that have the same base, you add their exponents: $b^m \cdot b^n = b^{m+n}$.

Example:

To simplify $7^x \cdot 7^2$ , you apply the product property to get $7^{x+2}$ .

Reflection over the y-axis

Criticality: 2

A transformation that flips a graph across the y-axis, effectively changing the sign of the x-coordinates.

Example:

The graph of $y=(1/3)^x$ is a reflection over the y-axis of the graph of $y=3^x$ .

Vertical dilation

Criticality: 2

A transformation that stretches or compresses a graph vertically from the x-axis, changing its height.

Example:

The function $y=4 \cdot 2^x$ represents a vertical dilation (stretch) of $y=2^x$ by a factor of 4.

kth root

Criticality: 2

A number that, when multiplied by itself *k* times, yields a given number.

Example:

The kth root of 32 when k=5 is 2, because $2^5 = 32$ .