Glossary

Base (of logarithm/exponential)

Criticality: 2

In $f(x) = a \log_b(x)$ or $f(x) = ab^x$, 'b' is the base. It must be greater than 0 and not equal to 1, determining the rate of growth or decay.

Example:

In the function $y = 5^x$ , the base is 5, indicating that the output multiplies by 5 for each unit increase in x.

Coefficient (of logarithm/exponential)

Criticality: 2

In $f(x) = a \log_b(x)$ or $f(x) = ab^x$, 'a' is the coefficient. It scales the function vertically and cannot be zero.

Example:

For the function $y = 3 \cdot 2^x$ , the coefficient 3 means the initial value or y-intercept is 3 when x=0.

Exponential Function

Criticality: 3

A function of the form $f(x) = ab^x$, where the input variable 'x' is in the exponent. It models rapid growth or decay.

Example:

The population growth of bacteria, doubling every hour, can be modeled by an exponential function like $P(t) = P_0 \cdot 2^t$ .

Exponential Growth

Criticality: 2

A pattern where output values change multiplicatively as input values change additively, resulting in a rapidly increasing curve.

Example:

The value of an investment earning compound interest exhibits exponential growth, increasing faster over time.

Horizontal Asymptote (for exponential functions)

Criticality: 2

A horizontal line that the graph of an exponential function approaches but never touches. For $f(x) = ab^x$, it is typically the x-axis ($y=0$).

Example:

The graph of $y = 2^x$ has a horizontal asymptote at $y=0$ , indicating that as x approaches negative infinity, the y-values get closer and closer to zero.

Identity Function

Criticality: 2

The function $h(x) = x$, which is a straight line with a slope of 1 passing through the origin. It acts as the line of reflection for inverse functions.

Example:

When graphing $y=x^2$ and its inverse $y=\sqrt{x}$ , the identity function $y=x$ serves as the mirror across which they reflect.

Inverse Relationship

Criticality: 3

The fundamental connection between exponential and logarithmic functions, where one 'undoes' the other. This means their x and y values are swapped.

Example:

Since $2^3 = 8$ , the inverse relationship tells us that $log_2(8) = 3$ .

Logarithmic Function

Criticality: 3

A function of the form $f(x) = a \log_b(x)$, which is the inverse of an exponential function. It helps determine the exponent to which a base must be raised to get a certain number.

Example:

If you want to find out what power you need to raise 2 to get 8, you'd use a logarithmic function like $log_2(8) = 3$ .

Logarithmic Growth

Criticality: 2

A pattern where output values change additively as input values change multiplicatively, resulting in a slowly increasing curve.

Example:

The perceived loudness of sound, measured in decibels, follows a logarithmic growth pattern, meaning a large increase in sound energy is perceived as a smaller increase in loudness.

Ordered Pairs (swapping)

Criticality: 3

A key characteristic of inverse functions where if a point $(s, t)$ is on the original function, then the point $(t, s)$ is on its inverse function.

Example:

If the point $(2, 4)$ is on the graph of $y = 2^x$ , then by ordered pairs (swapping), the point $(4, 2)$ must be on the graph of $y = \log_2(x)$ .

Reflection (of graphs)

Criticality: 3

The transformation where the graph of an inverse function is obtained by mirroring the original function's graph across the line $y=x$.

Example:

The graph of $y = \log_5(x)$ is a reflection of the graph of $y = 5^x$ over the line $y=x$ .

Vertical Asymptote (for logarithmic functions)

Criticality: 2

A vertical line that the graph of a logarithmic function approaches but never touches. For $f(x) = a \log_b(x)$, it is typically the y-axis ($x=0$).

Example:

The graph of $y = \log(x)$ has a vertical asymptote at $x=0$ , meaning it gets infinitely close to the y-axis but never crosses it.