Glossary

Additive Transformation

Criticality: 2

A transformation of a function $f(x)$ by adding a constant 'k' to its output, resulting in a new function $g(x) = f(x) + k$.

Example:

Changing $f(x) = 2^x$ to $g(x) = 2^x + 7$ is an additive transformation that shifts the graph upwards by 7 units.

Base (b)

Criticality: 3

In an exponential function $f(x) = ab^x$, 'b' is a positive constant (not equal to 1) that determines the rate at which the function grows or decays.

Example:

For an investment that doubles every year, the base of its exponential function would be 2, indicating a 100% growth rate.

Concave Down (exponential function)

Criticality: 2

The characteristic shape of an exponential function where the graph opens downwards, which the provided notes state occurs when the base 'b' is between 0 and 1.

Example:

While standard exponential functions with a positive initial value are always concave up, a function like $g(x) = -2(0.5)^x$ would be concave down, curving downwards as x increases.

Concave Up (exponential function)

Criticality: 2

The characteristic shape of an exponential function where the graph opens upwards, always occurring for standard exponential functions $f(x) = ab^x$ when 'a' > 0.

Example:

The graph of $y = 3^x$ always curves upwards, illustrating that it is concave up across its entire domain.

Decreasing (exponential function)

Criticality: 2

An exponential function is considered decreasing if its base 'b' is between 0 and 1, meaning its y-values continuously fall as x-values increase.

Example:

The function $A(t) = 100 \cdot (0.95)^t$ representing the remaining amount of a substance after decay is a decreasing exponential function.

Domain (of exponential function)

Criticality: 2

The set of all possible input values (x-values) for which an exponential function is defined, which for standard exponential functions is always all real numbers.

Example:

For any exponential function like $f(x) = 7^x$ , you can substitute any real number for x, meaning its domain is (-∞, ∞).

Exponential Decay

Criticality: 3

A characteristic of an exponential function $f(x) = ab^x$ where the base 'b' is between 0 and 1, causing the function's output to decrease rapidly as the input 'x' increases.

Example:

The reduction of a medication's concentration in the bloodstream over time often follows a pattern of exponential decay.

Exponential Function

Criticality: 3

A function of the form $f(x) = ab^x$, where 'a' is the initial value, 'b' is the base (a positive number not equal to 1), and the variable 'x' is in the exponent.

Example:

The population growth of a city modeled by $P(t) = 10000 \cdot (1.03)^t$ is an example of an exponential function, showing how the population changes over time.

Exponential Growth

Criticality: 3

A characteristic of an exponential function $f(x) = ab^x$ where the base 'b' is greater than 1, causing the function's output to increase rapidly as the input 'x' increases.

Example:

The spread of a highly contagious virus can be modeled by exponential growth, where the number of infected individuals increases at an accelerating rate.

Extrema (absence on open interval)

Criticality: 2

Maximum or minimum values of a function. Exponential functions do not have extrema on an open interval because they are continuously increasing or decreasing without bound in one direction and approaching an asymptote in the other.

Example:

The function $f(x) = 5^x$ will never reach a highest or lowest point across its entire domain, meaning it has no global extrema.

Increasing (exponential function)

Criticality: 2

An exponential function is considered increasing if its base 'b' is greater than 1, meaning its y-values continuously rise as x-values increase.

Example:

A function modeling the value of a rare collectible that appreciates over time, like $V(t) = 500 \cdot (1.1)^t$ , is always increasing.

Inflection Points (absence)

Criticality: 2

Points on a graph where the concavity changes. Exponential functions do not have inflection points because they maintain the same concavity (either concave up or concave down) throughout their entire domain.

Example:

Unlike a sine wave, the graph of $y = 4^x$ never changes its curvature, thus it has no inflection points.

Initial Value (a)

Criticality: 3

In an exponential function $f(x) = ab^x$, 'a' represents the y-intercept of the graph and the starting quantity or amount when the input variable 'x' is zero.

Example:

If a bacterial culture starts with 50 cells, the initial value in its growth model $N(t) = 50 \cdot b^t$ would be 50.

Limits (as x approaches infinity/negative infinity)

Criticality: 3

Describes the behavior of a function's output as its input approaches positive or negative infinity, indicating whether the function grows without bound, decreases without bound, or approaches a specific value (like zero).

Example:

For the exponential decay function $f(x) = 10(0.6)^x$ , the limit as x approaches infinity is 0, meaning the graph gets infinitely close to the x-axis.

Vertical Shifts

Criticality: 3

A type of additive transformation where the graph of a function is moved up or down by a constant amount 'k', affecting its y-intercept and horizontal asymptote.

Example:

If the base cost of a service is modeled by an exponential function, adding a fixed monthly fee would result in a vertical shift of the cost graph.