AP Precalculus: Unit 2A

Competing Function Model Validation

A process to determine which mathematical model (linear, quadratic, exponential) best represents a given data set.

Exponential Model

A mathematical representation where one variable changes at a constant percentage rate relative to another variable; change multiplies rather than adds. Growth or Decay accelerates.

Residual Plot

Graph that helps you judge whether a model (linear, quadratic, exponential) is a good fit for a set of data.

Error in the Model

Predicted values minus actual values in data analysis.

Model Validation

The process of confirming that a constructed model accurately reflects the data set it is based on.

Linear Function Model

A model that displays a constant rate of change between two variables (same amount each time).

Quadratic Function Model

A model that represents a relationship where the rate is changing at a constant rate.

Data Set

A collection of related data points used for analysis or modeling.

Underestimate / Overestimate

The concepts referring to the prediction being lower/higher than the actual value within a specific interval.

Regression Lines

Lines or Curves that fit a set of data.

Residual Value

Actual measured data point minus predicted value

Good regression model

Residual plot shows no clear pattern

Bad regression model

Residual plot shows some type of pattern such as a curve or a wave.

Exponential Functions

Functions that model growth patterns where successive output values over equal-length input-value intervals are proportional.

Proportional Growth Pattern

A situation where the output values increase or decrease in proportion to their input values, often modeled by exponential functions. A constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern.

Natural Base (e)

The constant approximately equal to 2.718, often used as the base in exponential functions.

Growth Factor (b)

In the exponential function f(x) = ab^x, the base b represents the growth factor related to percent change in context.

Exponential Regression

A method of constructing an exponential function model for a data set using technology.

Transformation of Functions

The process of modifying the function f(x) = ab^x based on characteristics of a contextual scenario or data set.

Exponential Model

A mathematical model represented by the function f(x) = ab^x, used for analyzing data sets or predicting values.

Percent Increase

Start with the initial amount “a” and grow with a % increase:

y=a(1 + % increase)

Percent Decrease

Start with the initial amount “a” and decay with a % decrease:

y=a(1 - % decrease)

Half Life

An initial amount a that shrinks by half every h: f(t) = a(1/2)^t/h

Doubling Time

An initial amount a that doubles every d: 

f(t) = a(2)^t/d 

Equivalent Forms

Reveal different properties of the function

sequence

An ordred list of numbers that could be finite or infinite. Each listed number is a term. A graph of a sequence contains discrete points, not a connected line or curve.

arithmetic sequence

A sequence in which each successive term has a common difference (or a constant rate of change).

equation for the nth term of an arithmetic sequence

a_n = a₀ + dn_, where a₀ is the initial value (zero term) and d is the common difference.

equation for an arithmetic sequence using ANY term (kth term)

a_n = a_k + d(n-k), where a_k is the kth term of the sequence.

geometric sequence

A sequence in which each successive term has a common ratio (or constant proportional change).

equation for the nth term of a geometric sequence

g_n = g₀ r^n, where g₀ is the initial value (zero term) and r is the common ratio

equation for an geometric sequence using ANY term (kth term)

g_n = g_kr^(n-k), where g_k is the kth term of the sequence.

Difference in growth between arithmetic and geometric sequences

Increasing arithmetic sequences increase equally with each step, while increasing geometric sequences increase by a larger amount with each successive step.

Similarities of linear functions f(x) = b+mx and arithmetic sequences a_n = a₀ + d_n

Both can be expressed as an initial value (b or a₀) plus repeated addition of a constant rate of change, the slope (m or d).

Similarities of linear functions and arithmetic sequences when expressed using a kth term / point? 

1. Arithmetic sequences for the kth term use a_n = a_k + d(n-k), which are based on a known difference (d), and a kth term.

2. Linear functions can be expressed in the form f (x) = y₁ + m(x-x_1), based on a known slope (m), and a point, (x₁, y₁).  Equation of a line = y-y₁ = m (x-x₁)

3. The point (x₁, y₁) for a linear function is similar to the term (k, a_k) of an arithmetic sequence.

Similarities of exponential functions f(x) = ab^x and geometric sequences g_n = g₀rⁿ

Both can be expressed as an initial value (a or g₀) times repeated multiplication by a constant proportion (b or r).

Similarities of exponential functions and geometric sequences when expressed using a kth term / point

1. Geometric sequences use the equation: g_n = g_kr^(n-k), which are based on a known ratio ( r ) and a kth term.

2. The shifted exponential function can be expressed using the equation: f(x) = y₁r^x-x₁ based on a known ratio( r ) and a point (x₁ y₁).

3. The point (x₁, y₁) for the exponential funtion is similar to the term (k, g_k)

True or False? The domain of a sequence is always the same as the domain of its corresponding function.

False; Sequences and their corresponding functions may have different domains.

Linear functions and exponential functions can be expressed analytically in terms of an initial value and a constant involved in change. There is a difference between the two though, what is that difference?

Linear functions are based on addition and exponential functions are based on multiplication.

Over equal-length input-value intervals, if the output values of a function change at constant rate, the function is _______.

Linear (Adding the slope)

Over equal-length input-value intervals, if the output values of a function change proportionally, the function is _______.

Exponential (Multiplying the ratio)

If you know a function is linear or exponential (for sequences that would be arithmetic or geometric), what do you need to come up with an equation (rule) for the function or sequence?

Two distinct sequence or function values

Product Property for Exponents

States that b^mbⁿ = b^m+n  

Power Property for Exponents

States that (b^m)ⁿ = b^mn .

Negative Exponent Property

States that b^-n = (1/b)ⁿ

Horizontal Translation

Every translation of an exponential function of the form f(x) = b^(x+k) is equivalent to a vertical dilation:

f(x) = b^(x+k) = b^xb^k = ab^x, where a = b^k.

kth Root of b

b^(1/k) is the kth root of b, where k is a natural number and the root exists

Horizontal Dilation

Every horizontal dilation of an exponential function, f(x) = b^(cx) is equivalent to a change of the base of an exponential function:

f(x) = (b^c)^x, where b^c is a constant and c ≠ 0.   

Exponential Function

A function in the form f(x) = ab^x, where a > 0 and b > 0.

Exponential Growth

When a > 0 and b > 1 in an exponential function.

Exponential Decay

When a > 0 and 0 < b < 1 in an exponential function.

Domain of Exponential Functions

The domain of an exponential function is all real numbers.

Output Values of Exponential Functions

These are proportional over equal-length input-value intervals.

Concavity of Exponential Functions

The graphs of exponential functions are always concave up (growth) or concave down (decay).

Extrema in Exponential Functions

Exponential functions do not have extrema except on a closed interval.

Points of Inflection in Exponential Functions

The graphs of exponential functions do not have points of inflection; they are always concave up or concave down.

Limit of Exponential Functions as x approaches infinity

As x increases, lim (x→∞) ab^x will increase or decrease without bound.

Limit of Exponential Functions as x approaches negative infinity

As x decreases, lim (x→−∞) ab^x approaches 0.

Additive Transformation of an exponential function

g(x) = f(x)+k is a vertical shift of the graph up or down, depending on the value of k. If the output values of g are proportional over equal-length input-values, then f is exponential.