AP Precalculus

~ 1.1-1.2: Rates of Change ~

1.1 Course Objectives

Describe how the input and output values of a function vary together by comparing function values.
Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.

A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value.
The set of input values is called the domain of the function, and the set of output values is called the range of the function.
The variable representing input values is called the independent variable, and the variable representing output values is called the dependent variable.
The input and output values of a function vary in tandem according to the function rule, which can be expressed graphically, numerically, analytically, or verbally.
A function is increasing over an interval of its domain if, as the input values increase, the output values always increase.
A function is decreasing over an interval of its domain if, as the input values increase, the output values always decrease.
The graph of a function displays a set of input-output pairs and shows how the values of the function’s input and output values vary.
A verbal description of the way aspects of phenomena change together can be the basis for constructing a graph.
The graph of a function is concave up on intervals in which the rate of change is increasing.
The graph of a function is concave down on intervals in which the rate of change is decreasing.
The graph intersects the x-axis when the output value is zero. The corresponding input values are said to be zeros of the function.

Compare the rates of change at two points using average rates of change near the points.
Describe how two quantities vary together at different points and over different intervals of a function.

The average rate of change of a function over an interval of the function’s domain is the constant rate of change that yields the same change in the output values as the function yielded on that interval of the function’s domain.
The average rate of change is the ratio of the change in the output values to the change in input values over that interval.
The rate of change of a function at a point quantifies the rate at which output values would change were the input values to change at that point.
The rate of change at a point can be approximated by the average rates of change of the function over small intervals containing the point, if such values exist.
The rates of change at two points can be compared using average rate of change approximations over sufficiently small intervals containing each point, if such values exist.
Rates of change quantify how two quantities vary together.
A positive rate of change indicates that as one quantity increases or decreases, the other quantity does the same.
A negative rate of change indicates that as one quantity increases, the other decreases.

The video for AP Precalculus, 1.1-1.2: Rates of Change will come out on Thursday, September 7, 2023.