六至八年级Analysis计划中
Graphs of change
Reading and sketching graphs that show how one quantity changes with another, such as distance against time, and interpreting steepness as rate of change.
math.analysis.graphs-of-change
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前置知识
8 项必需的前置知识（一直追溯到底）
你将学会
- Interpret distance–time and other real-world graphs, describing where a quantity increases, decreases or stays constant.
- Compute average rates of change between two points of a graph or table.
- Sketch a graph from a verbal description, and describe a situation from a graph.
定义
- The average rate of change of y between x = a and x = b is (y(b) − y(a))/(b − a).
- A graph is increasing on an interval if y gets larger whenever x gets larger within that interval.
定理
- On a distance–time graph, the slope of a segment is the average speed over that time interval.
- A horizontal segment means the quantity is not changing.
- Two situations can have the same average rate over an interval but behave differently within it.
常见误解
- Reading a distance–time graph as a picture of the path travelled, such as a hill.
- Thinking a steeper graph means a larger quantity rather than a faster change.
通向
可选发现
- Introductory calculus Shrinking the interval turns an average rate of change into an instantaneous rate: the derivative.