الصفوف 6–8Analysisالدرس متاح
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
يعتمد على
عدد المتطلبات السابقة الإلزامية (حتى الأساس): 8
ستتعلّم
- 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.
تعريف
- 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.
مبرهنة
- On a graph of distance from the start against time, the slope of a segment is the average velocity over that interval (negative when moving back towards the start); it equals the average speed when there is no turning back.
- 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.
مفاهيم خاطئة شائعة
- 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.
يقود إلى
اكتشافات اختيارية
- Introductory calculus Shrinking the interval turns an average rate of change into an instantaneous rate: the derivative.
قراءات إضافية
- Malcolm Swan and the Shell Centre for Mathematical Education, The Language of Functions and Graphs (1985)
- I. M. Gelfand, E. G. Glagoleva and E. E. Shnol, Functions and Graphs (1969)