📐 Mutual Position of Graphs of Linear Functions
Взаимное расположение графиков линейных функций
Grade 7 · Algebra
§1. Introduction
In previous lessons, you learned that every linear function has a graph that is a straight line. But what happens when we have two or more linear functions? How do their graphs relate to each other on the coordinate plane?
Key Question: Given two straight lines on a coordinate plane, what are the possible ways they can be positioned relative to each other?
It turns out there are exactly three possibilities:
✖ Intersecting
Пересекающиеся прямые
Lines cross at exactly one point
∥ Parallel
Параллельные прямые
Lines have no common points
≡ Coinciding
Совпадающие прямые
Lines are identical — infinitely many common points
§2. Review: The Linear Function y = kx + b
Recall that a linear function has the form:
y = kx + b
k — Slope (Угловой коэффициент)
- Determines the steepness and direction of the line
- Shows how much y changes when x increases by 1
- Controls the angle the line makes with the x-axis
b — Y-intercept (Свободный член)
- The point where the line crosses the y-axis
- When x = 0, then y = b
- Shifts the line up or down
Remember: Each linear function y = kx + b has a unique graph — a straight line completely determined by its slope k and y-intercept b.