91Ó°ÊÓ

Intercepts play a crucial role when graphing linear equations, as they provide key reference points for constructing the graph. To understand intercepts, let's break down the concept into two parts:

• Y-Intercept: This is the point where the graph intersects the y-axis. To find the y-intercept of a linear equation such as \( y = 200 - 4x \), set \( x \) to zero. The resulting y-value will be the y-intercept. For our example:
  \( y = 200 - 4(0) \) simplifies to \( y = 200 \), so the y-intercept is (0, 200).


• X-Intercept: This is the point where the graph crosses the x-axis. To find it, set \( y \) to zero and solve the equation for \( x \). For the equation \( y = 200 - 4x \), setting \( y \) to zero gives:
  \( 0 = 200 - 4x \), solve for \( x \): \( x = 50 \). Hence, the x-intercept is (50, 0).

These intercepts are plotted on the coordinate plane to help visualize the linear function.

The coordinate plane is a standard number grid that helps visualize mathematical concepts such as equations and geometrical shapes. It consists of two axes intersecting at right angles: the x-axis (horizontal) and the y-axis (vertical). Here’s how the coordinate plane assists with graphing linear functions:

• Every point on the plane is defined by a pair of numbers, known as coordinates, in the form \((x, y)\).
• The intersection of the axes is called the origin, represented by the point \((0, 0)\).


• When graphing the linear equation \( y = 200 - 4x \), the y-intercept (0, 200) and x-intercept (50, 0) are initially plotted on this plane.

By effectively using the coordinate plane, you can plot these points and then use them to accurately draw the graph of the equation.

A linear function is an algebraic equation that forms a straight line when graphed on a coordinate plane. Linear functions are expressed in the form of \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating its steepness and direction.
• \( b \) is the y-intercept, representing where the line crosses the y-axis.

For the equation \( y = 200 - 4x \), the slope \( m \) is \(-4\), meaning the line slopes downwards from left to right.
Linear functions are essential in understanding relationships between two variables as they provide a consistent rate of change. The graph of a linear function, identifiable by its straight-line shape, shows how one variable affects another in a constant manner.
By connecting the intercepts, you form the complete linear graph, demonstrating how the variables interact according to the linear function's equation.