Section 2.1: Graphing Linear Equations
A linear equation in two variables, \(x\) and \(y\), can be written in the form:
\[ y = mx + b \]
where:
- \( m \) is the slope (rate of change)
- \( b \) is the y-intercept (where the line crosses the y-axis)
Graphing a linear equation involves plotting points that satisfy the equation and drawing a straight line through them.
Two points are enough to graph a line, since a line is uniquely determined by two distinct points.
Example 1
Graph \( y = 2x + 1 \).
Step 1: Find two points. Let \( x = 0 \): \( y = 2(0) + 1 = 1 \) → Point (0,1) Let \( x = 2 \): \( y = 2(2) + 1 = 5 \) → Point (2,5)
Step 2: Plot the points and draw a straight line through them.
Example 2
Graph \( y = -x + 3 \).
Step 1: Pick points. \( x = 0 \): \( y = -0 + 3 = 3 \) → (0,3) \( x = 3 \): \( y = -3 + 3 = 0 \) → (3,0)
Step 2: Plot and connect the points with a straight line.
Practice Problems
- Graph \( y = x - 2 \).
- Graph \( y = 3x + 1 \).
- Graph \( y = -2x + 4 \).
- Find two points and graph \( y = \frac{1}{2}x - 1 \).
- Determine the slope and y-intercept of \( y = -3x + 5 \) and sketch its graph.