3-1 Graphing Linear Functions

Property

An ordered pair $(x, y)$ is plotted by moving horizontally first (along the $x$-axis) and then vertically (along the $y$-axis). The order matters: $(x, y) \neq (y, x)$ unless $x = y$.

Property

To graph a linear equation by plotting points:
Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

Examples

• To graph $y = x + 3$, find three solution points like $(0, 3)$, $(1, 4)$, and $(-1, 2)$. Plot these points and draw a line through them.
• To graph $y = \frac{1}{2}x - 1$, choose multiples of 2 for $x$ to avoid fractions. Good choices would be $(0, -1)$, $(2, 0)$, and $(4, 1)$.
• For $3x + y = 4$, first rewrite it as $y = -3x + 4$. Then find points by choosing values for $x$, such as $(0, 4)$, $(1, 1)$, and $(2, -2)$.

Explanation

This method is like an algebraic connect-the-dots. Find at least three coordinate pairs that solve the equation, place them on the graph, and draw a straight line through them. Using three points helps you catch any calculation mistakes.

Property

To graph a line using intercepts: find the x-intercept by setting $y = 0$, find the y-intercept by setting $x = 0$, plot both intercept points, then draw a straight line through them.

Examples