Graphing Linear Functions & Inequalities
Linear functions are the backbone of Algebra 2, and graphing them from is quick once you trust the slope and intercept. Checking whether a point lies on a line is even faster — just substitute.
Inequalities add one step: graph the boundary line, then shade the half-plane of points that make the inequality true. This lesson keeps both moves tight.
Graphing from slope-intercept form
In , plot the -intercept first, then use the slope as rise over run to step to a second point. Connect them.
To test whether a point is on the line, substitute its coordinates. If the equation holds true, the point lies on the graph.
Shading an inequality
Graph the boundary as if it were an equation. A strict inequality ( or ) uses a dashed line; or uses a solid line.
Then pick a test point not on the line — often — and shade the side that makes the inequality true.
The graph below shows : a solid boundary (the line is included) with the entire region above it shaded, since those points make the inequality true.
Worked examples
Example 1: is a point on the line
Which point lies on : or ?
Answer:
Example 2: choosing the boundary style
Should use a solid or dashed boundary line?
Answer: Dashed
Example 3: choosing which side to shade
Which side of the boundary line do you shade for ?
Answer: The side above the line
Try one yourself
Common questions
How do I check if a point is on a line?
Substitute the point's x into the equation. If the result equals the point's y, it is on the line; otherwise it is not.
When is the boundary line solid versus dashed?
Solid for or (the line is included), dashed for or (the line is excluded).
How do I know which side to shade?
Test a point not on the line, like . If it satisfies the inequality, shade its side; if not, shade the other.
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