MAT-144 · Mathematical Reasoning
Topic 01 · Linear Functions
Graphing a line
Plot a line from its slope-intercept equation. Find the intercept, then use slope to step out a second point.
01
Worked Example Video
Every step of Graphing a line explained on video. Pause anywhere; the embed scrolls independently of the page.
YOUTUBE
02
What This Looks Like on ALEKS
ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.
Graphing a line
Graph the line y = 2x + 3. What is the y-intercept as an ordered pair?
y-intercept =
✓
Right — b = 3, so the y-intercept is (0, 3). From there, use slope 2 (rise 2, run 1) to step to (1, 5).
✗
The y-intercept is at x = 0. In y = 2x + 3, b = 3, so the y-intercept is (0, 3).
Graphing a line — fractional slope
Graph the line y = (3/4)x − 2. What is the y-intercept as an ordered pair?
y-intercept =
✓
Right — b = −2, so the y-intercept is (0, −2). From there, slope 3/4 means rise 3, run 4 → second point (4, 1).
✗
The y-intercept is at x = 0. In y = (3/4)x − 2, b = −2, so the y-intercept is (0, −2).
Graphing a line — negative slope
Graph the line y = −x + 4. What is the y-intercept as an ordered pair?
y-intercept =
✓
Right — b = 4, so the y-intercept is (0, 4). Slope −1 means rise −1, run 1 → second point (1, 3).
✗
The y-intercept is at x = 0. In y = −x + 4, b = 4, so the y-intercept is (0, 4).
Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.
03
The Rule
Equation is y = mx + b
the y-intercept is (0, b)
Slope m = rise/run
from (0, b), go up rise, right run
Plot the y-intercept first — it’s always at x = 0. From there, use the slope as a rise-over-run recipe to step out a second point. Two points define the line. See it live: m = 3, b = −2 →
Common slip
Students swap rise and run. Slope 3/4 means rise = 3, run = 4 (numerator on top, denominator on bottom). A slope of just 2 is really 2/1 — rise 2, run 1.
04
Practice — Step by Step
Practice this problem step by step
Graph the line y = 3x − 2.
Find b (the y-intercept value)
In y = 3x − 2, what is b?
In y = mx + b, b is the number added (or subtracted) after mx. In y = 3x − 2, that’s the −2.
✓
Right — b = −2. The y-intercept is the point (0, −2).
✗
Not quite — try again.
Find m (the slope)
In y = 3x − 2, what is m? (Remember: 3 as a slope means 3/1 — rise 3, run 1.)
m is the coefficient of x — the number sitting in front of x.
✓
Right — m = 3, which means rise = 3 and run = 1.
✗
Not quite — try again.
Plot the line
Drag point A to the y-intercept (0, −2), then drag point B to a second point on y = 3x − 2. The line will follow. When both points sit on the line, click Check my graph.
Drag both points onto the correct positions.
Place point A at (0, −2), then place point B by going up 3, right 1 from A — that lands at (1, 1).
✓
Nailed it — both points land on y = 3x − 2. Two points, one line.
✗
Not quite. From the y-intercept (0, −2), slope 3 means up 3 and right 1 — landing at (1, 1). Try again.
🎉
You graphed the line end to end.
Same three moves every time: read b (the y-intercept), read m (rise/run), step out to the second point. Then draw the line through the two dots. ALEKS will use different values of m and b — but the move is identical.