Home›Stage 5: Essential Lines, Graphs and Slope›Finding Slope
Stage 5 · Essential Lines, Graphs and Slope
Finding Slope
Slope measures how steeply a line rises or falls. On the GED you'll find slope from a graph, from a table of values, and from two coordinate points — this lesson covers all three.
Slope measures how steep a line is — specifically, how much the line goes up or down for every step it moves to the right. That's the key idea: slope is always a rate — vertical change per horizontal change.
Before any numbers, just look at direction. A line going uphill has a positive slope. A line going downhill has a negative slope. A flat horizontal line has a slope of zero.
Positive slope (+)
Goes uphill left to right. y increases as x increases. The steeper the climb, the larger the number.
Negative slope (−)
Goes downhill left to right. y decreases as x increases. The steeper the drop, the larger the negative number.
Slope is a rate of change. A slope of 3 means "for every 1 step right, the line goes up 3." A slope of −½ means "for every 2 steps right, the line goes down 1." This is the same idea as unit rate — just applied to a graph.
Part 1 — Slope from a Graph
Always work left to right. Start at the left point and move to the right point. Count vertical change first, then horizontal change. This consistent routine prevents the most common sign errors.
1
Circle the left point. This is always your starting point.
2
Count the vertical change first. From the left point, move straight up or down to reach the same height as the right point. Going up? That's the rise — write it as a positive number. Going down? That's the drop — write it as a negative number.
3
Count the run second — move right to reach the right point. Always right, always positive.
4
Write the vertical change over the run. Rise over run for uphill. Drop over run for downhill. Simplify if needed.
Rise means going up. Drop means going down. If you're counting upward, you have a rise — write it as positive. If you're counting downward, you have a drop — write it as negative. The word you use tells you the sign automatically.
Worked Example — Positive Slope (uphill) — Rise over Run
The line goes uphill — so we'll have a rise. Start at the left point.
→ Circle the LEFT point: (−3, −2). This is where we start.
→ Count UP to reach the height of the right point (3, 2) — 4 squares up. Rise = +4.
→ Count RIGHT to reach the right point — 6 squares right. Run = +6.
→ Write rise over run: +4 ÷ +6 = 4/6. Simplify by dividing both by 2.
m = 2/3 (positive — line goes uphill ✓)
Worked Example — Negative Slope (downhill) — Drop over Run
The line goes downhill — so we'll have a drop, not a rise. Start at the left point.
→ Circle the LEFT point: (−3, 2). This is where we start.
→ Count DOWN to reach the height of the right point (3, −2) — 4 squares down. That's a drop of 4. Write it as −4.
→ Count RIGHT to reach the right point — 6 squares right. Run = +6.
→ Write drop over run: −4 ÷ +6 = −4/6. Simplify by dividing both by 2.
m = −2/3 (negative — line goes downhill ✓)
Don't reverse vertical and horizontal. Always count vertical first (rise or drop), then horizontal (run). The vertical change goes on top of the fraction, run goes on the bottom. Switching them gives you the wrong answer every time.
Part 2 — Slope from a Table
A table works the same way as a graph — slope is still the change in y divided by the change in x. Pick two rows and measure how much each column changed. Always subtract in the same order: second row minus first row for both y and x.
1
Pick two consecutive rows — the top row is row 1, the next one down is row 2.
2
Find the change in y — subtract: row 2 y minus row 1 y. Write the sign. Going up = positive, going down = negative.
3
Find the change in x — subtract: row 2 x minus row 1 x. x almost always increases, so this is usually positive.
4
Slope = (change in y) ÷ (change in x). The sign of the y change determines whether the slope is positive or negative.
Worked Example — Slope from a Table
For every 2 steps x increases, y increases by 3. That's the rate of change. The curved arrows show what we're counting.
→ Use rows 1 and 2 (highlighted). Row 2 y − Row 1 y = 5 − 2 = +3. Change in y = +3 (y went up).
→ Row 2 x − Row 1 x = 2 − 0 = +2. Change in x = +2 (x went right).
→ Slope = (change in y) ÷ (change in x) = 3 ÷ 2.
m = 3/2 — for every 2 right, y goes up 3.
The sign lives in the change in y. If y is going down, the change in y is negative and the slope is negative. You don't need to look at the whole table — just watch the direction y moves between any two rows.
Part 3 — Slope from Two Points
When you have two coordinate pairs and no graph or table, use the slope formula. This formula is on your GED formula sheet — you don't need to memorize it. It's the same idea as before: change in y on top, change in x on bottom.
m = y₂ − y₁x₂ − x₁
This is just (change in y) ÷ (change in x) written formally. y₂ − y₁ is the change in y. x₂ − x₁ is the change in x. The subscripts mean "second point minus first point" — always in that order for both.
1
Label the pairs — write them on your paper. You'll be looking at a screen, so copy the two coordinate pairs down and write (x₁, y₁) under the first and (x₂, y₂) under the second. Do this every time.
2
Write out the subtractions before calculating. y₂ − y₁ on top. x₂ − x₁ on bottom. Don't do it in your head.
3
Watch for subtracting a negative. If y₁ = −3, then y₂ − y₁ = y₂ − (−3) = y₂ + 3. Write it out explicitly.
4
Divide and simplify.
Worked Example — No Negative Coordinates
Find the slope of the line through (2, 5) and (6, 1).
This is on a screen — copy the two points onto paper before you start.
Step 1 — Label the pairs on your paper
Step 2 — Write the subtraction as a fraction (this is also how you enter it in the GED calculator)
Worked Example — Negative Coordinates
Find the slope of the line through (−2, 4) and (2, −4).
Copy the two points onto paper. Negative signs are easy to drop — writing it out protects you.
Step 1 — Label the pairs on your paper
Step 2 — Write the subtraction as a fraction
Subtracting a negative is the #1 sign error. When you see x₂ − x₁ and x₁ is negative — say x₁ = −3 — write it out: x₂ − (−3) = x₂ + 3. Never skip this step in your head.
🎯 All three methods — same idea
→Graph: Circle the left point. Count vertical change (rise if up, drop if down), then run (right). Divide.
→Table: Pick two rows. Change in y = row 2 y − row 1 y. Change in x = row 2 x − row 1 x. Divide.
→Two points: Label point 1 (smaller x) and point 2. m = (y₂ − y₁) ÷ (x₂ − x₁).
→In every case — change in y on top, change in x on bottom, sign matters.