Gradient and y-intercept

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The gradient measures the steepness and direction of a line — how much y changes for every 1 unit increase in x. Positive gradient → rises left to right. Negative gradient → falls left to right. Zero gradient → horizontal line.

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Gradient = rise ÷ run = 8 ÷ 2 = 4. The line goes up by 4 for every 1 unit to the right. This is a positive, fairly steep gradient.

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The line falls steeply — for every 1 unit moved right, y drops by 5. Steepness = |−5| = 5 (compare using absolute value). The negative sign means it slopes downward from left to right.

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Wrong — steepness uses absolute value: |−4| = 4 > |3| = 3. The line with gradient −4 is steeper. Never compare signed gradient values to decide steepness — always compare |m₁| and |m₂|.

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m = (y₂ − y₁) / (x₂ − x₁) The y-change (rise) goes on top. The x-change (run) goes on the bottom. Use the same pair order for both: subtract in the same direction.

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m = (9 − 1) / (7 − 3) = 8 / 4 = 2. y increased and x increased → positive gradient makes sense. ✓

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m = (−1 − 5) / (4 − (−2)) = −6 / 6 = −1. Key step: 4 − (−2) = 4 + 2 = 6. Subtracting a negative flips the sign.

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They have swapped Δy and Δx. The gradient formula is m = Δy/Δx, not Δx/Δy. Fix: always write the formula first — m = (y₂ − y₁)/(x₂ − x₁) — before substituting numbers.

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The y-intercept is the point where the line crosses the y-axis — the value of y when x = 0. In y = mx + c, the y-intercept is c, the constant term. Example: y = 4x − 7 has y-intercept = −7, so it crosses at (0, −7).

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m is the gradient — it is the coefficient of x. c is the y-intercept — it is the constant term. Example: y = −2x + 9 → gradient = −2, y-intercept = 9.

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Gradient m = −3. y-intercept c = 7. Coordinates of y-intercept: (0, 7).

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The equation is not in y = mx + c order. Rewrite: y = −3x + 5. Gradient m = −3, y-intercept c = 5. Always rearrange into y = mx + c form before reading off m and c.

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Horizontal line: gradient = 0 (no rise — Δy = 0). Vertical line: gradient is undefined — Δx = 0, so we would divide by zero.

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Compare the absolute values of their gradients. The line with the larger |m| is steeper. Example: |−5| = 5 > |2| = 2, so y = −5x is steeper than y = 2x.

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y-intercepts: A → c = 1, B → c = −5. Line A crosses higher. Steepness: |−3| = 3 vs |4| = 4. Line B is steeper. Two different comparisons — do them separately.

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They dropped the negative sign. The gradient is m = −1/3 (negative, because it is − times 1/3). The y-intercept is 9. Read the coefficient of x including its sign.