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All 16 Flashcards — Gradient and y-intercept
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Question
What does the gradient of a straight line measure?
Answer
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.
Question
A line goes up 8 units for every 2 units moved to the right. What is the gradient?
Answer
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.
Question
What does a gradient of −5 tell you about the line?
Answer
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.
Question
Exam trap: Lines have gradients −4 and 3. A student says gradient 3 is steeper because 3 > −4. Correct this.
Answer
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|.
Question
State the formula for gradient between two points (x₁, y₁) and (x₂, y₂).
Answer
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.
Question
Find the gradient of the line through (3, 1) and (7, 9).
Answer
m = (9 − 1) / (7 − 3) = 8 / 4 = 2. y increased and x increased → positive gradient makes sense. ✓
Question
Find the gradient of the line through (−2, 5) and (4, −1).
Answer
m = (−1 − 5) / (4 − (−2)) = −6 / 6 = −1. Key step: 4 − (−2) = 4 + 2 = 6. Subtracting a negative flips the sign.
Question
Exam trap: A student writes m = (x₂ − x₁)/(y₂ − y₁). What is the error and how do you avoid it?
Answer
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.
Question
What is the y-intercept of a straight line?
Answer
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).
Question
In y = mx + c, which letter is the gradient and which is the y-intercept?
Answer
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.
Question
State the gradient and y-intercept of y = −3x + 7. Then write down the coordinates of the y-intercept.
Answer
Gradient m = −3. y-intercept c = 7. Coordinates of y-intercept: (0, 7).
Question
Exam trap: A student reads y = 5 − 3x and writes gradient = 5, y-intercept = −3. What went wrong?
Answer
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.
Question
What is the gradient of a horizontal line? What about a vertical line?
Answer
Horizontal line: gradient = 0 (no rise — Δy = 0). Vertical line: gradient is undefined — Δx = 0, so we would divide by zero.
Question
How do you decide which of two lines is steeper?
Answer
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.
Question
Line A: y = −3x + 1. Line B: y = 4x − 5. Which crosses the y-axis higher? Which is steeper?
Answer
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.
Question
Exam trap: A student has y = −(1/3)x + 9. They write gradient = 1/3. What is wrong?
Answer
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.
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