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What is the gradient formula?
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All Flashcards in Topic 2.1
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2.1.110 cards
What is the gradient formula?
m = (y₂ − y₁)/(x₂ − x₁) = rise ÷ run. Subtract the coordinates in the same order top and bottom.
What does the sign of the gradient tell you?
m > 0 uphill, m < 0 downhill, m = 0 horizontal (y = c), vertical lines (x = a) have no gradient.
State the three forms of a straight line.
Gradient–intercept y = mx + c; point–gradient y − y₁ = m(x − x₁); general ax + by + d = 0.
In y = mx + c, what are m and c?
m is the gradient; c is the y-intercept (where the line crosses the y-axis).
How do you get the gradient from ax + by + d = 0?
Rearrange to y = mx + c — the gradient is m = −a/b.
How do you find a line from a gradient m and a point (x₁, y₁)?
Put m into y = mx + c, then substitute the point to find c. (Or use point–gradient form y − y₁ = m(x − x₁).)
How do you find a line through two points?
Find the gradient m = (y₂ − y₁)/(x₂ − x₁) first, then substitute one point into y = mx + c to find c.
How do you find the y-intercept of a line?
Set x = 0 (or, in y = mx + c, read off c).
How do you find the x-intercept of a line?
Set y = 0 and solve for x.
What are the equations of vertical and horizontal lines?
Vertical: x = a (gradient undefined). Horizontal: y = b (gradient 0).
2.1.22 cards
When are two lines parallel?
When they have the same gradient: m₁ = m₂ (with different y-intercepts).
How do you find a line through a point parallel to a given line?
Use the SAME gradient, put it into y = mx + c, then substitute the point to find c.
2.1.36 cards
When are two lines perpendicular?
When their gradients multiply to −1: m₁m₂ = −1, i.e. m₂ = −1/m₁.
How do you get the perpendicular gradient?
Take the negative reciprocal — flip the fraction and change the sign. E.g. ⅔ → −3/2.
Perpendicular gradient of 5?
Write 5 as 5/1; the perpendicular gradient is −1/5.
How do you find a line through a point perpendicular to a given line?
Use the negative-reciprocal gradient, put it into y = mx + c, then substitute the point to find c.
What is a normal to a curve?
The line perpendicular to the tangent at a point; its gradient is −1/(tangent gradient). Used in calculus.
Why doesn't m₁m₂ = −1 work for horizontal & vertical lines?
They are perpendicular, but a vertical line (x = a) has no gradient, so the product rule can't be applied — state it separately.
2.1.48 cards
What is a perpendicular bisector?
The line through the midpoint of a segment, perpendicular to it (negative-reciprocal gradient).
State the three steps to find a perpendicular bisector.
1) Midpoint of the endpoints. 2) Gradient of the segment, then its negative reciprocal. 3) Substitute the midpoint into y = mx + c.
Midpoint of (x₁, y₁) and (x₂, y₂)?
((x₁ + x₂)/2, (y₁ + y₂)/2) — average each coordinate.
Which gradient does the bisector use?
The negative reciprocal of the segment's gradient (flip the fraction and change the sign).
Which point does the bisector pass through?
The midpoint of the two endpoints — not either endpoint.
What is special about every point on the perpendicular bisector?
It is equidistant from the two endpoints (the same distance from A as from B).
Perpendicular bisector of A(1, 2) and B(5, 8)?
Midpoint (3, 5); m_AB = 3/2 → bisector gradient −2/3; y = −2/3 x + 7.
Bisector answer needs general form — what do you do?
Build y = mx + c first, then clear fractions and move everything to one side: ax + by + d = 0.
Topic 2.1 study notes
Full notes & explanations for Straight lines
Math AA exam skills
Paper structures, command terms & tips
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