Quadratics
𝑏 𝑏
● Completing the square: (general formula) 𝑥 2 + 𝑏𝑥 = (𝑥 + 2)2 − (2)2
𝑏 2 𝑏2
○ Or more generally: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑎(𝑥 + ) + (𝑐 − )
2𝑎 4𝑎
Graphs
Cubic graphs (greatest power x3)
● https://www.desmos.com/calculator/gwczj0eqba
−𝑏 ± √𝑏2 − 3𝑎𝑐
● For a cubic function ax3 + bx2 + cx + d, 𝑥𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 3𝑎
When discriminant of xcritical = b2 - 3ac < 0, function has no stationary points, m > 0 for
ALL x ∈ ℝ (function is continuously increasing)
𝑑𝑦 −𝑏 ± √𝑏2 − 3𝑎𝑐
○ 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑, = 3𝑎𝑥 2 + 2𝑏𝑥 + 𝑐 ≥ 0, 𝑥 = //
𝑑𝑥 3𝑎
Quartic graphs (`` x4)
𝑑𝑦
● 𝑦 = 𝑎𝑥 4 + 𝑏𝑥 3 + 𝑐𝑥 2 + 𝑑𝑥 + 𝑒, = 4𝑎𝑥 3 + 3𝑏𝑥 2 + 2𝑐𝑥 + 𝑑 ≥ 0, 𝑥𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =
𝑑𝑥
−𝑐 ± √𝑐 2 − 3𝑏𝑑
3𝑏
(real solutions, when a = 0, b ≠ 0)
Quintic graphs (`` x5)
𝑑𝑦
● 𝑦 = 𝑎𝑥 5 + 𝑏𝑥 4 + 𝑐𝑥 3 + 𝑑𝑥 2 + 𝑒𝑥 + 𝑓, 𝑑𝑥 = 5𝑎𝑥 4 + 4𝑏𝑥 3 + 3𝑐𝑥 2 + 2𝑑𝑥 + 𝑒 ≥ 0,
𝑥𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = . ..
https://www.wolframalpha.com/input?i=solve+5ax%5E4%2B4bx%5E3%2B3cx%5E2
%2B2dx%2Be%3D0+for+x
1
,Reciprocal graphs
●
Graph transformations
3 )
● Translation, eg translation ( −2 (3 units right on x-axis, 2 units down on y-axis)
(f(x) = x2, f(x - 3) - 2 = (x - 3)2 - 2)
2
, ● Reflection, eg reflection through y-axis
(f(x) = x2, -f(x) = -x2)
● Stretch, eg stretch x direction s.f. 3
● Rotation (¿swap x & y in an equation)
3
, ●
○ af(x) for a < 0: reflection over x-axis as well
○ Eg f(x) = x2, 4f(x) = f(2x), 9f(x) = f(3x) etc.
○ Eg f(x) = x3, 8f(x) = f(2x), 27f(x) = f(3x) etc. etc..
●
Straight lines graphs
● y = mx + c
𝑦 −𝑦
○ m (gradient): 𝑥2 − 𝑥1
2 1
○ c (y-intercept)
■ To find x/y-intercepts, subs y/x = 0 into known equation
● y - y1 = m(x - x1): new equation formed with known gradient & 1 known point/2
known points (gradient ALSO found with 2 known points)
𝑥1 + 𝑥2 𝑦1 + 𝑦2
● Midpoint of (x1, y1), (x2, y2): ( 2
, 2 )
Circle theorem
● Equation of circle with centre (a, b) and radius r: (x - a)2 + (y - b)2 = r2
● In order to prove intersection, subs linear equation into circle equation, then find
discriminant...
● To prove a known point on whether it lies inside/on/outside the circle:
○ subs x & y values into circle equation
4
𝑏 𝑏
● Completing the square: (general formula) 𝑥 2 + 𝑏𝑥 = (𝑥 + 2)2 − (2)2
𝑏 2 𝑏2
○ Or more generally: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑎(𝑥 + ) + (𝑐 − )
2𝑎 4𝑎
Graphs
Cubic graphs (greatest power x3)
● https://www.desmos.com/calculator/gwczj0eqba
−𝑏 ± √𝑏2 − 3𝑎𝑐
● For a cubic function ax3 + bx2 + cx + d, 𝑥𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 3𝑎
When discriminant of xcritical = b2 - 3ac < 0, function has no stationary points, m > 0 for
ALL x ∈ ℝ (function is continuously increasing)
𝑑𝑦 −𝑏 ± √𝑏2 − 3𝑎𝑐
○ 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑, = 3𝑎𝑥 2 + 2𝑏𝑥 + 𝑐 ≥ 0, 𝑥 = //
𝑑𝑥 3𝑎
Quartic graphs (`` x4)
𝑑𝑦
● 𝑦 = 𝑎𝑥 4 + 𝑏𝑥 3 + 𝑐𝑥 2 + 𝑑𝑥 + 𝑒, = 4𝑎𝑥 3 + 3𝑏𝑥 2 + 2𝑐𝑥 + 𝑑 ≥ 0, 𝑥𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =
𝑑𝑥
−𝑐 ± √𝑐 2 − 3𝑏𝑑
3𝑏
(real solutions, when a = 0, b ≠ 0)
Quintic graphs (`` x5)
𝑑𝑦
● 𝑦 = 𝑎𝑥 5 + 𝑏𝑥 4 + 𝑐𝑥 3 + 𝑑𝑥 2 + 𝑒𝑥 + 𝑓, 𝑑𝑥 = 5𝑎𝑥 4 + 4𝑏𝑥 3 + 3𝑐𝑥 2 + 2𝑑𝑥 + 𝑒 ≥ 0,
𝑥𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = . ..
https://www.wolframalpha.com/input?i=solve+5ax%5E4%2B4bx%5E3%2B3cx%5E2
%2B2dx%2Be%3D0+for+x
1
,Reciprocal graphs
●
Graph transformations
3 )
● Translation, eg translation ( −2 (3 units right on x-axis, 2 units down on y-axis)
(f(x) = x2, f(x - 3) - 2 = (x - 3)2 - 2)
2
, ● Reflection, eg reflection through y-axis
(f(x) = x2, -f(x) = -x2)
● Stretch, eg stretch x direction s.f. 3
● Rotation (¿swap x & y in an equation)
3
, ●
○ af(x) for a < 0: reflection over x-axis as well
○ Eg f(x) = x2, 4f(x) = f(2x), 9f(x) = f(3x) etc.
○ Eg f(x) = x3, 8f(x) = f(2x), 27f(x) = f(3x) etc. etc..
●
Straight lines graphs
● y = mx + c
𝑦 −𝑦
○ m (gradient): 𝑥2 − 𝑥1
2 1
○ c (y-intercept)
■ To find x/y-intercepts, subs y/x = 0 into known equation
● y - y1 = m(x - x1): new equation formed with known gradient & 1 known point/2
known points (gradient ALSO found with 2 known points)
𝑥1 + 𝑥2 𝑦1 + 𝑦2
● Midpoint of (x1, y1), (x2, y2): ( 2
, 2 )
Circle theorem
● Equation of circle with centre (a, b) and radius r: (x - a)2 + (y - b)2 = r2
● In order to prove intersection, subs linear equation into circle equation, then find
discriminant...
● To prove a known point on whether it lies inside/on/outside the circle:
○ subs x & y values into circle equation
4
