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Straight Line y = mx + c

Terminology Median – midpoint Bisector – midpoint Perpendicular – Right Angled Altitude – right angled. Distance between 2 points. Form for finding line equation y – b = m(x - a). m > 0. m 1 .m 2 = -1. m < 0. Possible values for gradient. Straight Line y = mx + c. m = 0.

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Straight Line y = mx + c

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  1. Terminology Median – midpoint Bisector – midpoint Perpendicular – Right Angled Altitude – right angled Distance between 2 points Form for finding line equation y – b = m(x - a) m > 0 m1.m2 = -1 m < 0 Possible values for gradient Straight Line y = mx + c m = 0 Parallel lines have same gradient (a,b) = point on line m = gradient c = y intercept (0,c) m = undefined For Perpendicular lines the following is true. m1.m2 = -1 m = tan θ θ

  2. f(x) f(x) f(x) + f(x) flip in y-axis Move vertically up or downs depending on k - Stretch or compress vertically depending on k y = f(x) ± k f(x) f(x) y = f(-x) Remember we can combine function !! y = kf(x) Graphs & Functions y = -f(x) y = f(kx) y = f(x ± k) Stretch or compress horizontally depending on k flip in x-axis - + Move horizontally left or right depending on k

  3. g(f(x)) g(x) = g(x) = But y = f(x) is x2 - 4 f(x) = x2 - 4 g(f(x)) = x y = f(x) Restriction Domain Range x2 - 4 ≠ 0 A complex function made up of 2 or more simpler functions Similar to composite Area (x – 2)(x + 2) ≠ 0 1 1 1 1 1 1 1 x ≠ 2 x ≠ -2 Composite Functions x2 - 4 y x x x x = + But y = g(x) is f(g(x)) f(x) = x2 - 4 f(g(x)) = x2 2 - 4 x y = g(x) y2 - 4 Rearranging - 4 x2 ≠ 0 Restriction Domain Range

  4. Format for Differentiation / Integration Surds Indices Basics before Integration Division Working with fractions Adding Subtracting Multiplication

  5. Nature Table Equation of tangent line Leibniz Notation x -1 2 5 - + f’(x) 0 Straight Line Theory Max Gradient at a point f’(x)=0 Stationary Pts Max. / Mini Pts Inflection Pt Graphs f’(x)=0 Derivative = gradient = rate of change Differentiation of Polynomials f(x) = axn then f’x) = anxn-1

  6. Completing the square f(x) = a(x + b)2 + c Easy to graph functions & graphs -2 1 4 5 2 -2 -4 -2 Factor Theorem x = a is a factor of f(x) if f(a) = 0 1 2 1 0 (x+2) is a factor since no remainder f(x) =2x2 + 4x + 3 f(x) =2(x + 1)2 - 2 + 3 f(x) =2(x + 1)2 + 1 If finding coefficients Sim. Equations Discriminant of a quadratic is b2 -4ac Polynomials Functions of the type f(x) = 3x4 + 2x3 + 2x +x + 5 Degree of a polynomial = highest power Tangency b2 -4ac > 0 Real and distinct roots b2 -4ac < 0 No real roots b2 -4ac = 0 Equal roots

  7. Limit L is equal to Given three value in a sequence e.g. U10 ,U11 , U12 we can work out recurrence relation U11 = aU10 + b U12 = aU11 + b Use Sim. Equations b L = a = sets limit b = moves limit Un = no effect on limit (1 - a) Recurrence Relations next number depends on the previous number Un+1 = aUn + b |a| > 1 |a| < 1 a > 1 then growth a < 1 then decay Limit exists when |a| < 1 + b = increase - b = decrease

  8. Remember to change sign to + if area is below axis. Remember to work out separately the area above and below the x-axis . f(x) g(x) b A= ∫ f(x) - g(x) dx Integration is the process of finding the AREA under a curve and the x-axis a Area between 2 curves Finding where curve and line intersect f(x)=g(x) gives the limits a and b Integration of Polynomials IF f’(x) = axn Then I = f(x) =

  9. Special case

  10. Double Angle Formulae sin2A = 2sinAcosA cos2A = 2cos2A - 1 = 1 - 2sin2A = cos2A – sin2A Addition Formulae sin(A ± B) = sinAcosB cosAsinB cos(A ± B) = cosAcosB sinAsinB 90o A S 180o 0o C T 270o The exact value of sinx Trig Formulae and Trig equations xo 4 4 3cos2x – 5cosx – 2 = 0 2 Let p = cosx 3p2 – 5p - 2 = 0 (3p + 1)(p -2) = 0 sinx = 2sin(x/2)cos(x/2) cosx = 2 p = cosx = 1/3 sinx = 2 (¼ + √(42 - 12) ) x = no soln x = cos-1( 1/3) sinx = ½ + 2√15) x = 109.5o and 250.5o

  11. Rearrange into sin = • Find solution in Basic Quads • Remember Multiple solutions Exact Value Table 90o S A 180o 0o C T 270o degrees radians 3 Basic Strategy for Solving Trig Equations then Xπ ÷180 2 Trigonometry sin, cos , tan 1 1 1 Amplitude 0 Complex Graph 0 0 Period 90o 180o 360o 270o 360o -1 360o sin x Basic Graphs -1 Amplitude -1 Amplitude Period 1 ÷π then x180 0 Period y = 2sin(4x + 45o) + 1 90o 180o Period -1 Max. Value =2+1= 3 Period = 360 ÷4 = 90o cos x tan x Amplitude = 2 Mini. Value = -2+1 -1

  12. same for subtraction Addition 2 vectors perpendicular if Scalar product Component form Magnitude Basic properties scalar product Vector Theory Magnitude & Direction Q Notation B P a Component form A Vectors are equal if they have the same magnitude & direction Unit vector form

  13. b Tail to tail θ a Angle between two vectors properties Vector Theory Magnitude & Direction Section formula C B n C c A B Points A, B and C are said to be Collinear if B is a point in common. m b A a O

  14. Trig. Harder functions Use Chain Rule Rules of Indices Polynomials Differentiations Factorisation Graphs Real life Stationary Pts Mini / Max Pts Inflection Pts Meaning Rate of change of a function. Straight line Theory Gradient at a point. Tangent equation

  15. y = logax To undo log take exponential y = ax y To undo exponential take log y (a,1) loga1 = 0 a0 = 1 (1,a) (0,1) logaa = 1 x (1,0) a1 = a x Basic log graph Basic exponential graph log A + log B = log AB A log A - log B = log B log (A)n = n log A Logs & Exponentials Basic log rules y = axb Can be transformed into a graph of the form y = abx Can be transformed into a graph of the form log y log y log y = x log b + log a log y = b log x + log a (0,C) (0,C) Y = mX + C Y = mX + C log x x Y = (log b) X + C Y = bX + C C = log a m = log b C = log a m = b

  16. f(x) = a sinx + b cosx Compare coefficients compare to required trigonometric identities Square and add then square root gives a = k cos β b = k sin β f(x) = k sin(x + β) = k sinx cos β + k cosx sin β Process example Divide and inverse tan gives Wave Function a and b values decide which quadrant transforms f(x)= a sinx + b cosx into the form Write out required form OR Related topic Solving trig equations

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