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Higher Maths. Revision Notes. Get Started. Functions and Graphs. goodbye. Functions and graphs You should know the meaning of the terms domain and range of a function;. Recognise the probable form of a function from its graph. f : x → sin ( ax + b ), - PowerPoint PPT Presentation

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  • Get StartedgoodbyeRevision NotesFunctions and GraphsHigher Maths

  • Functions and graphs You should know the meaning of the terms domain and range of a function;f : x sin (ax + b), f : x cos (ax + b)f: x ax (a > 1 and 0 < a < 1, x R) f: x logax (a > 1, x > 0)Recognise the probable form of a function from its graphGiven the graph of f(x) draw the graphs of related functions, where f(x) is a simple polynomial or trigonometric function Polynomial functions functions with restricted domainInverse of a functionComposite functionComplete the square.Radian measure.

  • In both graphsa = 360 wavelength. the number of waves in 360.b = shift to the left a.Test Yourself?

  • Exponential functionsLogarithmic functionsNote:An exponential function is the inverse of the corresponding logarithmic function.When a = e = 271828the function is called the exponential function.Test Yourself?

  • Polynomial functionsIn general a polynomial of order n will have at most n real roots and at most (n 1) stationary points.

    e.g. a cubic can have, at most, 3 real roots and 2 s.p.sIf a cubic has 3 real roots a, b and c then its equation will be of the form y = k(x a)(x b)(x c) where k is a constant.If a, b and c are known then k can be calculated if one point the curve passes through is also known.When the brackets are expanded the cubic will have the form y = px3 + qx2 + rx + s, where p, q, r, and s are constants and p 0

    Similar statements can be made of the other polynomials.Test Yourself?Constant: y = alinear: y = ax + bquadratic: y = ax2 + bx + ccubic: y = ax3 + bx2 + cx + dquartic: y = ax4 + bx3 + cx2 + dx + e

  • Reciprocal functionsRestricted domainThe denominator can never equal zero.So a value of x which makes this happen is not in the domain.For exampleSquare root functionThe term within the radical sign must always be 0So any value of x which makes this negative is not in the domain.For exampley = tan(x) then x 90, 270 or 90 + 180n where n is an integer.

    y = log(x) then x > 0

    Others

    y = sin1(x)y = cos1(x) then 1 x > 1

    Test Yourself?

  • inverseIf f(g(x)) = x and g(f(x)) = x for all x in the domain then we say that f is the inverse of g and vice versa.

    The inverse of f is denoted by f1.Examples [over suitable domains]f(x) = x2 f1(x)= x f(x) = sin(x) f1(x) = sin1(x) f(x) = 2x + 3 f1(x) = (x 3)/2 f(x) = loga(x) f1(x) = ax f(x) = ex f1(x) = ln(x)For the Higher exam you need not know how to find the formula for the inverse of any function.composites

  • Composite functionsExampleSuppose we have two functions: f(x) = 3x + 4 and g(x) = 2x2 + 1.We can use these definitions to create new functions:1f(f(x))= f(3x + 4)= 3(3x + 4) + 4= 9x + 162g(g(x))= g(2x2 + 1)= 2(2x2 + 1)2 + 1= 8x4 + 8x2 + 33f(g(x))= f(2x2 + 1)= 3(2x2 + 1) + 4= 6x2 + 74g(f(x))= g(3x + 4)= 2(3x + 4)2 + 4= 18x2 + 48x + 36Things to note:A composition can be made from more than two functionsConsidering examples 1 and 2 leads to recurrence relations e.g. f(f(f(f(x)))))In general f(g(x)) g(f(x)) the order in which you do things are important.If either f or g have restrictions on their domain, this will affect the domain of the composite function.If f(g(x)) = x for all x in the domain then we say that f is the inverse of g it can be denoted by g1Test Yourself?

  • Related functionsy = f(x) + ay = f(x + a)y = af(x)y = f(ax)y = f(x)y = f(x)y = f1(x)y = f (x)[The inverse][The derivative][x-translation of a][y-translation of a][reflection in x-axis][reflection in y-axis][stretch in y-direction][squash in x-direction]Test Yourself?

  • Completing the squareWe can use this identity to simplify quadratic expressions.Example 1Express x2 + 6x + 1 in the form (x + a)2 + b

    Given x2 + 6x + 1 By inspection a = 6 2 = 3So x2 + 6x + 1 = (x + 3)2 32 + 1= (x + 3)2 8

    Note: a = 3 and b = 8Example 2(a)Express 3x2 + 12x + 1 in the form a(x + b)2 c(b)Find the smallest value the expression can take.

    (a)Given 3x2 + 12x + 1, Take 3 out as a common factor leaving the coefficient of x2 as 1So 3(x2 + 4x) +1 focus on the red text.

    By inspection a = 4 2 = 2So we get 3(x2 + 4x) +1 = 3[(x + 2)2 22] + 1= 3(x + 2)2 12 + 1= 3(x +2)2 11

    The smallest a perfect square can be is zero.So the smallest the expression can be is 0 11 = 11.Ths happens when x = 2.Test Yourself?

  • radiansWe can measure angle size using the degree (90 - 1 right angle)We can measure angle size using the grad (100 grads - 1 right angle)1 radianRRRMathematicians find it convenient to use the radian.(/2 radians = 1 right angle.degree 180radiandegree 180radianThe values are often given in terms of . Test Yourself?

  • [x-translation of a]y = f(x + a)1

  • y = f(x) + a[y-translation of a]5

  • y = f(x)[reflection in x-axis]

  • y = f(x)[reflection in y-axis]

  • y = f (x)[The derivative]++++++++Gradient of the function is shown in redStationary points of the function correspond to zeros of the derived function.Positive gradients of the function correspond to Parts below the axis on the derived function.Negatve gradients of the function correspond to Parts above the axis on the derived function.

  • y = f1(x)[The inverse]When a function has an inverse then, if (x, y) lies on the graph of the function,(y, x) lies on the graph of the inverse function. one is the reflection of the other in the line y = x.Note that the example function does not have an inverse. The reflection in y = x has, for example, 3 values corresponding to x = 5.

  • y = f(ax)[squash in x-direction]

  • y = af(x)[stretch in y-direction]

  • Using suitable units, the distance of the tip of the rotorto the tail of a helicopter can be calculated using a formula of the formD = 3sin(ax + b) + 10. The graph is shown below.

    What are the values of a and breveal

  • Using suitable units, the distance of the tip of the rotorto the tail of a helicopter can be calculated using a formula of the formD = 3sin(ax + b) + 10. The graph is shown below.

    What are the values of a and bNote:The 10 translates the sine wave 10 units up.The 3 stretches the wave by a factor of 3 in the y-direction.The wavelength, by inspection, is 120.a = 360 120 = 3

    One would expect the first peak of y = sin(3x) to occur at 90 3 = 30.It occurs at 15. Thus the shift to the left is 30 15 = 15.So b = shift a = 15 3 = 45.

    The equation is:D = 3sin(3x + 45) + 10

  • The profile of the Eiffel tower can be modelled by the formula y = a ln(bx) where a and b are constants and y m is the height of a spot on the profile and x is its distance measured horizontally from the centre.When y = 0, x = 63. When y = 42, x = 40.Find the values of a and b.reveal

  • The profile of the Eiffel tower can be modelled by the formula y = a ln(bx) where a and b are constants and y m is the height of a spot on the profile and x is its distance measured horizontally from the centre.When y = 0, x = 63. When y = 42, x = 40.Find the values of a and b.When y = 0, x = 63

    y = a ln(bx) 0 = a ln (63b) ln(63b) = 0 63b = 1 b = 1/63

    When y = 42, x = 40 y = a ln(bx) 42 = a ln (40 63) 42 = a 0454255 (calculator)a = 92 (to nearest whole number)

    Eiffel tower can be modelled by

    y = 92 ln(x/63)

  • As can be seen in the graph, there is a simple cubic function which for 1 > x > 1, and working in radians, behaves almost the same as the sine wave. i.e.sin(x) px3 + qx2 + rx + s where p, q, r, and s are constants.The roots of this cubic are 6 and 0. Express the cubic in terms of its factors viz. k(x a)(x b)(x c).We know sin(/6) = 1/2. Use this to find k as a simple fraction with a unit numerator.It fits where it touches.reveal

  • As can be seen in the graph, there is a simple cubic function which for 1 > x > 1, and working in radians, behaves almost the same as the sine wave. i.e.sin(x) px3 + qx2 + rx + s where p, q, r, and s are constants.The roots of this cubic are 6 and 0. Express the cubic in terms of its factors viz. k(x a)(x b)(x c).We know sin(/6) = 1/2. Use this to find k as a simple fraction with a unit numerator.It fits where it touches.(a)(b)

  • A function is defined by f: x (x2 x 2)

    Find the largest possible domain for the function.reveal

  • A function is defined by f: x (x2 x 2)

    Find the largest possible domain for the function.The function within the radical sign must be greater than or equal to zero. The sketch of this quadratic tells us that x 2 or x 1.

    The sketch on the left shows the function in question.

  • f(x) = 2x 1 and g(x) = x2 + 2.

    Find an expression for f(g(x).In general f(g(x)) g(f(x)).However, in this case there are two values of x for which f(g(x)) = g(f(x)).Find these values.reveal

  • f(x) = 2x 1 and g(x) = x2 + 2.

    Find an expression for f(g(x).In general f(g(x)) g(f(x)),however, in this case there are two values of x for which f(g(x)) = g(f(x)).Find these values.f(g(x) = f(x2 + 2)= 2(x2 + 2) 1= 2x2 + 3

    g(f(x)) = g(2x 1)= (2x 1)2 + 2= 4x2 4x + 3 g(f(x)) = f(g(x) 4x2 4x + 3 = 2x2 + 32x2 4x = 0x(x 2) = 0x = 0 or x = 2

  • revealThe sketch shows part of the function y = f(x)

    Draw a sketch of (i) y = f(x) (ii) y = f(1 x)Make a sketch of y = f(x)(075, 1)

  • The sketch shows part of the function y = f(x)

    Draw a sketch of (i) y = f(x) (ii) y = f(1 x)Make a sketch of y = f(x)(075, 1)(a (i))(a (ii))(b)

  • Q1Find the maximum value of the function defined by:Q2Prove that y = 3x3 + 3x2 + 5x + 1 is an increasing function.Where completing the square is usefulreveal