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Higher Unit 1

Higher Unit 1. What is a set. Recognising a Function in various formats. Composite Functions. Exponential and Log Graphs. Graph Transformations. Trig Graphs. Connection between Radians and degrees & Exact values. Solving Trig Equations. Basic Trig Identities. Exam Type Questions.

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Higher Unit 1

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  1. Higher Unit 1 What is a set Recognising a Function in various formats Composite Functions Exponential and Log Graphs Graph Transformations Trig Graphs Connection between Radians and degrees & Exact values Solving Trig Equations Basic Trig Identities Exam Type Questions www.mathsrevision.com

  2. Sets & Functions Notation & Terminology SETS: A set is a collection of items which have some common property. These items are called the members or elements of the set. Sets can be described or listed using “curly bracket” notation.

  3. Sets & Functions eg {colours in traffic lights} = {red, amber, green} DESCRIPTION LIST eg {square nos. less than 30} = { 0, 1, 4, 9, 16, 25} NB: Each of the above sets is finite because we can list every member

  4. Sets & Functions We can describe numbers by the following sets: = {1, 2, 3, 4, ……….} N = {natural numbers} W = {whole numbers} = {0, 1, 2, 3, ………..} Z = {integers} = {….-2, -1, 0, 1, 2, …..} Q = {rational numbers} This is the set of all numbers which can be written as fractions or ratios. eg 5 = 5/1 -7 = -7/1 0.6 = 6/10 = 3/5 55% = 55/100 = 11/20 etc

  5. Sets & Functions R = {real numbers} This is all possible numbers. If we plotted values on a number line then each of the previous sets would leave gaps but the set of real numbers would give us a solid line. We should also note that N “fits inside” W W “fits inside” Z Z “fits inside” Q Q “fits inside” R

  6. Sets & Functions N W Z Q R When one set can fit inside another we say that it is a subset of the other. The members of R which are not inside Q are called irrational numbers. These cannot be expressed as fractions and include  , 2, 35 etc

  7. Sets & Functions To show that a particular element/number belongs to a particular set we use the symbol . eg 3  W but 0.9  Z Examples { x  W: x < 5 } = { 0, 1, 2, 3, 4 } { x  Z: x  -6 } = { -6, -5, -4, -3, -2, …….. } { x  R: x2 = -4 } = { } or  This set has no elements and is called the empty set.

  8. Functions & Mappings Defn: A function or mapping is a relationship between two sets in which each member of the first set is connected to exactly one member in the second set. If the first set is A and the second B then we often writef: A  B The members of set A are usually referred to as the domain of the function (basically the starting values or even x-values) while the corresponding values or images come from set B and are called the range of the function (these are like y-values).

  9. Functions & Mapping Functions can be illustrated in three ways: 1) by a formula. 2) by arrow diagram. 3) by a graph (ie co-ordinate diagram). Example Suppose that f: A  B is defined by f(x) = x2 + 3x where A = { -3, -2, -1, 0, 1}. FORMULA then f(-3) = 0 , f(-2) = -2 , f(-1) = -2 , f(0) = 0 , f(1) = 4 NB: B = {-2, 0, 4} = the range!

  10. Functions & Mapping ARROW DIAGRAM A B f(x) f(-3) = 0 f(-2) = -2 f(-1) = -2 f(0) = 0 f(1) = 4 -3 0 -2 -2 -1 -2 0 0 0 1 4

  11. Functions & Graphs In a GRAPH we get : NB: This graph consists of 5 separate points. It is not a solid curve.

  12. Functions & Graphs Recognising Functions A B a b c d e f g A B e f g a bc d YES Not a function two arrows leaving b!

  13. Functions & Graphs A B Not a function - d unused! a b c d e f g A B a bc d e f g h YES

  14. Functions & Graphs Recognising Functions from Graphs If we have a function f: R  R (R - real nos.) then every vertical line we could draw would cut the graph exactly once! This basically means that every x-value has one, and only one, corresponding y-value!

  15. Function & Graphs Y Function !! x

  16. Function & Graphs Y Not a function !! Cuts graph more than once ! x must map to one value of y x

  17. Functions & Graphs Not a function !! Y Cuts graph more than once! X

  18. Functions & Graphs Y Function !! X

  19. Composite Functions COMPOSITION OF FUNCTIONS ( or functions of functions ) Suppose that f and g are functions where f:A  B and g:B  C with f(x) = y and g(y) = z where xA, yB and zC. Suppose that h is a third function where h:A  C with h(x) = z .

  20. Composite Functions ie A B C g f y z x h We can say that h(x) = g(f(x)) “function of a function”

  21. g(4)=42 + 1 =17 f(2)=3x2 – 2 =4 Composite Functions f(5)=5x3-2 =13 g(2)=22 + 1 =5 Example 1 f(1)=3x1- 2 =1 f(1)=3x1 - 2 =1 Suppose that f(x) = 3x - 2 and g(x) = x2 +1 (a) g( f(2) ) = g(4) = 17 g(26)=262 + 1 =677 g(5)=52 + 1 =26 (b) f( g (2) ) = f(5) = 13 (c) f( f(1) ) = f(1) = 1 (d) g( g(5) ) = g(26) = 677

  22. Composite Functions Suppose that f(x) = 3x - 2 and g(x) = x2 +1 Find formulae for (a) g(f(x)) (b) f(g(x)). (a) g(f(x)) = g(3x-2) = (3x-2)2 + 1 = 9x2 - 12x + 5 (b) f(g(x)) = f(x2 + 1) = 3(x2 + 1) - 2 = 3x2 + 1 NB: g(f(x))  f(g(x)) in general. CHECK g(f(2)) = 9 x 22 - 12 x 2 + 5 = 36 - 24 + 5 = 17 f(g(2)) = 3 x 22 + 1 = 13

  23. Composite Functions Let h(x) = x - 3 , g(x) = x2 + 4 and k(x) = g(h(x)). If k(x) = 8 then find the value(s) of x. k(x) = g(h(x)) Put x2 - 6x + 13 = 8 = g(x - 3) then x2 - 6x + 5 = 0 = (x - 3)2 + 4 or (x - 5)(x - 1) = 0 = x2 - 6x + 13 So x = 1 or x = 5 = 8 CHECK g(h(5)) = g(2) = 22 + 4

  24. Composite Functions Choosing a Suitable Domain (i) Suppose f(x) = 1 . x2 - 4 Clearly x2 - 4  0 So x2 4 So x  -2 or 2 Hence domain = {xR: x  -2 or 2 }

  25. Composite Functions x = 0 (0 + 4)(0 - 2) = negative (ii) Suppose that g(x) = (x2 + 2x - 8) x = 3 (3 + 4)(3 - 2) = positive We need (x2 + 2x - 8)  0 x = -5 (-5 + 4)(-5 - 2) = positive Suppose (x2 + 2x - 8) = 0 Then (x + 4)(x - 2) = 0 -4 2 So x = -4 or x = 2 Check values below -4 , between -4 and 2, then above 2 So domain = { xR: x  -4 or x  2 }

  26. Exponential (to the power of) Graphs Exponential Functions A function in the form f(x) = ax where a > 0, a ≠ 1 is called an exponential function to base a . Consider f(x) = 2x x -3 -2 -1 0 1 2 3 f(x) 1 1/8 ¼ ½ 1 2 4 8

  27. Graph The graph of y = 2x (1,2) (0,1) Major Points (i) y = 2x passes through the points (0,1) & (1,2) (ii) As x ∞ y ∞ however as x -∞ y 0 . (iii) The graph shows a GROWTH function.

  28. Log Graphs ie x 1/8 ¼ ½ 1 2 4 8 y -3 -2 -1 0 1 2 3 To obtain y from x we must ask the question “What power of 2 gives us…?” This is not practical to write in a formula so we say “the logarithm to base 2 of x” y = log2x or “log base 2 of x”

  29. Graph (2,1) (1,0) The graph of y = log2x NB: x > 0 Major Points (i) y = log2x passes through the points (1,0) & (2,1) . • As x ∞ y ∞but at a very slow rate and as x  0 y  -∞ .

  30. Exponential (to the power of) Graphs The graph of y = axalways passes through (0,1) & (1,a) It looks like .. Y y = ax (1,a) (0,1) x

  31. Log Graphs The graph of y = logaxalways passes through (1,0) & (a,1) Y It looks like .. (a,1) (1,0) x y = logax

  32. Graph Transformations • We will investigate f(x) graphs of the form • 1. f(x) ± k • f(x ±k) • -f(x) • f(-x) • kf(x) • f(kx) Each moves the Graph of f(x) in a certain way !

  33. Graph of f(x) ± k Transformations 3 y = f(x) y = x2 2 y = f(x) ± k y = x2-3 1 y = x2+ 1 Mathematically y = f(x) ± k moves f(x) up or down Depending on the value of k + k move up - k move down -2 -1 0 1 2 -1 -2 -3

  34. Mathematically y = f(x ± k) moves f(x) to the left or right depending on the value of k -k move right + k move left Graph of -f(x ± k) Transformations y = f(x) y = x2 y = f(x ± k) y = (x-1)2 y = (x+2)2 3 2 1 -4 -3 -2 -1 0 1 2 3

  35. Graph of -f(x) Transformations y = f(x) y = x2 y = -f(x) y = -x2 Mathematically y = –f(x) reflected f(x) in the x - axis

  36. Graph of -f(x) Transformations y = f(x) y = 2x + 3 y = -f(x) y = -(2x + 3) Mathematically y = –f(x) reflected f(x) in the x - axis

  37. Graph of -f(x) Transformations y = f(x) y = x3 y = -f(x) y = -x3 Mathematically y = –f(x) reflected f(x) in the x - axis

  38. Graph of f(-x) Transformations y = f(x) y = x + 2 y = f(-x) y = -x + 2 Mathematically y = f(-x) reflected f(x) in the y - axis

  39. Graph of f(-x) Transformations y = f(x) y = (x+2)2 y = f(-x) y = (-x+2)2 Mathematically y = f(-x) reflected f(x) in the y - axis

  40. Graph of k f(x) Transformations y = f(x) y = x2-1 3 y = k f(x) y = 4(x2-1) 2 y = 0.25(x2-1) 1 Mathematically y = k f(x) Multiply y coordinate by a factor of k k > 1  (stretch in y-axis direction) 0 < k < 1  (squash in y-axis direction) -2 -1 0 1 2 -1 -4

  41. Graph of -k f(x) Transformations y = f(x) y = x2-1 y = k f(x) y = -4(x2-1) 4 y = -0.25(x2-1) Mathematically y = -k f(x) k = -1 reflect graph in x-axis k < -1  reflect f(x) in x-axis & multiply by a factor k (stretch in y-axis direction) 0 < k < -1  reflect f(x) in x-axis multiply by a factor k (squash in y-axis direction) 1 -2 -1 0 1 2 -1 -4

  42. Graph of f(kx) Transformations 4 y = f(x) y = (x-2)2 3 y = f(kx) y = (2x-2)2 2 y = (0.5x-2)2 1 0 1 2 3 4 5 6 Mathematically y = f(kx) Multiply x – coordinates by 1/k k > 1  squashes by a factor of 1/k in the x-axis direction k < 1  stretches by a factor of 1/k in the x-axis direction

  43. Graph of f(-kx) Transformations 4 y = f(x) y = (x-2)2 3 2 y = f(-kx) y = (-2x-2)2 1 y = (-0.5x-2)2 -4 -3 -2 -1 0 1 2 3 4 Mathematically y = f(-kx) k = -1 reflect in y-axis k < -1  reflect & squashes by factor of 1/k in x direction -1 < k > 0  reflect & stretches factor of 1/k in x direction

  44. Trig Graphs The same transformation rules apply to the basic trig graphs. NB: If f(x) =sinx then 3f(x) = 3sinx and f(5x) = sin5x Think about sin replacing f ! Also if g(x) = cosx then g(x) – 4 = cosx – 4 and g(x + 90) = cos(x + 90)  Think about cos replacing g !

  45. Trig Graphs Sketch the graph of y = sinx - 2 If sinx = f(x) then sinx - 2 = f(x) - 2 So move the sinx graph 2 units down. 1 0 90o 180o 270o 360o -1 -2 y = sinx - 2 -3

  46. Trig Graphs Sketch the graph of y = cos(x - 50) If cosx = f(x) then cos(x - 50) = f(x - 50) So move the cosx graph 50 units right. 50o 1 0 90o 180o 270o 360o -1 -2 y = cos(x - 50)o -3

  47. Trig Graphs Sketch the graph of y = 3sinx If sinx = f(x) then 3sinx = 3f(x) So stretch the sinx graph 3 times vertically. 3 2 1 0 90o 180o 270o 360o -1 -2 y = 3sinx -3

  48. Trig Graphs Sketch the graph of y = cos4x If cosx = f(x) then cos4x = f(4x) So squash the cosx graph to 1/4 size horizontally 1 0 90o 180o 270o 360o -1 y = cos4x

  49. Trig Graphs Sketch the graph of y = 2sin3x If sinx = f(x) then 2sin3x = 2f(3x) So squash the sinx graph to 1/3 size horizontally and also double its height. 3 2 1 0 90o 180o 270o 360o -1 -2 y = 2sin3x -3

  50. Radians Radian measure is an alternative to degrees and is based upon the ratio of arc length radius a θ θ(radians) = r θ- theta

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