STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs
Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic Functions... 9 Hbrid Functions... 11 Absolute Value Functions... 13 Inverse Functions... 15 Graphs and Transformations... 17
FUNCTIONS AND RELATIONS Relations A relation is a set of ordered pairs. (1, ), (, 6), (3, ), (, ), (,) are ordered pairs and {(1, ), (, 6)}, {(, 6), (3, ), (,)} and {(1, ), (, 6), (3, ), (, ), (,) } are relations. The domain of a relation is the set of first elements or the -values of the ordered pairs. For the relation {(1, ), (, 6), (3, ), (, ), (,)} domain D = {1,, 3, }. The range of a relation is the set of second elements or the -values of the ordered pairs. For the above ordered pairs the range R = {, 6,, }. A rule that links the domain and range ma be used to define the relation: S, :, R The set of ordered pairs the rule joining the domain and range. an restrictions placed on the domain or range. This relation, called S, includes all ordered pairs that have greater than. (1,), (1,100), and ( 1,0) are three of the ordered pairs in this relation. The domain of S is D = R and the range of S will also be R. Functions A function is a special tpe of relation. Each point in the domain of a function has a unique value in the range; each value of ma have onl one value of. The relation {( 1, ), ( 1, ), (1, 6), (, 8), (3, 10)} is not a function because there are two distinct ordered pairs with = 1. The relation {( 1,1), (0,), (1,3), (,5), (3,5)} is a function because each value has onl one corresponding value. NB: If we have the graph of a relation it will be a function if an vertical line passes through onl one point. Most functions are defined b a rule that links the and values, is a function of, or = f(). The ordered pairs (0, 3), (, 5), (, 1) and man more belong to the function {(,): = f() = + 3}. The graph of a function is wa to show all the ordered pairs that belong to the function. 1
Eamples 1. State the domain and range: {(,): = }. Is this relation a function?. The graph of + = is shown. State the domain and range of this relation. Is the relation a function? Eercises 0 15 10 5 - - - - - - State the domain and range of the following relations, and whether each is also a function. 1. {(1, 0), (0,1)}. {(3, ) (5, 5), (1, 5), (3, 9), (5, 10)} 3. {(, ):, 0}. {(, ): = sin() } 1-1 - -3 6 8 10 Because can be an real number and we can see from the graph that it etends indefinitel to the left and right: D = R We can also see from the graph that the smallest value of will be 0 but the graph etends indefinitel upwards: R {: 0} A vertical line drawn anwhere will pass through onl one point of the graph so the relation is a function. D = {: - } R : = {: - } The relation is not a function because there are man places where a vertical line would pass through two points eg: (0, ) and (0, -) both belong to the relation. 5. {(, ): = e } 6. {(, ): + 9, 0} 1 0.5-6 -0.5-1 Answers 1. D = {0, 1} R = {0, 1}, function. D = {1, 3, 5} R = {, 5, 9, 10}, not a function 3. D {: 0} R = {: }, function. D = R R = {: -1 }, function 5. D = R R = {: >0}, function 6. D = {: 0 } R = { : 3 }, not a function
INTERVAL NOTATION Often the domain of a function will be restricted to a subset of R, with specific endpoints. This subset is called an interval. Closed intervals If the end points are included it is called a closed interval. When representing a closed interval on a real number line solid circles are used for the endpoints to indicate the are included: a and [, ] are other was to denote the same interval. b Open intervals If the endpoints are not included, the interval is called an open interval. When representing an open interval on a real number line open circles are used for the endpoints to indicate the are not included: a < < b and (a, b) are other was to denote the same interval Eamples a b Interval Notation [, ) Inequalit Notation Line Graph a b [, ) a (, ) NB: In interval notation the smaller number is alwas written to the left i.e. [, ) (, ] The smbol (infinit) is not a numeral. is the concept of continuing indefinitel to the right and is the concept of continuing indefinitel to the left. Hence we cannot write [, ], [, ] or, etc. Eamples 1. Write the following in inequalit notation and graph on a real number line. (a) [, ) (b) (, ] Inequalit notation Inequalit notation b Graph Graph - 3 3 3
Multiple intervals Two (or more) subsets of R, with end points a and b, and c and d, respectivel, can also be represented on a real number line. Eample 5 1 This is written in interval notation as (, ] (, ]. Eercises 1. Write the following inequalities in interval notation and graph on a real number line. (a) 0 (b) (c). Use interval notation and inequalit notation to represent the following intervals. (a) (b) -5 3. Graph the following on a real number line and write in inequalit notation. -3 0 (a) (, ) (, ] (b) [, ] [,9] (c) (, ] (, ) Answers 1. (a) [1, 10 ] 1 10 (b) [ 6, ) - 6 - (c) (5, ) 5. (a) (, 5 ], 5 (b) ( 3, 0 ), 3 < < 0 3. (a) < < 3 with 8 < 13-3 8 13 (b) 1 with 6 9-1 6 9 (c) - < 3 with 6 < < 3 6
GRAPHS: LINEAR FUNCTIONS Cartesian Plane The Cartesian plane is defined b a pair of mutuall perpendicular coordinate aes or number lines. The horizontal ais is the ais and the vertical ais is the ais. Their point of intersection is called the origin. Points are defined b coordinate pairs (, ). Coordinate pairs are ordered pairs: (3,1) is not the same point as (1,3) Linear Graphs If the rule that generates a function can be arranged in the form = m + c then the function is a linear function. The graph of a linear function is a straight line. Ever point on the line represents an ordered pair (,) that satisf the rule for the function. Just two of these ordered pairs are needed to graph the function. The intercepts of a line are the points (0,) and (,0) where the line crosses the aes. These points are useful for graphing a linear function. The graph of the line represented b = is shown together with the intercepts. Eamples 1. Sketch the graph of the function {(,): = 3 + 6} We need to find an two points on the line i.e. points that satisf the rule = 3 + 6. The - and -intercepts are usuall the easiest: -intercept: = 0 = 3(0) + 6 = 6 -intercept: = 0 0 = 3 + 6 = The line passes through the points (0, 6) and (, 0) 5
. Sketch the graph of {(, ): 3 = 0} -intercept: = 0 3(0) = 0 = 0 = 0. The point (0,0) is on the line. But the intercept is also the -intercept so a different value is chosen to find a second point = 3() = 0 6 = 0 = 6 = The point (, ) is on the line. Eercise Sketch the graphs of the following linear functions. (a) {(,): = +3}. (b) {(,): 3 + = 9} (c) {(,): + = 0} Answers (a) (b) (c) 6
LINES AND GRADIENTS The gradient of a line gives a measure of its slope, usuall denoted m. This value ma be positive: or negative: The gradient can be found: directl from the graph ( ) b using the formula where (, ) and (, ) are two points on the line. b rearranging the equation of the line to the form = m +c, then m is the gradient. The graph of the line = + is shown. We can see = = OR rise The points (,) and (-,0) are on the line: m m 0 = = run NB: It doesn t matter which of the points are chosen as (1, 1) and (, ) OR + and m Eamples 1. Find the gradient of the line joining the two points (1,3) and (,5). = = NB: when a line is in the form m c, c is the - intercept. Find the gradient of the line Because the equation is in the form = m + c we can determine b inspection that 3. Find the gradient of the line shown in the graph The gradient of the line will be negative ( ) Drawing a rise and run triangle on the diagram enables us to see = = 7
Parallel and perpendicular lines Parallel lines have the same gradient. Perpendicular lines have gradients that are negative reciprocals. = 3 1 and = 3 + 3 are parallel lines. = 3 1 and = + 7 are perpendicular lines. m 1 = m m 1 or m 1 m = 1 Horizontal lines have m = 0 and an equation of the form = c. Vertical lines have a gradient which is undefined and an equation of the form = k. Eample Draw the graph of the line that passes through (1,1) and is perpendicular to the line. Therefore the gradient of the new line is. Equation of new line is of the form = + c. Substitute (1,1) to find c: 1 = (1) + c c = 3 Equation of line is = + 3 The line can be plotted using intercepts or b using = and the point (1,1). Eercise 1. Find the equations of the lines shown in the graphs (a) (b) (c). Graph the line that passes through the point (1,1) and is parallel to the line = + 7. Answers 1(a) = 3 (b) = + 5 (c) = 0.5 +. 8
GRAPHS: QUADRATIC FUNCTIONS A quadratic function has the form f() = a + b + c, a 0. The graph of a quadratic function is called a parabola. Sketching a parabola Find the orientation a < 0, a > 0. the turning point or verte -intercept [put = 0] -intercepts ( if an) [put = 0, then solve the quadratic equation] A parabola is smmetrical about a vertical line through the verte. Turning point The -coordinate of the verte is given b =. To find the co-ordinate substitute the value of the - co-ordinate in the equation. Eample: Find the turning point of the parabola = 3. First rearrange to the form f() = a + b + c: = + 3 a = 1, b=, c= 3 = = = 1 is the -coordinate of the turning point = 3 ( 1) ( 1) = The turning point is ( 1, ) If the quadratic function can be rearranged to the form = a( h) + k b completing the square Then (h, k) is the turning point. Eg: = 6 + 13 = ( 3) +. The turning point of the parabola is (3,) Intercepts The intercept is the value of c in the equation = a + b + c. The intercepts can be found b solving the equation a + b + c = 0 9
Eamples 1. Sketch = + 3 a = 1, b = 0, c = 3 Orientation: a > 0 the parabola opens upwards Turning point: = + 3 = ( + 0) + 3 (0, 3) is the turning point. -intercept: c = 3 -intercepts: + 3 = 0 has no solutions so there are no -intercepts. Sketch = + 8 a = 1, b =, c = 8 Turning point: = = = 1 X = 1 ( 1) ( ) 9 The turning point is ( 1, 9) -intercept: c = 8 -intercepts: + 8 = 0 ( + )( ) = 0 or = NB: Because of smmetr (, 8) is also a point on the graph Eamples 1. State the turning point of the following graphs: (a) (b) (c) ( ) (d) ( ). Sketch the graphs of: (a) (b) ( ) (c) Answers 1. (a) (1, 5) (b) (0,3) (c) (6,0) (d) ( 5, 3). (a) (b) (c) 10
HYBRID FUNCTIONS Functions with a restricted domain Functions which have different rules for each subset of the domain are called hbrid functions. Sometimes the are referred to as piecewise defined functions. Consider the following functions and their graphs noting the restricted domains: =, 1 = 1,, - - - - - - - - - - - - Functions defined in pieces These pieces can be put together to form the hbrid function, = f() = {,, and the graph of = f(): - - - - Eample Draw a sketch graph of = f() = {, 0, 0 - - - - 11
Eercise Draw a sketch graph of, 0 1. f() = {, 0. f() = { 3. f() = { 0,,, 0, 0,. f() = {,,, Answers 1.. - - - - - - - - 3.. - - - - - - - - 1
ABSOLUTE VALUE FUNCTIONS The absolute value of a number gives a measure of its size or magnitude regardless of whether it is positive or negative. If a number is plotted on a number line then its absolute value can be considered to be the distance from zero. Eamples (i) = (ii) = (iii) = = 1 (iv) = 8 + 1 = 9 (v) = 7 = 7 or = 7 Graph of the absolute value function The absolute value function is a hbrid function defined as follows:, 0 f:r R where f() = = {, 0 with graph NB: The domain of f() = is R and the range is R + {0} - - - - The graph of = ma be translated in the same wa as the graphs of other functions. Compare the graphs of the following functions with that of = : 1. =. = + 1 3. = 3 = + 3 = translated horizontall = translated verticall = reflected in the -ais units to the right one unit up followed b a vertical shift of 3 units To sketch the graph of = ( ) we need to sketch the graph of = f() first and then reflect in the -ais the portion of the graph which is below the -ais. Eample Sketch { (,): = } The graph of this function is the graph of = 1 with the portion below the -ais reflected across the -ais. - - - - 13
Equations and inequalities involving ( ) Because = ( ) is a hbrid function two cases must be considered when solving equations and inequalities. For inequalities it is helpful to know that: < a a < < a > a > a or < a Eamples 1. Solve = 3 = 3 = 3 or ( ) = 3 ie = 5 or = 1 With an absolute value epression on each side of the equation it is easier to square both sides:. Solve = = (+1) = ( 5) + + 1 = 10 + 5 3 + 1 = 0 (3 )( + 6) = 0 = or = 6 NB: Care must be taken when multipling or dividing b a negative to reverse an inequalit. 3. Solve < < < 1 1 < < 1 1 < < 10 1 > > 10 or 10 < < 1 Eercise 1. Evaluate (a) (b) 9 (c) (d) (e) 0. Sketch the graph of (a) = (b) = (c) = 3. Find for R (a) { : } (b) { : } Answers 1. a) 11 b) 5 c) 1 d) 9 e) 6. (a) (b) (c) - - - - - - - - - - - - 3. (a) { 6,6} (b) {: 1} {: 5} 1
INVERSE FUNCTIONS Definition of an inverse function If () is the inverse function of a one-to-one function f () then () is the set of ordered pairs obtained b interchanging the first and second elements in each ordered pair. So if (a,b) f then (b,a) f 1 and if f (a) = b then f 1 (b) = a The domain of f is the range of f 1 and the range of f is the domain of f 1. For eample, the function f :R R defined b = f () = has an inverse function with rule () = + 1. So (3,1) belongs to f and (1,3) belongs to f 1, and ( 7, ) belongs to f and (, 7) belongs to f 1. Graph of an inverse function The graphs of an one-to-one function f and its inverse f 1 are smmetric about the line =. Finding an inverse function for = f() To obtain the rule for an inverse function swap the and coordinates in f and rearrange to epress in terms of : Eample Find the inverse function of f where f ( )= 3 = 3 = 3 [swap and ] = 3 + = 3 = () = [rearrange to make the subject] 15
Eercise Find the inverse of each of the following one-to-one functions: 1) = + 5 ) = 3) = ) =, ½ Answers 1) () = 5 ) () = 3) () = ) () =, 0 16
GRAPHS AND TRANSFORMATIONS The known graphs of some simple functions and relations can be used to sketch related, but more complicated functions. Some graphs that would be useful to remember are: a b c -intercepts: a, b, c 17
1 Circle of radius log 1 To sketch a graph look for: 1 a and intercepts turning points behaviour as tends to asmptotes (e.g. when the denominator of a fraction = 0) Graphs should be named, aes labelled and an intercepts, turning points or asmptotes marked. Reflections If ( ) then ( ) is the reflection of the graph of about the -ais ( ) is the reflection of the graph of about the -ais Eamples ( ) ( ) 10 5 - - -5-10 18
( ) ( ) ( ) 3. If ( ), ( ). The graph of is the graph of reflected about the -ais. Translations A translation ma be a horizontal shift or a vertical shift. Horizontal The graph of ( ) is a shift of the graph ( ) units to the right. The graph of ( ) is a shift of the graph ( ) units to the left Vertical The graph of ( ) is a shift of the graph ( ) units up. The graph of ( ) is a shift of the graph ( ) units down. Eamples log( ) Basic graph is log Replacing with ( 1) shifts the graph 1 unit to the right log( 1) 6 8 shifted 1 unit right - 19
( ) 10 5-6 - - Basic graph is Replacing with ( + 3) shifts the graph 3 units to the left ( ) -5-10 shifted 3 units left Asmptote is at = 3, when + 3 = 0 15 Basic graph is Adding shifts the graph up 10 5 shifted units up - - -5 Dilations A dilation is a stretching or a squashing. The graph of ( ) is a dilation of the graph of ( ) b a factor of units parallel to the - ais. The graph of ( ) is a dilation of the graph of ( ) b a factor of units parallel to the - ais. Eamples Stretched b a factor of parallel to the -ais 3 1 - -1 1-1 Each value of is obtained b multipling the corresponding value of b. e.g. = 1 gives = 1 for and = for 0
( ) ( ) ( ) Dilated b a factor of 1/ parallel to the -ais In this case values that differ b a factor ½ give the same values ( ) e.g The value of = is given b = 3 for ( ) and b for ( ) Combinations More complicated graphs can be sketched b using combinations of dilation, reflection and translation. Dilations must alwas be considered before reflection and translation. Eamples 10 8 6 Basic graph is Multipling b stretches the graph b a factor of parallel to the ais Adding shifts the graph up units - -1 1 3 - Stretched b factor verticall shifted units up Basic graph is ( ) - - 6 - - -6 Replacing with ( + ) shifts the graph units to the left. Adding 1 shifts the graph 1 unit down The negative sign before the bracket reflects the graph about the ais ( ) -8 reflected about ais Shifted units left shifted 1 unit down 1
Basic graph is 6 Replacing with reflects the graph about the ais. Adding + shifts the graph up units. -8-6 - - reflected about ais Shifted units up - Eercises Sketch the following graphs. 1. ( ). log( ) 3.. ( ) ( ) 5. ( ) 6. Answers 10 5 - - 6 - - - -5 - -10 1 3-6 5 6