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1 Functions Assessment statements.. Concept of a function as a mapping... Domain and range. Mapping diagrams.. Linear functions and their graphs, for eample f : m c.. The graph of the quadratic function: f () a b c... Properties of smmetr; verte; intercepts... The eponential epression: a b ; b... Graphs and properties of eponential functions. f () a ; f () a ; f () ka c... Growth and deca; basic concepts of asmptotic behaviour. Overview B the end of this chapter, ou will be able to: understand the concepts of relation and function draw and interpret mapping diagrams find the domain and range graph linear functions graph quadratic functions b using properties of smmetr simplif epressions of the form a b, b understand the concept of asmptotic behavior and find equations of lines of asmptote graph eponential functions graph linear, quadratic and eponential functions using a GDC solve real-world problems involving linear, quadratic and eponential models Relations, functions, and mapping diagrams Concept of a function as a mapping. Domain and range. Mapping diagrams. In our world there are man relationships in which one thing depends upon another thing. Our height, at least for a while, depends upon how old we are. The price of a house depends on the economic idea of suppl and demand. Weight loss depends on reducing calorie intake. Another popular wa to epress these ideas is to use the word function. A good golf score is a function of long hours of practice. A health relationship is a function of compromise. Mathematicall, we define relations and functions as follows: A relation is an set of ordered pairs. For a brief look at a histor of functions, visit www. heinemann.co.uk/hotlinks, enter the epress code 0P and click on weblink.. When using our TI calculator, press MODE and make sure that our calculator is set up as in the diagram below: NORMAL FLOAT RADIAN FUNC SCI ENG DEGREE PAR POL SEQ CONNECTED DOT SEQUENTIAL SIHUL REAL a+bi reˆ0i FULL HORIZ G-T 09/0/07 :0PM Look in the fourth row: FUNC Function PAR Parametric POL Polar SEQ Sequence For this chapter make sure that ou set our calculator in FUNC mode. 6

2 Functions Downloaded from Figure. The graph of ( ). For eample: A {(, ), (, )} is a relation since it is a set of two ordered pairs. B {(, ), (, ), (, 7)} is also a relation consisting of three ordered pairs. Now consider sets of ordered pairs in graphic form. The graphs of ( ) (Figure.) and (Figure.) are also relations, since graphs reall consist of infinitel man ordered pairs (see Chapter ). ( ) (, ) (, ) 0 Figure. Can ou see how this set of ordered pairs ( ) differs from the set in Figure.? (, ) (.,.669) (, ) (.7,.90) To determine whether or not a graph can be considered a function, use the vertical line test. Draw a vertical line on the graph. If the line intersects the graph in two or more places, the graph is not a function. This is because two ordered pairs would have the same first elements. 6 A function is defined as a set of ordered pairs (a relation) such that no two ordered pairs have the same first element. In other words, for a set of ordered pairs to be considered a function, all ordered pairs must have different first elements. Hence, set A above is a function, but set B is not, since is a first element in two of the ordered pairs. The graph, in Figure., of ( ) does not represent a function, since if, then will equal both and. In other words, (, ) and (, ) both eist on the graph and therefore, b definition, the set of ordered pairs represented b the graph is not considered a function. The answer to the question posed in Figure. is that the set of ordered pairs represented b the graph is a function, since ever -value will be paired with a one and onl one -value.

3 The domain of a function (or relation) is the set of independent values that are allowed. The range of a function (or relation) is the set of dependent values that are allowed. The independent values are often thought of as the -values or as the first elements and the dependent values as the -values or the second elements. A mapping diagram is a drawing depicting the pairing of the independent and dependent values. Eample. Given the mapping named as f, a) list the elements in the domain of f b) list the elements in the range of f c) list the ordered pairs that describe the relation f d) is f a function? Independent values a) Domain {,, 0,, } b) Range {0,, } c) {(, ), (, ), (0, 0), (, ), (, )} d) This mapping diagram does describe a functional relation since all of the first elements are different. 0 f Dependent values 0 There are different tpes of mappings. For eample, there is a mapping that maps real numbers to ordered pairs: e.g. 6 (,. ) The fact that some of the second elements are the same has no bearing on whether or not the relation, f, is called a function. Eaminer s hint: Do not list the elements of the range more than once. Eercise.. Eplain the mathematical concept of a relation.. Eplain the mathematical concept of a function.. Using the word function, describe a relationship between dail calorie intake and weight.. Describe a non-mathematical relationship between two things.. List a set of five ordered pairs that describe a relation that is not a function. 6. List a set of five ordered pairs that describe a function. 7. Draw a mapping diagram for the relation, where {,, 0,, }. a) Is this relation a function? Wh? b) List the elements in the domain. c) List the elements in the range. _ 8. Draw a mapping diagram for the relation, where {0,,, 9}. a) Is this relation a function? Wh? b) List the elements in the domain. c) List the elements in the range. 9. Determine if each of the following sets of ordered pairs is a function. Give a reason for each answer. a) {(, ), (, ), (, )} b) {(, ), (, ) (, 8), (, 8), (, 7), (, 7)} c) {(, ), (, ), (0, ), (, ), (, )} d) {(7, ), (7, ), (7, 0), (7, ), (7, )} 6

4 Functions Downloaded from e) 0 f) 0 g) 0 h) Sketch the graph of a non-functional relation. Eplain, in words and b drawing lines on the sketch, wh the graph is not a function.. Sketch the graph of a functional-relation. Eplain, in words and b drawing lines on the sketch, wh the graph is a function. 6

5 . Sa whether each of the following relations is a function or not. Eplain wh. a) b) c) d) _ e) f) g) 6 h). a) Represent the function, where {,, 0,,, }, b a mapping diagram. b) List the elements of the domain of this function. c) List the elements of the range of this function. International Baccalaureate Organization 00.. Domain and range.. Domain and range. Mapping diagrams. The concept of domain and range is finding, either algebraicall or graphicall, which -values are allowed to be used when finding -values, and which -values will result after those calculations are made. In other words, what ordered pairs will eist on the graph. Finding domain algebraicall The domain of a function is the set of allowable values the independent variable ma take on. There are two situations that ou need to be concerned with when finding the domain of a function:. Division b zero.. Taking the square-root of a negative number. For eample, consider the function: g() If it helps ou, replace g() with. In other words, think:. Ask ourself what value of will produce a division b zero. The domain will then be restricted from that value. If,. Since division b zero is undefined, the domain 0 must be restricted from the value of. Another wa to epress the domain of g is to use set notation: Domain of g D(g) { :, }. This notation is read: The domain of g is equal to the set of all such that is an element of the set of real numbers and is not equal to. The independent variable s generic name is abscissa. Would earl humans have had a need for the number 0? How long has the number 0 been used? 6

6 Functions Downloaded from For more insight on dividing b zero, visit co.uk/hotlinks, enter the epress code 0P and click on weblink.. When the domain is the set of real numbers, the notation ma be (and usuall is) omitted. Finding the range of a function algebraicall is beond the scope of this course. However, below are the basic steps used to find the range of simple functions. Solve for in terms of. Appl the methods used in eamples. and/or.. Eaminer s hint: Draw means draw accuratel on graph paper. Sketch means give a general shape of the graph. Eample. Let f (). Determine the domain of f. Since the square root of a negative number is not an element of the set of real numbers, 0. Therefore,. Hence, D(f ) { :, } Finding domain and range graphicall The domain and range of a function (or relation) can alwas be found b eamining the graph. Step : Draw an accurate graph. Step : To find the domain, draw vertical lines on the graph paper to the -ais. If the vertical line intersects the graph, the -value on the - ais is part of the domain. If the vertical line does not intersect the graph, the -value on the -ais is not part of the domain. Step : To find the range, draw horizontal lines on the graph paper to the -ais. If the horizontal line intersects the graph, the on the -value on the -ais is part of the range. If the horizontal line does not intersect the graph, the -value on the -ais is not part of the range. Eample. Find the domain and range f () shown in the graph below. f()

7 B drawing vertical lines (see the solid lines here) ou can see that the leftmost -value is 6 and the rightmost -value is 7. All vertical lines drawn between those -values will intersect the graph and the -ais. Therefore, the domain is: D(f ) { : 6 7, }. B drawing horizontal lines (see dashed lines) ou can see that the lowest -value is and the highest -value is 6. All horizontal lines drawn between those -values will intersect the graph and the -ais. Therefore, the range is: R(f ) { : 6}. 7 6 f() Eercise. For each of the sets A to D: a) state the domain b) state the range c) state if the relation is a function and eplain our answer.. A {(0, ), (, ), (, ), (, 8), (, 6)}. B {(, 0), ( 0, ), (, ), ( 60, ), (90, 0), ( 0, ), (, ), ( 0, ), (80, 0) }. C {(, ), (, ), (0, 0), (, ), (, )}. D {(, ), (, ), (0, 0), (, ), (, )} For each of the mapping diagrams and 6: a) state the domain b) state the range c) state if the relation is a function and eplain our answer.. f 6. g Determine the domain for each of the following functions f () g(). f :. g : 67

8 Functions Downloaded from For each graph 6, determine the domain and range If f (t) 000( 0.07) t, where t represents elapsed time in ears and f (t) represents an amount of mone after t ears, find the domain and range of f. 8. If the perimeter of a regular polgon is given b P(), where is the length of a side, find the domain and range of P. 68

9 . Function notations. Linear functions and their graphs, for eample f : m c. There are man notations used to describe ordered pairs that satisf a given function. For eample:. A table of values:. In words: If, then. If, then.. In function notation: f( ). This can be read as f at equals. f ( ). This can be read as f of equals.. In the mapping notation: f :. This can be read as: The function f that pairs with the value. f :. This can be read as: The function f that assigns to the value. The above notations are all describing the set of ordered pairs: {(, ) and (, )}. Each notation is useful in its own right. Perhaps the most popular is the notation used in number. This is the independent variable Can Mathematics be considered a language unto itself? Wh are smbols used instead of writing the ideas as words? f () } This is the dependent variable. This is the value. Therefore we can write, f (). Eample. Write each of the following in words and describe the concept for the notation. a) f () b) g ( ) c) h() d) r : e) k : t t a) f at equals. (, ) is an ordered pair on the graph of f. b) g of equals. (, ) g. c) h of equals. (, ) is an ordered pair on the graph of h. d) The function r that pairs with the value. (, ) lies on the graph of r. e) The function k that assigns to t the value t. (t, t ) is an ordered pair on the graph of k. When evaluating a function for a given value, simpl substitute the value for the variable. One reason that the function notation is so powerful is that it can be embedded. A function of a function can then be epressed as: f (g ()). For eample, if f () and g (), f (g ()) f ( ). 69

10 Functions Downloaded from Eaminer s hint: In the IB eaminations, students are epected to be able to read the IB notation, but ma write answers with the maths language and notations the are comfortable with. Eample. Given that f (), evaluate each of the following: a) f () b) f ( ) c) f (k) d) f ( h) a) f () 9 7 b) f ( ) ( ) c) f (k) k d) f ( h) ( h) ( h h ) h h h h Eercise.. Write each of the following in words: a) f () b) r ( ) c) g (a) b d) u : 6 0 e) f : s s. Describe the concept for each notation: a) f () 9 b) g ( ) 7 c) v (c) d d) h : 0..7 e) f : a a. Given that f (), evaluate each of the following: a) f () b) f (0) c) f (a) d) f ( h) e) f ( ). Given that g : t t, evaluate each of the following: a) g () b) g ( ) c) g () d) g (r) e) g ( h). If f (), find f ( h). 6. If g(), find g ( h) g(). 7. If f (), and g (), find f (g()). Hint: Find g() and then find f at that value. 8. If h() and k (), find (h k)(). Hint: Rewrite (h k)() as h(k()). 9. If f () 000 ( 0.08 ), find f (0). 0. If A(r) r, find A(0) correct to significant figures. Note f (g()) is called a composite function. It is a function of a function. It is read as f at g at. It is a notation that requires a double substitution. The first requires finding g() and the second requires finding f at the g() value. For Questions 6, let f (), g(), and h(), and find each of the following:. f (g()). f (h()). g(f( )). h(g()). f (g(h(0))) 6. h (f (g())) f g is another notation used to denote a composite function. It is read as f operation g or, simpl, f op g. (f g)() can be thought of as f (g()). For Questions 7, use the same function definitions as in 6 above and find each of the following: 7. (g f )( ) 8. (h g)( ) 9. (f f )() 0. (h f )(0.). (h g)(). (g h)() 70

11 . Linear functions. Linear functions and their graphs, for eample f : m c. As the name suggests, a linear function is a relation in which all of the ordered pairs form a straight line. Recall from Chapter that a line is an undefined term in geometr, but we think of it as being made up of infinitel man points joined so closel together that there is no space between them. Linear functions can be written in several was. Eamples are given below. f () f : Each notation epresses the idea of ordered pairs that lie on the graph. From geometr, we know that we can draw a unique line through an two points. Therefore, in order to draw the graph of a linear function, at least two ordered pairs must be produced and the points plotted. There are several was to produce a set of at least two ordered pairs: producing a table of values finding function values finding the - and -intercepts using one ordered pair and the gradient. Producing a table of values Eample.6 Produce a table of values, without a calculator, for. 0 Figure. Here ou see man ordered pairs that satisf. Between each pair of them are infinitel man more. This set of points is called a line, and the equation is called a linear function. When ou use the everda epression varies directl as, ou are describing a linear relationship or a linear function through the origin. For eample: simple interest varies directl as the number of ears, or the circumference of a circle varies directl as the radius. Let, then. Let 0, then 0. Let, then. (, ) (, ) 0 (0, ) (, ) Eample.7 Produce a table of values with the TI calculator for. Method I: Tpe the following kestrokes: Y,, ND WINDOW, TblStart, Tbl, Indpnt: Auto, Depend: Auto, ND GRAPH You will now see a table of values. Each ordered pair satisfies. Use the or to see more ordered pairs. TABLE SETUP TblStart=- Tbl= Indpnt: Depend: Auto Auto Ask Ask - 0 X X= Y

12 Functions Downloaded from The ND ke is coloured blue and therefore activates all of the blue function kes. For eample, tping the kes ND, WINDOW accesses the function TBLSET. Method II: Tpe the following kestrokes: Y,, ND WINDOW, TblStart, Tbl, Indpnt: Ask, Depend: Auto, ND GRAPH You should now a see table with no values in it. If there are values present, use the DEL ke to delete them. Tpe, press ENTER and the calculator will return. Tpe 0, press ENTER and the calculator will return. Tpe, press ENTER and the calculator will return. Continue the above process to produce more ordered pairs. When the table fills up onl the last entr will change. TABLE SETUP TblStart=- Tbl= Indpnt: Depend: Auto Auto Ask Ask Finding functional values (, ) f means that (, ) lies on the graph of f. Eample.8 Find ordered pairs for {, 0, } where f (). f ( ) (, ) f. f (0) 0 (0, ) f. f () (, ) f. Finding - and -intercepts The -intercept is the point at which the graph crosses the -ais. All -intercepts have the form of (, 0). To find the -intercept, let 0 and then solve for. The -intercept is the point at which the graph crosses the -ais. All -intercepts have the form of (0, ). To find the -intercept, let 0 and then solve for. Eample.9 Find the - and -intercepts for the linear function f (). When the graph crosses the -ais at (0, ), it is common to sa that is the -intercept. Think of f () as. If 0, then 0.. Hence, the -intercept is (, 0 ). If 0, then 0. the -intercept is (0, ). 7

13 Using one ordered pair and the gradient The gradient of a linear function is the same as the slope of the line. There are several notations for the gradient of a line: m, where means the change in. rise (from one point to another) run gradient slope m change in change in rise run Eample.0 Describe the meaning of a gradient of. A gradient means that the slope of a line is constant from an one point to another point on the line. It means that, from point A, the net point could be found b rising units ( units down) and running units ( units to the right). Eample. If a linear function has a gradient of and point A has coordinates (, 6), find two more points that lie on the line. From (, 6): rise units from 6 (6 ) and then run units from ( 7). another point on the line would be (7, ). From (, 6): rise units from 6 (6 8) and then run units from ( ). another point on the line would be (, 8). See the diagram below (, ) (, 8) A (, ) (, 6) (, ) (7, )

14 Functions Downloaded from General form of a linear function The general form of a linear function is defined as: m c, where m is the gradient c is the -intercept. Eample. Given f () _, find the gradient and the -intercept. Using the above definition, the gradient _ and the -intercept. Eaminer s hint: Make sure ou number the interval marks on the aes of our graph and label the - and -aes. Graphing linear functions To draw the graph of a linear function: Step : Rewrite the function as an equation in terms of. Step : Use one of the methods previousl outlined to plot at least two ordered pairs (two points). Step : Connect the points. Step : Pa attention to the domain if it is required. Step : Draw the graph using IB -mm graph paper and a straight edge. Eample. Draw the graph of f :. The following steps will result in an accuratel drawn graph. Step : Rewrite the function notation as. Step : Use -mm graph paper. Step : Let cm unit. Step : Label the - and -aes. Step : Use one of the previous methods to produce and plot at least two ordered pairs. Step 6: Connect the points with a straight edge. Step 7: Label the graph as: f :, or f (), or. (Note: Both intercepts and an etra (table) value were found.) (0, ) (, 0) 0 (, ) 7

15 Writing linear functions You need to know how to write a linear function when ou know the gradient and one ordered pair when ou know two ordered pairs. Eample. Write a linear function in the form f () m c when m and the ordered pair (, ) lies on the graph of f. Think of f () m c as m c. Substitute for m and (, ) for (, ). Solve for c and back-substitute to write the equation. c 6 c Hence, c 6. Therefore, 6, or f () 6. Eample. Given that (, ) and (, ) are ordered pairs that satisf a linear function, epress that function as f () m c. Since the ordered pairs lie on a line, write m c Calculate m ( ) Hence, b wa of substitution, 8 c.. In order to find c, the -intercept, substitute either of the points that lie on the line for and. Using (, ) we have c c b substitution, f (). c. Applications of the linear function Eample.6 It is known that water freezes at F (Fahrenheit) and 0 C (Celsius). It is also known that water boils at F and 00 C. a) Write a linear function to convert Fahrenheit degrees to Celsius degrees. b) Find the Celsius temperature that corresponds to a temperature of 0 F. 7

16 Functions Downloaded from a) The general ordered pair is in the form (F, C). Therefore we know that (, 0) and (, 00) satisfies the function that relates F and C. The general form of a linear function is f () m c. In this application the function will take the form: f (F) m F c, where C f (F). Hence, m c c f (F) F 60 or C F b) f (7) 7 9 ( 60 9 ).9 C to significant figures. Eercise.. For an two points that lie on a line, how man points are between them?. List three different was that a linear function can be written.. List four methods for finding a set of ordered pairs.. If, produce a table of values under the condition that and.. If produce a table of values, using our GDC, under the following conditions: TblStart 0, Tbl 0., and If.6.7, produce a table of values, using our GDC, under the following conditions: Indpnt: Ask, 0., and If, produce a table of values when: Indpnt: Ask,, and If f (), find: f ( ), f (0), f (). 9. If g() _, find: g( ), g(), g(6). 0. A linear function has a gradient of _ and a point (, ) that lies on its graph. Find two ordered pairs, one on each side of (, ), that also lie on the graph.. A linear function has a slope of and the ordered pair (, 6) lies on its graph. Find two other ordered pairs, one on each side of (, 6), that also lie on the graph.. Find the - and -intercepts for each of the following linear functions. a) b) f () c) g : 7 d) f () d h. a) If is the -intercept, what ordered pair is associated with and on what ais does that point eist? b) If is the -intercept, what ordered pair is associated with and on what ais does that point eist?. a) Write an eample of a linear function that does not have an -intercept. a) What is the equation of the of the -ais? b) Wh doesn t the -ais represent a function? 76

17 . Draw the graph of each of the following under the following conditions: Use -mm graph paper (if possible). Let cm unit. Label the - and -aes. Plot at least two ordered pairs. Connect the points with a straight edge. Label the graph. a) b) c) f () d) g().. e) g : _ f ) h : 6. Write a linear function in the form f () m c for each of the following, under the given set of conditions. a) m, c 7. b) m, (0, ) lies on the graph of f. c) m, (, ) lies on the graph of f. d) m, (, ) f. e) Both (, ) and (, 7) lie on the graph of f. f ) Both (, ) and (, 8) lie on the graph of f. g) The -intercept is and the -intercept is. h) The -intercept is and the -intercept is. i) Both the - and -intercept are 0. (, ) f. 7. The functions f and g are defined for all values of b: f : k g : a) Write down the definition of (i) f g (ii) g f. b) If f g g f, find the value of k. International Baccalaureate Organization The cost of making a coat is, within a certain range, a linear function of the number of coats made. For 0 coats, the cost is $0. For 0 coats, the cost is $90. What is the cost of making 00 coats? International Baccalaureate Organization The dail cost in Swiss francs, C(), of producing chairs is given b: C(). 00. If each chair sells for.7 francs, how man chairs must be produced so that the cost of production equals the amount of mone obtained from sales? International Baccalaureate Organization A function f () is defined as: if 0 f () 9 if a) Cop and complete the table below. 0 6 f () 0 b) Using -mm graph paper, draw the graph of f with the scale unit cm on both the - and -aes. International Baccalaureate Organization

18 Functions Downloaded from The sketch below shows the graph of a function f () for between 0 and 6. if 0, The function f () is defined as follows: f () a if, m c if 6, where a, c and m are constants a) State the value of a. b) Find two equations for the constants m and c. c) Hence or otherwise, calculate the values of m and c. International Baccalaureate Organization 00.. Quadratic functions... The graph of the quadratic function: f () a b c. Properties of smmetr; verte; intercepts. Apollonius of Perga (6 BC 90 BC) was known as The Great Geometer. In his book, Conics, he introduced the term parabola. To find out more, visit hotlinks, enter the epress code 0P and click on weblinks.,. and.. In this course, a, b and c will be limited to the set of rational numbers. 78 The quadratic function that we will stud in this section is called a parabola. It is one of the four conic sections. Like all functions, a quadratic function is a set of ordered pairs. The first word, quadratic, defines that set of ordered pairs as a second degree polnomial function of the form: f () a b c, where a, b, c, a 0. All quadratic functions have similar characteristics. The are -shaped up or -shaped down. The have a minimum value or a maimum value. The point at which the maimum or minimum value occurs is called the verte or the turning point. The have smmetr with respect to a vertical line called the ais of smmetr. At least three points, but preferabl five points, are required to draw the graph. There is alwas a -intercept. Sometimes there are no -intercepts. The graph is called a parabola.

19 Eample.7 Draw the graph of f (). Describe the characteristics of the graph. a) It has a minimum value: 9 8. b) The ais of smmetr is. c) At least three points have been plotted. d) The -intercept is. e) The -intercepts are 0. and. This section will eamine how to draw the graph of a quadratic function and find the common characteristics listed above. There are several methods b which the graph of a quadratic function ma be drawn: point-plotting using a table of values plotting the - and -intercepts and the verte using the ais of smmetr and its properties. ( 0., 0) (, 0) 0 (0, ) (, 8 ) 8 Point-plotting using a table of values Since a quadratic function is a set of ordered pairs and since we alread know its general shape, or, simpl plot a sufficient number of points so that the graph reveals itself. Eample.8 Draw the graph of f () using the point-plotting method. The easiest method to produce a set of ordered pairs is to make a table of values using the method in Section.. TABLE SETUP TblStart=- Tbl=0. Indpnt: Depend: Auto Auto Ask Ask X X=- Y X 0... X= Y

20 Functions Downloaded from Plotting the - and -intercepts and the verte Once ou have knowledge of the shape of a parabola, ou will be able to draw a reasonable graph using onl three points. One of those points will need to be the verte, or turning point, and two other ver good choices will be the -intercepts. An eas fourth point to use would be the -intercept. There are several was to find the -intercepts, if the eist. There will alwas be a -intercept and onl one wa to find to find it. There are several was to find the verte. In Chapter ou learned to solve quadratic equations b factoring, b using the quadratic formula, and b using our GDC. For eample, ou learned that the solutions for the equation 0 are or. Think of the above equation as 0. Now, replace 0 with and write:. Finall, let f (). Therefore, f (). As ou can see, there is a clear connection between the equation and the associated function. The solutions, or, are called: answers to the equation roots to the equation. The are also called zeros of the function. The reason that the are called zeros is that, when the are substituted for in f (), f () will be 0. In other words, f ( ) ( ) ( ) 0. This implies that (, 0) f. And, f () () 0. This implies that (, 0) f. The ordered pairs, (, 0) and (, 0) are called the -intercepts. To find the -intercepts of a quadratic function, let f () 0 and solve the resultant equation b factorizing using the quadratic formula using our GDC. To find the -intercept of a quadratic function, let 0 and solve for. Eaminer s hint: The formula b is in the IB Informational a Booklet. See Equation of ais of smmetr. Eample.9 Given the quadratic function f (), find the - and -intercepts. Let f () 0 and solve for. 0 ( )( ) 0 0 or 0 or Hence, the -intercepts are (, 0) and (, 0). Let 0 and solve for f (). f () 0 (0). the -intercept is (0, ). 80

21 The verte or turning point occurs at the highest or lowest point of the parabola. There are several was to find the verte:. Find the mean average of the -intercepts to find the abscissa and then substitute that value into the function to find the ordinate.. Use ( b a, f ( b a )), where f () a b c.. Tpe the following kestrokes on our TI calculator: Y (for eample) ZOOM 6 (to see if the verte is the highest or lowest point) ND, TRACE (which accesses CALC) :minimum (if the verte is the lowest point) or :maimum (if the verte is the highest point),, or,, etc. (until the cursor is to the left of the verte) ENTER,, or,, etc. (until the cursor is to the right of the verte) ENTER, or (until the cursor is between the left- and right-bound pointers) ENTER the verte is ( _, _ ). Plot Plot Y= X +X- Y= Y= Y= Y= Y6= Y7= Plot ZOOM MEMORY :ZBo :Zoom In :Zoom Out :ZDecimal :ZSquare 6:ZStandard 7 ZTrig CALCULATE :value :zero :minimum :maimum :intersect 6:d/d 7: f()d Y-X+X- Y-X+X- Right Bound? X=-.096 X=-.76 Guess? X= Y=-.978 Minimum X= Y=-.. Use a calculus method. (See Chapter.). Completing the square. (This method is beond the scope of this course.) Eample.0 Find the verte of the parabola f (). The abscissa is : b ( ). a The ordinate is: f ( b a ) f () () 9. Therefore, the verte is (, 9). 8

22 Functions Downloaded from A function is said to be even if f ( ) f (), for all in the domain of f. A function is said to be odd if f ( ) f (), for all in the domain of f. Eample. Draw the graph of f () b plotting the - and -intercepts and the verte. : See above for the required points. (, 0) (, 0) (0, ) (, ) (, 9) Verte When answering the question about the ais of smmetr, the answer must be given as an equation. For eample,. The ais of smmetr The idea of smmetr, as it applies to quadratic functions, is that for ever -value there are two different -values that are paired with it. For eample, in Eample. (0, ) for corresponds to (, ). The are said to be smmetrical points. All points, ecept the verte, will have a corresponding point. f() This vertical line is the ais of smmetr. Its equation is:. (0, ) (, ) Verte (, 9) 0 A quadratic function can be classified as an even function. An even function is a set of ordered pairs that are smmetric with respect to a vertical line. This line is called the ais of smmetr. In a parabola: the ais of smmetr must pass through the verte the equation of the ais of smmetr is: b a, where f () a b ca 0 the equation can also be found b observing the -part of the verte. 8

23 Eample. Find the equation for the ais of smmetr for f () ( ). f () ( ) 6 9 f () 6 0 Since b a ( 6), the equation for the ais of smmetr is:. Eample. If f () and (, 7) lies on the graph of f, but is not the verte, what other point must lie on the graph of f? Since f is a quadratic and hence an even function, there must be an ordered pair whose -value is 7. Therefore, ( )( ) 0 0. Therefore, the other ordered pair that must eist is (., 7). Graphing quadratic functions using translations The primar quadratic function is: f (). The graph is shown on the right. Another wa to graph a quadratic function is to compare the given function to f () and then make the necessar horizontal and/or vertical translations. f() 0 To translate a point is to change its coordinates b moving either horizontall or verticall or both. We will call a horizontal translation a horizontal slide. We will call a vertical translation a vertical shift. 8

24 Functions Downloaded from Eample. Draw the graph of b comparing it to. Since, then ou ma correctl think of as the -value. Therefore can be thought of as one more than the -value. Hence the -value, for ever ordered pair that lies on the graph of, will be one more than the -value for each ordered pair that lies on the graph of. In other words, the entire graph of will be shifted verticall up one unit. 0 In Eample., we sa that has a vertical shift of (when compared to ). A horizontal slide occurs when the function looks like: ( ) or ( ). Subtracting will slide the graph of two units to the right. Adding will slide the graph of three units to the left. ( ) ( ) 0 8

25 Eample. For each of the following functions, give the horizontal slide and the vertical shift when compared to. a) b) ( ) c) ( ) a) Horizontal slide: None. Vertical shift: units down. b) Horizontal slide: units to the left. Vertical shift: None. c) Horizontal slide: unit to the right. Vertical shift: units up. Eample.6 Draw the graph of ( ) b comparing it to. Ever ordered pair on the graph of f () will be translated horizontall unit to the right and verticall units up. Therefore, choose at least three ordered pairs that lie on the graph of and translate them accordingl. (0, 0) will translate to 0 (0, 0 ) (, ). 9 (, ) will translate to 8 (, ) (, ). 7 (, ) will translate to 6 (,, ) (0, ) ( ) 0 Graphing quadratic functions that are stretched A stretch occurs when the term is multiplied b a constant. is stretched b a factor of when compared to. _ is stretched b a factor of _ when compared to. is stretched b a factor of when compared to. is stretched b a factor of when compared to. _ is stretched b a factor of _ when compared to. When the coefficient of is greater than (or less than ), the resulting graph will be narrower than the graph of (or ). When the coefficient of is between 0 and (or 0 and ), the resulting graph will be wider than the graph of (or ). 0 8

26 Functions Downloaded from Solving quadratic equations using accuratel drawn graphs There are several methods that can be used to solve quadratic equations. Three of these methods were studied in Chapter. The were: using the quadratic formula factorization using a GDC. Another method that can be used to approimate the solutions to a quadratic equation involves using an accuratel drawn graph that is associated with the equation. Eample.7 B drawing an accurate graph, approimate the solutions for correct to the nearest tenth. Think of two functions, one being and the other being. Step : Carefull draw the graph for both functions on the same -mm graph paper. Step : Draw a vertical line from each point of intersection to the -ais. Step : Read the approimate answer for where the vertical lines touch the -ais. Therefore,.6 or.6 to the nearest tenth. Eaminer s hint: Drawing lines on our graph is an ecellent wa to show our work and to score points on Paper A GDC can be used to check the above solution. 86 Eample.8 Solve b using a GDC. Use the following kestrokes: Y Y Y ZOOM 6 (or ZOOM ) CALC (ND TRACE) : intersect

27 ,, (until ou get close to the intersection point) ENTER, ENTER, ENTER.6 to significant figures. Repeat the above sequence of kestrokes and use the arrow kepad to get close to the other intersection point..6 to significant figures. Intersection X=.68 Y= Applications of the quadratic function Some of the real-world phenomena that are associated with parabolas are: stringing cables that hold up suspension bridges, kicking a football, throwing a baseball, dropping a rock from the top of a building, firing a pellet from a pellet gun, making a reflective surface for a flashlight, making a satellite disk to receive a signal from space. Intersection X=-.6 Y= Eample.9 A well-known formula from phsics for describing the height H, in metres, of an object thrown upward with an initial velocit V, in metres per second (m s ), for T number of seconds, and from a starting height B, in metres, is given b: H.9T VT B a) Find the height that a ball will reach if it is thrown verticall upward, if the starting height is m, the initial velocit is 0 m s and the ball stas in the air second. b) A to rocket is shot verticall into the air, from ground level, with an initial velocit of 9. m s. How man seconds will it take to reach its maimum height? a) In the function H.9T VT B, let V 0 and B. Therefore, H.9T 0T. Hence, find H when T : H.9() 0() 7. m. b) In the function H.9T VT B, let V 9. and B 0 since the rocket left at ground level. H.9T 9.T 0.9T 9.T 0.9T 9.T (This is true since H will be 0 again when the to rocket hits the ground.) 0 T(.9T 9.) 0 T or 0.9T 9. Hence,.9T 9. Therefore, T 9. 6 seconds. (This is the total time for the rocket to.9 go up and come back down.) Therefore, the time to reach the maimum height is 6 seconds. This is not the shortest solution. You could find the verte and give the abscissa as the answer, or use an of the other methods described in the section. 87

28 Functions Downloaded from Eercise.. Sketch the general shape of a b c where: a) a 0 b) a 0.. At least how man points are needed to draw a fairl accurate graph of a quadratic function?. What shape will a parabola have if it has a maimum value?. Wh can t a quadratic function have this shape? 0. Sketch a parabola that does not have an -intercepts. 6. Sketch a quadratic function that turns downward and has onl one -intercept. 7. Sketch a quadratic function that has two zeros. 8. Sketch a quadratic function that turns up, has onl one zero and passes through (0, ). Draw the ais of smmetr. 9. Sketch a quadratic function that turns down, has two zeros at and and a -intercept at Sketch a quadratic function that turns up, has two zeros at and and a - intercept at.. Sketch a quadratic function that has the following characteristics: a) has two zeros at and b) has the -intercept at 6 c) shows the ais of smmetr as a solid line d) has a verte at approimatel (0., 6).. Sketch a quadratic function that has the following three characteristics: a) has two -intercepts at 0 and b) has a maimum at (., 6.) c) shows the ais of smmetr as a solid line. N Find each of the following for Questions and : 0 (i) the -intercept (ii) the -intercept M (iii) the ais of smmetr (iv) the verte (v) one etra ordered pair on each side of the ais of smmetr.. a) f () 6 7 b) g() c) 8 d) 8. a) ( ) b) ( ) c) f () _ ( ) d) g : ( ). Using -mm graph paper and using the scale units cm on both aes, draw the graph of each function in Question. 6. Using -mm graph paper and using the scale unit cm on both aes, draw the graph of each function in Question. 88

29 7. If g() 6 8 and (, ) lies on the graph of g, but is not the verte, what other ordered pair must lie on the graph? 8. If f : and (, 7) lies on the graph of f, but is not the verte, what other ordered pair must lie on the graph? 9. For each of the following problems, first draw the graph of and then, on the same coordinate plane, draw the graph of the given function under the three conditions: (i) Use the scale of unit cm. (ii) Draw the given quadratic function b using translations and/or stretches. (iii) Show at least three points on the translated graph. a) b) c) ( ) d) ( ) e) ( ) f) ( ) g) h) i) _ ( ) j) ( ) 0. Draw an accurate graph of each function. Use -mm graph paper and the scale unit cm on both aes. Locate all relevant information and plot at least five significant ordered pairs. a) _ _ b) c) d) 7 e) 7 f). Using the method described in Eample.7, solve each of the following using the graphs from Question 0. a) _ _ b) 6 c) d) 7 6 e) 7.6 f) 8.. Using the method described in Eample.8, solve each equation in Question with our GDC.. The diagram below shows part of the graph of a. The line is the ais of smmetr. M and N are points on the curve, as shown. a) Find the value of a. b) Find the coordinates of (i) M N (ii) N 0 M International Baccalaureate Organization

30 Functions Downloaded from A rectangular dog pen was built against the back wall of a house as shown below. The total amount of fencing used was 00 feet. W wall L a) Write a linear equation describing the length, L, in terms of W. b) Write a quadratic equation describing the area, A, in terms of W. c) Find the width of the pen which will maimize the amount of area. d) Find the maimum amount of area that can be fenced in.. A ball is shot verticall upward with an initial velocit of 0 m s from a 0 m building. Give all answers correct to decimal place. (Hint: Use our GDC.) a) Sketch a diagram that describes the information given. b) Find the maimum height the ball will reach. c) Find the time it will take to reach the maimum height. d) Find the time it will take the ball to reach the ground. 6. A compan makes and sells calculators per week. The weekl cost in dollars, C, of making calculators is C The weekl selling price for one calculator, p dollars, is p 0 00 for 700. a) Show that the weekl amount, R dollars, received b the compan for selling calculators is a quadratic function in. b) Epress the weekl profit, F dollars, made b the compan in making and selling in terms of. International Baccalaureate Organization The diagram below is an enlarged portion of the curve near the point P (, 6). A point Q, with coordinates (.0, 6.080), near to P is shown. Q Q P M A a) Show that P and Q are, indeed, points on the curve. b) Find the gradient of the chord [PQ]. Q is a point on the curve whose coordinate is.00. c) Find (i) the coordinate of Q (ii) the gradient of the chord [PQ ] (iii) the equation of the chord [PQ ] International Baccalaureate Organization

31 8. The diagram shows the graph of 8. The graph crosses the -ais at the point A and has a verte at B. a) Factorize 8. b) Write down the coordinates of points A and B. A B 9 Consider the graphs of the following functions. (i) 7 (ii) ( )( ) (iii) (iv) Which of these graphs a) has a -intercept below the -ais? b) passes through the origin? c) does not cross the -ais? d) could be represented b the following diagram? International Baccalaureate Organization 000. International Baccalaureate Organization Eponential functions The eponential epression: a b ; b. Graphs and properties of eponential functions: f() a ; f() a ; f() ka c. Growth and deca; basic concepts of asmptotic behaviour. You might have heard the epression: The population is increasing eponentiall. In laman s terms, it means that the population is getting larger b larger amounts. The objective in this section is to eplain the concept in mathematical terms. Eponential epressions An eponential epression is of the form a b, where a b R, a, 0,, and b. 9

32 Functions Downloaded from Even though ( 8) _, since ( 8) _ ( 8), ( 8) _ /, since ( 8) _ ( 8) / i. 6i comple number sstem, but not the real number sstem. To find out more about real numbers, visit www. heinemann.co.uk/hotlinks, enter the epress code 0P and click on weblink.6. The following are eamples of eponential epressions: ( ) ( ( _ ) ) The following eamples are not eponential epressions: ( ) _, since ( ) _ 0. An important algebraic law of eponents is: a m n n a m, where For eample: _ and. n a m. Eample.0 State whether each eponential epression is a real number. If it is, simplif it. a) ( ) b) ( 6) _ a) ( ) is a real number. ( ) 6. 8 b) ( 6) _ is not a real number since ( 6) 096 and there is no real number that can be multiplied b itself in order to get 096. Figure. Increasing eponentiall. The graph above is said to be increasing or to be an increasing function. is increasing is increasing Eponential functions An eponential function is a set of ordered pairs in which the independent variable is in the eponent position. f () is an eample of such a function. There are two general shapes for the graphs of eponential functions. One shape is when the -value increases as the -value increases from left to right, as in Figure.. The other shape is when the -value decreases as the -value increases from left to right, as in Figure.. Figure. Decreasing eponentiall. is decreasing is increasing The graph right is said to be decreasing or to be a decreasing function. 9 In both graphs above, the -ais acts as a boundar line for the curve. The boundar line is a line that the curve approaches but never touches or crosses.

33 The name of the boundar line is asmptote. (The p is silent in the pronunciation.) Another wa to describe the curve is to sa that it is asmptotic to the -ais. There are man was to find the equation of an asmptote: graph the function and observe whether the graph approaches a line put our TI calculator into Ask mode (see Eample.7) and observe the -values for larger and larger -values in either the positive direction or in the negative direction observe patterns for specific functions that have asmptotes use techniques described in Chapter use calculus techniques (these are beond the scope of this course). Eample. Find the equation of the horizontal asmptote for. With our GDC: Y Y TBLSET Indpnt: Ask TABLE Now enter the following -values one at a time:,,, 6, 8, 0, As ou can see in the diagram below, the Y -values seem to be approaching. This is enough information to write the equation of the horizontal asmptote as:. The graph in Figure.6 supports the answer for the worked eample. Plot Y=ˆX+ Y= Y= Y= Y= Y6= Y7= TABLE SETUP TblStart= Tbl= Indpnt: Depend: X= X Plot Plot Auto Auto Y Ask Ask Figure.6 The graph for Eample

34 Functions Downloaded from The smbol (lamda) is a Greek lower case letter. Arguabl the most important eponential graph is e. The constant e is like in that it is irrational. e.788 to decimal places. e 0 Leonhard Euler (707-78), a Swiss mathematician for whom e is honoured, was perhaps the most influential mathematician of the eighteenth centur. He is responsible for the following notations: f (),, i, and e, among others. The phrase increases at a faster rate is readil understandable at an intuitive level and will be eamined in more detail in Chapter when ou stud calculus. Figure.7 Graphing eponential functions f () a is a general form of an eponential function, where a 0, a. f () is one eample. See graph below. A function in this form is called the primar function, against which other functions are compared when using translations. f () a, where, is one form of a function that can be compared to the primar function. f () is one eample. When is positive, then the eponential function is called a growth function and is similar in nature to the primar graph. f () f () 0 For eample f () is similar to the primar graph. Both graphs pass through (0, ). Both graphs are increasing. Both graphs have 0 as the horizontal asmptote. The graph of f () increases at a faster rate than f (). When is negative, then the eponential function is called a deca function. One of its primar functions is f (). Its graph is shown in Figure.7. An eample of a graph that is similar in nature to f () is f ().. f () f (). 0 In Figure.8, a function of the form f () ka, where k, a,, represents a graph that is similar to the primar graph, f () a, in that k stretches the graph much like the coefficient of the term for quadratic functions (see Section.). 9

35 f () f () Figure.8 0 A function of the form f () ka c, where k, a,, c,, represents a graph that is stretched and shifted verticall with respect to the primar graph, f () a. For eample, f () is a graph that has been stretched b a factor of and has a vertical shift of when compared to the primar graph f (). See Figure.9. f () f () Figure.9 0 From the ordered pair (0, ) on f (), the -value,, was first multiplied b, putting a point at (0, ). was then subtracted from leaving an ordered pair at (0, ). And so ever ordered pair on f () would be translated in the same manner. The net result is the graph of the function f (). Now that ou know the basic shape of an eponential function, ou can using one of several methods to draw its graph: b point-plotting until the shape becomes evident using our GDC and a table of values b using knowledge of a primar graph and identifing stretches and shifts. Eample. Draw the graph of f () (0.). We know several bits of information: The horizontal asmptote is, since the vertical shift is unit up. The graph is similar in shape to the primar graph f (). 9

36 Functions Downloaded from It is increasing, since It is stretched b a factor of. With the calculator in Ask mode, the following ordered pairs have been produced: NORMAL FLOAT RADIAN FUNC SCI ENG DEGREE PAR POL SEQ CONNECTED DOT SEQUENTIAL SIHUL REAL a+bi reˆ0i FULL HORIZ G-T 08//07 9:PM TABLE SETUP TblStart= Tbl= Indpnt: Depend: Auto Auto Ask Ask X= X Y The completed graph is shown below f () Eample. B drawing an accurate graph, approimate the solution for 6 to the nearest tenth. Think of two functions: and 6. Now, carefull graph both functions on the same set of aes. Step : Draw a vertical line from the point of intersection to the - ais. Step : Read the answer where the vertical line touches the ais. Therefore,.6 to the nearest tenth

37 The Guess and Check method and logarithms The Guess and Check method is an ecellent wa to solve a variet of problems. Almost all great science and mathematical discoveries have come b wa of taking a guess and then verifing that guess. Eaminer s hint: Even though ou don t need to know how to use logarithms, the can make finding the solutions to eponential equations more efficient. Eample. B using the Guess and Check method, solve 7, correct to decimal places. Step : Draw a T chart. Guess Check Step : Take a guess and write it in the Guess column. Step : Check our guess b substituting that value for in 7 and write the answer in the Check column. Step : If the answer is too small, write S b the number; if it is too large, write L b it. Step : Choose the net Guess number between the last guess that was too small ( S ) and the net guess that was too large ( L ) or vice versa. Step 6: Continue this process until the guess is correct to decimal places. Guess 0. Check 0 S S S 8 L..669 S S S 6.96 S 7.6 L 7.00 L 7.07 L 7.08 L 6.96 S S 7.00 L (Note: If 0 was the answer, then 0 would be 7, but it isn t; therefore 0 is not the answer!) (Note:. is halfwa between and. A calculator was used to approimate..) (Note: is read as approimatel equal to.) (Note: The better guesser ou are the fewer times ou will have to guess.) Instead of alwas retping, for eample,,^,.7, use the kestrokes ND ENTER. This accesses the ENTRY ke which returns the last equation entered. That equation can now be edited. Since the too small S number is.807 and the net too large L number is.808, we can deduce that the solution to the equation 7 must be.8 correct to decimal places. Although the above Guess and Check solution produces the correct answer, it is somewhat laborious. The logarithm method described below is much faster and ou can learn it quickl. For a detailed eplanation of wh it works, ask our teacher or visit hotlinks, enter the epress code 0P and click on weblink.7 or.8. 97