(完整版)A-Level-数学4课件.ppt

1、 Boardworks Ltd 20051 of 53 Boardworks Ltd 20051 of 53AS-Level Maths:Core 1for EdexcelC1.4 Algebra and functions 4This icon indicates the slide contains activities created in Flash.These activities are not editable.For more detailed instructions,see the Getting Started presentation.Boardworks Ltd 20

2、052 of 53Contents Boardworks Ltd 20052 of 53Plotting and sketching graphsGraphs of functionsUsing graphs to solve equations Transforming graphs of functionsExamination-style questionsPlotting and sketching graphs Boardworks Ltd 20053 of 53Plotting graphsSuppose we wish to plot the graph of y=x3 7x+2

3、 for 3 x 3.We can find the coordinates of any number of points that satisfy the equation using a table of values.For example:These values of x and y correspond to the coordinates of points that lie on the curve.x x3 7x+2 y=x3 7x+23210123278101827+21+14+70 7 14 21+2+2+2+2+2+2+24882448 Boardworks Ltd

4、20054 of 53Plotting graphsThe points given in the table are plotted x021312324246810y and the points are then joined together with a smooth curve.The shape of this graph is characteristic of a cubic function.x3210123y=x3 7x+24882448 Boardworks Ltd 20055 of 53Sketching graphsTo help us sketch a graph

5、 given its equation we can find:Points where the curve intercepts the x-axisThese are found by putting y=0 in the equation of the graph.Points where the curve intercepts the y-axisThese are found by putting x=0 in the equation of the graph.Turning pointsA turning point is a point where the gradient

6、of a graph changes from being positive to negative or vice versa.It can be a maximum or a minimum.The value of y when x is very large and positiveThe value of y when x is very large and negative When the general shape of a graph is known it is more usual to sketch the graph.Boardworks Ltd 20056 of 5

7、3Sketching graphsFor example:Sketch the curve of y=x3+2x2 3x.When x=0 we havey=03+2(0)2 3(0)=0So the curve passes through the point(0,0).When y=0 we havex3+2x2 3x=0Factorizing givesx(x2+2x 3)=0 x(x+3)(x 1)=0 x=0,x=3 or x=1So the curve also passes through the points(3,0)and(1,0).Boardworks Ltd 20057

8、of 53Sketching graphsWe can plot these three points on our graph.x02131231212345y467When x is very large and positive,y is very large and positive.We can write this as:as,.xy We can use this to sketch in this part of the graph:Also:as,.xy We can now produce a sketch of y=x3+2x2 3x.We can use this to

9、 sketch in this part of the graph:Boardworks Ltd 20058 of 53Contents Boardworks Ltd 20058 of 53Plotting and sketching graphsGraphs of functionsUsing graphs to solve equations Transforming graphs of functionsExamination-style questionsGraphs of functions Boardworks Ltd 20059 of 53Graphs of cubic func

10、tionsy=ax3+bx2+cx+d (where a 0)A cubic function in x can be written in the form:Graphs of cubic functions have a characteristic shape depending on the values of the coefficients:When the coefficient of x3 is positive the shape isWhen the coefficient of x3 is negative the shape isororCubic curves hav

11、e rotational symmetry of order 2.Boardworks Ltd 200510 of 53Graphs of factorized cubic functionsWhen a cubic function is written in the form y=a(x p)(x q)(x r),it will cut the x-axis atthe points(p,0),(q,0)and(r,0).p,q and r are the roots of the cubic function.In general:To sketch the graph of a cub

12、ic function given in factorized form,Find the roots of the function and plot these on the x-axis.Find the y-intercept by putting x equal to 0 in the equation.Look at the coefficient of x3 to decide whether the curve is N-shaped or -shaped.Boardworks Ltd 200511 of 53The graphs of y=x2 and y=x3You sho

13、uld be familiar with the graphs of y=x2 and y=x3:0 xy0 xyy=x3y=x2This is a quadratic functionThis is a cubic function Boardworks Ltd 200512 of 53Graphs of the form y=kxn Boardworks Ltd 200513 of 530 xy1=yxThe graph of y=1/xThis is a reciprocal functionNotice that the curve gets closer and closer to

14、the x-and y-axes but never touches them.The x-and y-axes form asymptotes.1=xyYou should also be familiar with the graph of .The graph of is an example of a discontinuous function.1=xy Boardworks Ltd 200514 of 53Graphs of the form y=kxn Boardworks Ltd 200515 of 53The graph of y=This graph can only be

15、 drawn for positive values of x.This is because we cannot find the square root of a negative number.=yxAnother interesting graph is .The curve is therefore only drawn in the first quadrant.0 xy=yxCompare this to the graph of y2=x.y2=xAlso,remember that is defined as the positive square root of x.=yx

16、x Boardworks Ltd 200516 of 53Graphs of the form y=knx Boardworks Ltd 200517 of 53Contents Boardworks Ltd 200517 of 53Plotting and sketching graphsGraphs of functionsUsing graphs to solve equations Transforming graphs of functionsExamination-style questionsUsing graphs to solve equations Boardworks L

17、td 200518 of 53Using graphs to solve equationsBy sketching an appropriate graph find the solutions to the equation 2x2 5=3x.We can do this by considering the left-hand side and the right-hand side of the equation as two separate functions.2x2 5=3xy=2x2 5 y=3xThe points where these two functions inte

18、rsect will give us the solutions to the equation 2x2 5=3x.Boardworks Ltd 200519 of 53Using graphs to solve equations123401234246246810y=2x2 5y=3x(1,3)(2.5,7.5)The graphs of y=2x2 5 and y=3x intersect at the points:The x-values of these coordinates give us the solutions to the equation 2x2 5=3x as(1,

19、3)and (2.5,7.5).x=1and x=2.5 Boardworks Ltd 200520 of 53Using graphs to solve equationsAlternatively,we can rearrange the equation so that all the terms are on the left-hand side:The line y=0 is the x-axis.This means that the solutions to the equation 2x2 3x 5=0 can be found where the function y=2x2

20、 3x 5 crosses the x-axis.2x2 3x 5=0y=2x2 3x 5 y=0These points represent the roots of the function y=2x2 3x 5.Boardworks Ltd 200521 of 53Using graphs to solve equations123401234246246810y=2x2 3x 5y=0(1,0)(2.5,0)The graph of y=2x2 3x 5 crosses the x-axis at the points:(1,0)and (2.5,0).The x-values of

21、these coordinates give us the same solutions:x=1and x=2.5 Boardworks Ltd 200522 of 53Using graphs to solve equationsUse a graph to solve the equation x3 3x=1.This equation does not have any rational solutions and so the graph can only be used to find approximate solutions.A cubic equation can have u

22、p to three solutions and so the graph can also tell us how many solutions there are.Again,we can consider the left-hand side and the right-hand side of the equation as two separate functions and find thex-coordinates of their points of intersection.x3 3x=1y=x3 3x y=1 Boardworks Ltd 200523 of 5312340

23、1234246246810Using graphs to solve equationsy=x3 3xy=1The graphs of y=x3 3x and y=1 intersect at three points.This means that the equation x3 3x=1 has three solutions.Using the graph these solutions are approximately:x=1.5x=0.3x=1.9 Boardworks Ltd 200524 of 53Contents Boardworks Ltd 200524 of 53Plot

24、ting and sketching graphsGraphs of functionsUsing graphs to solve equations Transforming graphs of functionsExamination-style questionsTransforming graphs of functions Boardworks Ltd 200525 of 53Transforming graphs of functionsGraphs can be transformed by translating,reflecting,stretching or rotatin

25、g them.The equation of the transformed graph will be related to the equation of the original graph.When investigating transformations it is most useful to express functions using function notation.For example,suppose we wish to investigate transformations of the function f(x)=x2.The equation of the

26、graph of y=x2,can be written as y=f(x).Boardworks Ltd 200526 of 53xVertical translationsThis is the graph of y=x2 7.What do you notice?Here is the graph of y=x2.yThe graph of y=f(x)+a is the graph of y=f(x)translated by the vector .0aThe graph of y=x2 has been translated 7 units down.If the original

27、 graph is written as y=f(x)then the translated graph can be written as y=f(x)7.In general:Boardworks Ltd 200527 of 53Translating quadratic functions vertically Boardworks Ltd 200528 of 53Translating cubic functions vertically Boardworks Ltd 200529 of 53Translating reciprocal functions vertically Boa

28、rdworks Ltd 200530 of 53xHorizontal translationsThe graph of y=f(x+a)is the graph of y=f(x)translated by the vector .a0yAgain,here is the graph of y=x2.This is the graph of y=(x+3)2.What do you notice?The graph of y=x2 has been translated 3 units to the left.If the original graph is written as y=f(x

29、)then the translated graph can be written as y=f(x+3).In general:Boardworks Ltd 200531 of 53Translating quadratic functions horizontally Boardworks Ltd 200532 of 53Translating cubic functions horizontally Boardworks Ltd 200533 of 53Translating reciprocal functions horizontally Boardworks Ltd 200534

30、of 53xReflections in the x-axisThe graph of y=f(x)is the graph of y=f(x)reflected in the x-axis.Here is the graph of y=x2 2x 2.yThis is the graph of y=x2+2x+2.What do you notice?The graph of y=x2 2x 2 has been reflected in the x-axis.If the original graph is written as y=f(x)then the translated grap

31、h can be written as y=f(x).In general:Boardworks Ltd 200535 of 53Reflecting quadratic functions in the x-axis Boardworks Ltd 200536 of 53Reflecting cubic functions in the x-axis Boardworks Ltd 200537 of 53Reflecting reciprocal functions in the x-axis Boardworks Ltd 200538 of 53Reflections in the y-a

32、xisHere is the graph of y=x3+4x2 3.The graph of y=f(x)is the graph of y=f(x)reflected in the y-axis.xyThis is the graph of y=(x)3+4(x)2 3.What do you notice?The graph of y=x3+4x2 3 has been reflected in the y-axis.If the original graph is written as y=f(x)then the translated graph can be written as

33、y=f(x).In general:Boardworks Ltd 200539 of 53Reflecting quadratic functions in the y-axis Boardworks Ltd 200540 of 53Reflecting cubic functions in the y-axis Boardworks Ltd 200541 of 53Reflecting reciprocal functions in the y-axis Boardworks Ltd 200542 of 53Vertical stretchesWe can produce the graph

34、 of y=2x2 6 by doubling the y-coordinate of every point on the original graph y=x2 3.This has the effect of stretching the graph in the vertical direction.Lets start with the graph of y=x2 3 and add the graph ofy=2x2 6.The graph of y=af(x)is the graph of y=f(x)stretched parallel to the y-axis by sca

35、le factor a.xyIf the original graph is written as y=f(x)then the translated graph can be written as y=2f(x).In general:Boardworks Ltd 200543 of 53Stretching quadratic functions vertically Boardworks Ltd 200544 of 53Stretching cubic functions vertically Boardworks Ltd 200545 of 53Stretching reciproca

36、l functions vertically Boardworks Ltd 200546 of 53Horizontal stretchesThe graph of y=f(ax)is the graph of y=f(x)stretched parallel to the x-axis by scale factor .a1xyLets start with the graph of y=x2+3x 4 and add the graph ofy=(2x)2+3(2x)4.We can produce the second graph by halving the x-coordinate

37、of every point on the original graph.This has the effect of compressing the graph in the horizontal direction.If the original graph is written as y=f(x)then the translated graph can be written as y=f(2x).In general:Boardworks Ltd 200547 of 53Stretching quadratic functions horizontally Boardworks Ltd

38、 200548 of 53Stretching cubic functions horizontally Boardworks Ltd 200549 of 53Stretching reciprocal functions horizontally Boardworks Ltd 200550 of 53Contents Boardworks Ltd 200550 of 53Plotting and sketching graphsGraphs of functionsUsing graphs to solve equations Transforming graphs of functions

39、Examination-style questionsExamination-style questions Boardworks Ltd 200551 of 53Examination-style questionThis diagram shows the graph of y=f(x)which has a minimum point at(2,3).y(2,3)a)Sketch the following graphs on separate sets of axes,indicating the turning point in each case.i)y=f(x+4)ii)y=f(

40、2x)b)Given that f(x)=ax2+bx+5 find the values of a and b.x Boardworks Ltd 200552 of 53Examination-style questiona)i)y=f(x+4)ii)y=f(2x)y(2,3)xy(1,3)x Boardworks Ltd 200553 of 53Examination-style questionb)f(x)is quadratic and so it can be written in the form a(x+p)2+q where(p,q)are the coordinates of the vertex.The vertex is at the point(2,3)sof(x)=a(x 2)2 3=a(x2 4x+4)3=ax2 4ax+4a 3Butax2 4ax+4a 3=ax2+bx+5So4a 3=5a=2b=8