ALevel数学4

Boardworks Ltd 2005,53,of 53,Click to edit Master title style,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*, Boardworks Ltd 2005,1,of 53,AS-Level Maths:,Core 1,for,Edexcel,C1.4 Algebra and functions 4,This icon indicates the slide contains activities created in Flash. These activities are not editable.,For more detailed instructions, see the,Getting Started,presentation.,Contents, Boardworks Ltd 2005,2,of 53,Plotting and sketching graphs,Graphs of functions,Using graphs to solve equations,Transforming graphs of functions,Examination-style questions,Plotting and sketching graphs,Plotting graphs,Suppose we wish to,plot,the graph of,y,=,x,3, 7,x,+ 2,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,x,3, 7,x,+ 2,y,=,x,3, 7,x,+ 2,3,2,1,0,1,2,3,27,8,1,0,1,8,27,+ 21,+14,+ 7,0, 7, 14, 21,+ 2,+ 2,+ 2,+ 2,+ 2,+ 2,+ 2,4,8,8,2,4,4,8,Plotting graphs,The points given in the table are plotted ,x,0,2,1,3,1,2,3,2,4,2,4,6,8,10,y, and the points are then joined together with a smooth curve.,The shape of this graph is characteristic of a,cubic function,.,x,3,2,1,0,1,2,3,y,=,x,3, 7,x,+ 2,4,8,8,2,4,4,8,Sketching graphs,To help us sketch a graph given its equation we can find:,Points where the curve intercepts the,x,-axis,These are found by putting,y,= 0 in the equation of the graph.,Points where the curve intercepts the,y,-axis,These are found by putting,x,= 0 in the equation of the graph.,Turning points,A,turning point,is a point where the gradient 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 positive,The 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.,Sketching graphs,For example:,Sketch the curve of,y,=,x,3,+ 2,x,2, 3,x,.,When,x,= 0 we have,y,= 0,3,+ 2(0),2, 3(0),= 0,So the curve passes through the point (0, 0).,When,y,= 0 we have,x,3,+ 2,x,2, 3,x,= 0,Factorizing gives,x,(,x,2,+ 2,x, 3) = 0,x,(,x,+ 3)(,x, 1) = 0,x,= 0,x,= 3 or,x,= 1,So the curve also passes through the points (3, 0) and (1, 0).,Sketching graphs,We can plot these three points on our graph.,x,0,2,1,3,1,2,3,1,2,1,2,3,4,5,y,4,6,7,When,x,is very large and positive,y,is very large and positive. We can write this as:,We can use this to sketch in this part of the graph:,Also:,We can now produce a sketch of,y,=,x,3,+ 2,x,2, 3,x,.,We can use this to sketch in this part of the graph:,Contents, Boardworks Ltd 2005,8,of 53,Plotting and sketching graphs,Graphs of functions,Using graphs to solve equations,Transforming graphs of functions,Examination-style questions,Graphs of functions,Graphs of cubic functions,y,=,ax,3,+,bx,2,+,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,x,3,is,positive,the shape is,When the coefficient of,x,3,is,negative,the shape is,or,or,Cubic curves have rotational symmetry of order 2.,Graphs of factorized cubic functions,When 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 cubic 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,x,3,to decide whether the curve is,N,-shaped or -shaped.,The graphs of,y,=,x,2,and,y,=,x,3,You should be familiar with the graphs of,y,=,x,2,and,y,=,x,3,:,0,x,y,0,x,y,y,=,x,3,y,=,x,2,This is a,quadratic,function,This is a,cubic,function,Graphs of the form,y,=,kx,n,0,x,y,The graph of,y,=,1,/,x,This is a,reciprocal,function,Notice that the curve gets closer and closer to the,x,- and,y,-axes but never touches them.,The,x,- and,y,-axes form,asymptotes,.,You should also be familiar with the graph of .,The graph of is an example of a,discontinuous,function.,Graphs of the form,y,=,kx,n,The graph of,y,=,This graph can only be drawn for positive values of,x,.,This is because we cannot find the square root of a negative number.,Another interesting graph is .,The curve is therefore only drawn in the first quadrant.,0,x,y,Compare this to the graph of,y,2,=,x,.,y,2,=,x,Also, remember that is defined as the positive square root of,x,.,Graphs of the form,y,=,k,Contents, Boardworks Ltd 2005,17,of 53,Plotting and sketching graphs,Graphs of functions,Using graphs to solve equations,Transforming graphs of functions,Examination-style questions,Using graphs to solve equations,Using graphs to solve equations,By sketching an appropriate graph find the solutions to the equation 2,x,2,5,=,3,x,.,We can do this by considering the left-hand side and the right-hand side of the equation as two separate functions.,2,x,2, 5,=,3,x,y,= 2,x,2,5,y,=,3,x,The points where these two functions intersect will give us the solutions to the equation 2,x,2,5,=,3,x,.,Using graphs to solve equations,1,2,3,4,0,1,2,3,4,2,4,6,2,4,6,8,10,y,= 2,x,2,5,y,=,3,x,(1,3),(2.5, 7.5),The graphs of,y,= 2,x,2,5,and,y,=,3,x,intersect,at the points:,The,x,-values of these coordinates give us the solutions to the equation,2,x,2,5 = 3,x,as,(1, 3),and,(,2.5,7.5,),.,x,= 1,and,x,= 2.5,Using graphs to solve equations,Alternatively, 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,2,x,2,3,x, 5,=,0 can be found where the function,y,= 2,x,2,3,x, 5,crosses the,x,-axis.,2,x,2,3,x,5,=,0,y,= 2,x,2,3,x,5,y,=,0,These points represent the roots of the function,y,=,2,x,2,3,x, 5,.,Using graphs to solve equations,1,2,3,4,0,1,2,3,4,2,4,6,2,4,6,8,10,y,= 2,x,2,3,x, 5,y,=,0,(1,0),(2.5, 0),The graph of,y,= 2,x,2, 3,x,5,crosses the,x,-axis at the points:,(1,0,),and,(,2.5,0,),.,The,x,-values of these coordinates give us the same solutions:,x,= 1,and,x,= 2.5,Using graphs to solve equations,Use a graph to solve the equation,x,3, 3,x,=,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 up 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 the,x,-coordinates of their points of intersection.,x,3,3,x,=,1,y,=,x,3,3,x,y,=,1,1,2,3,4,0,1,2,3,4,2,4,6,2,4,6,8,10,Using graphs to solve equations,y,=,x,3,3,x,y,=,1,The graphs of,y,=,x,3,3,x,and,y,=,1,intersect,at,three points.,This means that the equation,x,3, 3,x,=,1 has three solutions.,Using the graph these solutions are approximately:,x,= 1.5,x,= 0.3,x,= 1.9,Contents, Boardworks Ltd 2005,24,of 53,Plotting and sketching graphs,Graphs of functions,Using graphs to solve equations,Transforming graphs of functions,Examination-style questions,Transforming graphs of functions,Transforming graph,s of functions,Graphs can be transformed by translating, reflecting, stretching or rotating 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,) =,x,2,.,The equation of the graph of,y,=,x,2, can be written as,y,=,f,(,x,).,x,Vertical translations,This is the graph of,y,=,x,2, 7,.,What do you notice?,Here is the graph of,y,=,x,2,.,y,The graph of,y,=,f,(,x,),+,a,is the graph of,y,=,f,(,x,),translated by the vector .,0,a,The graph of,y,=,x,2,has been,translated,7 units down.,If the original graph is written as,y,=,f,(,x,),then the translated graph can be written as,y,=,f,(,x,) 7,.,In general:,Translating quadratic functions vertically,Translating cubic functions vertically,Translating reciprocal functions vertically,x,Horizontal translations,The graph of,y,=,f,(,x,+,a,),is the graph of,y,=,f,(,x,),translated by the vector .,a,0,y,Again, here is the graph of,y,=,x,2,.,This is the graph of,y,= (,x,+ 3),2,.,What do you notice?,The graph of,y,=,x,2,has been,translated,3 units to the left.,If the original graph is written as,y,=,f,(,x,),then the translated graph can be written as,y,=,f,(,x,+ 3),. In general:,Translating quadratic functions horizontally,Translating cubic functions horizontally,Translating reciprocal functions horizontally,x,Reflections in the,x,-axis,The graph of,y,= ,f,(,x,),is the graph of,y,=,f,(,x,),reflected in the,x,-axis.,Here is the graph of,y,=,x,2, 2,x, 2,.,y,This is the graph of,y,= ,x,2,+ 2,x,+ 2,.,What do you notice?,The graph of,y,=,x,2, 2,x, 2,has been,reflected,in the,x,-axis.,If the original graph is written as,y,=,f,(,x,),then the translated graph can be written as,y,= ,f,(,x,),.,In general:,Reflecting quadratic functions in the,x,-axis,Reflecting cubic functions in the,x,-axis,Reflecting reciprocal functions in the,x,-axis,Reflections in the,y,-axis,Here is the graph of,y,=,x,3,+ 4,x,2, 3,.,The graph of,y,=,f,(,x,),is the graph of,y,=,f,(,x,),reflected in the,y,-axis.,x,y,This is the graph of,y,= (,x,),3,+ 4(,x,),2, 3,.,What do you notice?,The graph of,y,=,x,3,+ 4,x,2, 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,y,=,f,(,x,),.,In general:,Reflecting quadratic functions in the,y,-axis,Reflecting cubic functions in the,y,-axis,Reflecting reciprocal functions in the,y,-axis,Vertical stretches,We can produce the graph of,y,= 2,x,2, 6,by doubling the,y,-coordinate of every point on the original graph,y,=,x,2, 3,.,This has the effect of,stretching,the graph in the vertical direction.,Lets start with the graph of,y,=,x,2, 3,and add the graph of,y,= 2,x,2, 6,.,The graph of,y,=,a,f,(,x,),is the graph of,y,=,f,(,x,),stretched parallel to the,y,-axis by scale factor,a,.,x,y,If the original graph is written as,y,=,f,(,x,),then the translated graph can be written as,y,= 2,f,(,x,),.,In general:,Stretching quadratic functions vertically,Stretching cubic functions vertically,Stretching reciprocal functions vertically,Horizontal stretches,The graph of,y,=,f,(,a,x,),is the graph of,y,=,f,(,x,),stretched parallel to the,x,-axis by scale factor .,a,1,x,y,Lets start with the graph of,y,=,x,2,+ 3,x, 4,and add the graph of,y,= (2,x,),2,+ 3,(2,x,) 4,.,We can produce the second graph by halving the,x,-coordinate 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,(2,x,),.,In general:,Stretching quadratic functions horizontally,Stretching cubic functions horizontally,Stretching reciprocal functions horizontally,Contents, Boardworks Ltd 2005,50,of 53,Plotting and sketching graphs,Graphs of functions,Using graphs to solve equations,Transforming graphs of functions,Examination-style questions,Examination-style questions,Examination-style question,This diagram shows the graph of,y,=,f,(,x,) which has a minimum point at (2, 3).,y,(2, 3),Sketch the following graphs on separate sets of axes, indicating the turning point in each case.,i),y,=,f,(,x,+ 4),ii),y,=,f,(2,x,),b) Given that,f,(,x,) =,ax,2,+,bx,+ 5 find the values of,a,and,b,.,x,Examination-style question,a) i),y,=,f,(,x,+ 4),ii),y,=,f,(2,x,),y,(2, 3),x,y,(1, 3),x,Examination-style question,b),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) so,f,(,x,) =,a,(,x, 2),2, 3,=,a,(,x,2, 4,x,+ 4) 3,=,ax,2, 4,ax,+ 4,a, 3,But,ax,2, 4,ax,+ 4,a, 3,=,ax,2,+,bx,+ 5,So4,a, 3 = 5,a,= 2,b,= 8,