x2 t04 03 cuve sketching - addition, subtraction, multiplication and division

(D) Addition & Subtraction of Ordinates

(D) Addition & Subtraction of Ordinates

y = f(x) + g(x) can be graphed by first graphing y = f(x) and y = g(x) separately and then adding their ordinates together.

(D) Addition & Subtraction of Ordinates

y = f(x) + g(x) can be graphed by first graphing y = f(x) and y = g(x) separately and then adding their ordinates together.NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and g(x)=0, then plot further points by addition and subtraction of ordinates and finally locate the position of stationary points.

(D) Addition & Subtraction of Ordinates

y = f(x) + g(x) can be graphed by first graphing y = f(x) and y = g(x) separately and then adding their ordinates together.NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and g(x)=0, then plot further points by addition and subtraction of ordinates and finally locate the position of stationary points.

xxy 1 e.g.

(D) Addition & Subtraction of Ordinates

y = f(x) + g(x) can be graphed by first graphing y = f(x) and y = g(x) separately and then adding their ordinates together.

y

x

NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and g(x)=0, then plot further points by addition and subtraction of ordinates and finally locate the position of stationary points.

xxy 1 e.g.

xy

xy 1

(D) Addition & Subtraction of Ordinates

y = f(x) + g(x) can be graphed by first graphing y = f(x) and y = g(x) separately and then adding their ordinates together.

y

x

NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and g(x)=0, then plot further points by addition and subtraction of ordinates and finally locate the position of stationary points.

xxy 1 e.g.

xy

xy 1

(D) Addition & Subtraction of Ordinates

y = f(x) + g(x) can be graphed by first graphing y = f(x) and y = g(x) separately and then adding their ordinates together.

y

x

NOTE: First locate points on y = f(x) + g(x) corresponding to f(x)=0 and g(x)=0, then plot further points by addition and subtraction of ordinates and finally locate the position of stationary points.

xxy 1 e.g.

xy

xy 1

xxy 1

y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x) separately and then adding the ordinates together.

y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x) separately and then adding the ordinates together.

xxy sin e.g.

y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x) separately and then adding the ordinates together.

xxy sin e.g.

y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x) separately and then adding the ordinates together.

xxy sin e.g.

y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x) separately and then adding the ordinates together.

xxy sin e.g.

y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x) separately and then adding the ordinates together.

xxy sin e.g.

y = f(x) – g(x) can be graphed by first graphing y = f(x) and y = – g(x) separately and then adding the ordinates together.

xxy sin e.g.

(E) Multiplication of FunctionsThe graph of y = f(x). g(x) can be graphed by first graphing y = f(x) and y= g(x) separately and then examining the sign of the product. Special note needs to be made of points where f(x) = 0 or 1, or g(x) = 0 or 1.

NOTE: The regions on the number plane through which the graph must pass should be shaded in as the first step.

32 11 e.g. xxxy

32 11 e.g. xxxy

32 11 e.g. xxxy

32 11 e.g. xxxy

32 11 e.g. xxxy

32 11 e.g. xxxy

32 11 e.g. xxxy

32 11 e.g. xxxy

32 11 e.g. xxxy

(F) Division of Functions

by; graphed becan ofgraph Thexgxfy

(F) Division of Functions

by; graphed becan ofgraph Thexgxfy

Step 1: First graph y = f(x) and y = g(x) separately.

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

(F) Division of Functions

by; graphed becan ofgraph Thexgxfy

Step 1: First graph y = f(x) and y = g(x) separately.Step 2: Mark in vertical asymptotes

12

21 e.g.

xxxxy

(F) Division of Functions

by; graphed becan ofgraph Thexgxfy

Step 1: First graph y = f(x) and y = g(x) separately.Step 2: Mark in vertical asymptotesStep 3: Shade in regions in which the curve must be (same as

multiplication.

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

(F) Division of Functions

by; graphed becan ofgraph Thexgxfy

Step 1: First graph y = f(x) and y = g(x) separately.Step 2: Mark in vertical asymptotesStep 3: Shade in regions in which the curve must be (same as

multiplication.Step 4: Investigate the behaviour of the function for large values of x

(find horizontal/oblique asymptotes)

(F) Division of Functions

by; graphed becan ofgraph Thexgxfy

Step 1: First graph y = f(x) and y = g(x) separately.Step 2: Mark in vertical asymptotesStep 3: Shade in regions in which the curve must be (same as

multiplication.Step 4: Investigate the behaviour of the function for large values of x

(find horizontal/oblique asymptotes)

221

22

1221

2

2

2

xxx

xxxx

xxxxy

(F) Division of Functions

by; graphed becan ofgraph Thexgxfy

Step 1: First graph y = f(x) and y = g(x) separately.Step 2: Mark in vertical asymptotesStep 3: Shade in regions in which the curve must be (same as

multiplication.Step 4: Investigate the behaviour of the function for large values of x

(find horizontal/oblique asymptotes)

221

22

1221

2

2

2

xxx

xxxx

xxxxy

1:asymptote horizontal y

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy

12

21 e.g.

xxxxy