2. 1 College of Engineering MATHEMATICS I Simple transformation and Curve Sketching Dr Fuad M. Shareef

3. 2 Graph Sketching The knowledge of simple/standard graph is used to sketch the graph of some more complicated graphs. The method is called Transformations. Vertical Shift: Example: The graph of can be used to sketch the graph of where a is an integer

4. 3 Vertical Shift This is a translation of f(x) along the y-axis by a units, a is an integer. f(x)=x 2 f(x)=x 2 -3 f(x)=x 2 +2

5. 4 Horizontal Shift This is a translation of f(x) along the x-axis direction by a units, a is an integer. Positive x-direction, if a <0 Negative x-direction, if a>0 f(x)=x 2 f(x)=(x-3) 2 f(x)=(x+3) 2

6. 5 Stretches For any curve f(x), and any value a>0, y=(a)f(x) is obtained from f(x) by a stretch of scale factor a parallel to the y-axis y=x 2 y=3x 2

7. 6 Stretches For any curve f(x), and any value a>0, y=f(ax) is obtained from f(x) by a stretch of scale factor 1/a parallel to the x-axis y=f(x)=x 2 f(x)=(3x) 2

8. 7 Stretches For any curve f(x), and any value a>0, y=f(x/a) is obtained from f(x) by a stretch of scale factor a parallel to the x-axis Y=f(x)=x 2 y=f(x/3)=(x/3) 2

9. 8 Reflections Reflection about the x-axis: Given f(x), the graph of g(x)=-f(x) is obtained by reflecting the f(x) about the x-axis f(x)=x 2 g(x)=-f(x)= -(x) 2

10. 9 Reflections Reflection about the y-axis: Given f(x), the graph of g(x)=f(-x) is obtained by reflecting the f(x) about the y-axis f(x)=e x f(x)=e - x

11. 10 Special Reflection Reflection about the line y=x Given f(x), the reflection of f(x) about the line y=x gives the inverse function g(x). f(x)=e x y=x f -1 (x)=lnx

12. 11 Combined Transformation Example: Any quadratic curve can be related to that of y=x 2. Write the given quadratic form as: Using completing square. That is finding a,p and q.

13. 12 Combined Transformation Example: Show how the graph of y =f(x)=1+4x-x 2 can be obtained from the graph of y =x 2, by succession of transformations.

14. 13 Combined Transformation Start with y = x 2 Obtain y = [(x-2) 2 -5] by applying the translations 2 and (-5) in the x-axis and y-axis. Obtain y = - [(x-2) 2 -5] by applying a reflection in the x-axis.

15. 14 Combined Transformation Start with y=x 2 Obtain y= [(x-2) 2 -5] by applying the translations 2 and (-5) in the x-axis and y-axis. Obtain y= -[(x-2) 2 -5] by applying a reflection in the x-axis. y=x 2 y=(x-2) 2 y=(x-2) 2 -5 y= - [(x-2) 2 -5]

16. 15 Graphical solution of simultaneous equations Example: Solve the following pair of equation graphically: xy = 9 and x-2y =3. Sketch and Obtain the point of intersections of the straight line with the curve. x=6, y=3/2 x=6, y=3/2 x= -3, y=-3

17. 16 Tutorial Exercise, Assignment and Lecture notes Visit: Course website at