ASSIGNMENT 2

BY

SHADRECK S CHITSONGA

EXPLORING GRAPHS OF QUADRATIC FUNCTIONS

1. Let us consider the cases when a, b and c all greater than zero.

 

 

 

 

 

All the graphs drawn above are graphs of quadratic functions. There are a number of things we can highlight from the exercise. These include the following:

     a. In all the cases above we see that constants a, b and c are all positive.

     b. The coefficients are in the same ratio, a:b:c =1:2:4

          c. All the graphs have the same axis of symmetry that is x=-1

          d. All the graphs are facing upwards (i.e. concave upwards)

 

Possible questions to be answered by students from this exercise:

    i).  From the graph which letter determines the point where the graph crosses the y-axis

         ( in  our case the coefficient of y is one throughout).

   ii). Which graph has the smallest coefficient of x^2. Is there any connection between the 

         width of the graph and the coefficient of x^2?

   iii). In all the four cases considered above find the value of x=-b/2a. Does the answer

        surprise you. X=-b/2a is a straight line. Say something about this line in connection

         with the graphs above.

   iv). Explain in you own words why the graphs have the same line of symmetry.

   vi).  If we have the function y=ax^2+bx+c, where a,b,c  all not equal to zero, explain

      what  happens when a ,b, and c  are divided or multiplied by the same constant(>0).

 

Your explanation should answer the following questions:

        i). Does the orientation of the graph change?

       ii). Does the line of symmetry move?

          iii). Does the y-intercept remain the same?

           iv). Does the width of the graph change?

 

2.Graphs of quadratic functions when two coefficients are kept constant and the third one

   varies.

 

A. First case

   a and c are constant but b varies . The values of b used in the functions below are. -3,-

   2,-1,-0.5 and -0.5. To avoid confusing this exercise with others that will be done later

    on, the signs  are the same.

   

 

                                                                                                 Y

 X

 

The exercise above shows that as we vary the values of b the shape of the graph in terms of its width remains the same. The orientation of the graph remains unchanged. In this particular case all the graphs face upwards. All the graphs pass through the same common point, which is 2 in this case. There is something that changes and that is the line of symmetry. All the graphs have different lines of symmetry, i.e. have different vertices.

 

Questions:

Write down three equations whose graphs have the same size; same orientation, same y-intercept but the graphs have different vertices. Students should be encouraged to test their graphs by drawing them or explaining why their graphs meet the conditions stated in the question

 

B.  Second case: b and c are constant but  a changes.

 

                                                                          

 

All the five graphs pass through the same point in the y-axis i.e. y = 2. The graphs do not have the same width. The graph with the highest coefficient of x is the narrowest and is on the inside. The graph with the least value of a is the widest and is on the outside.

 

Conclusion

The conclusion here is that the greater the value of a, the wider is the graph and vice versa. If the graphs are drawn between the same axes the one with the least a-value will be on the outside and the one with the greatest value will be inside.

Though the value of b was kept constant we noticed that the vertex of the graphs does not remain the same. Refer to the first exercise and establish the reason why changing the value a affects the position of the line of symmetry even though the value of b is kept constant.

 

Possible questions:

á      Given a set of graphs drawn on the same axes with the same values for b and c, ask students to match the graphs corresponding to different values of a.

á      Write down equations of graphs that have the same y intercept, the same values of b, but have different vertices.

 

C. Third case: a and b remain the same but c changes

 

                                             

            Y

 X

 

 

When a and b are kept constant and c varies, all the curves have the same line of symmetry, they are all concave upwards.

They are all congruent. We can conclude that changing the value of  c only changes the y-intercept. Of course we must note here that though all the curves have the same line of symmetry they have different minimum points. The curve whose c value is the greatest has the highest minimum point.

END