ASSIGNMENT 2
BY
SHADRECK S
CHITSONGA
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