Construct graphs for the parabola
for different values of a, b and c. (a , b and c can be any rational numbers)
I begin with the simplest type of quadratic function,
This gives me a parabola with vertex (0,0). if a > 0, the vertex is the point with the minimum y-value on the graph. If a < 0, the vertex is the point with the maximum y-value on the graph. We note that y=-ax is a mirror image of y=ax about thr x-axis for the same values of a.
Sketching of Quadratic Functions
Let us compare the following graphs:
We notice that with greater values of a, the graph shrinks by a factor a, creating a narrower parabola.
If a < 0 , the parabola opens more widely with smaller values of a. Hence, we conclude that the coefficient of a determines how widely the parabola
Next, I investigate how the various values of c affect the shape of
where c can take any rational numbers. From my observations, the point where the parabolas cut the y-axis will be (0,c). the point (0,c) is not the vertex of the parabola. Let a =1 and b = 1. We then have,
We construct this parabola using c =-6,-4,-2,0,2,4,6.
We obeserve that the parabola
cuts the y-axis at (0,c).
cuts the y-axis at (0,-c). This point is however not the vertex.
I investigated the movement of the parabola of the general form .
I discovered that the movement of the parabola traced another parabola of the form
Next, I investigate what happens when 2 parabolas of the form overlay/intercept. I set a = 1 or -1 and c = -5, 5.
Summary of Results
The 2 graphs intercept at 2 points. One of the points is (0,5) for all values of n. When x=0, the 2 graphs has one point of intersection at (0,5). The area of intersection is greater for greater values of n. The graphs move together for the same values of n. When n> 0, graphs move along the positive x-axis. When n<0, the graph moves along the negative x-axis. the following pairs of parabolas produce the same kind of interceptions.
The 2 parabolas intercept at at only 1 point (0,5) for all values of n and they are symmetrical about the point (0,5). They move in opposite directions for the same values of n. The pairs of parabolas that produce similiar interception patterns are
Most of the pairs of parabolas intercept in the above manner.
The 2 parabolas never neet and they move together.
Other pairs of parabola that intercept in the same manner is
If we put a is negative, the above set of parabolas still move together but with the cup facing downwards.
Vertex of Parabola
The vertex of a parabola of the form
is obtained when
Roots of Parabola
The solutions of a quadratic equation in the general for are given by
To find the total number of roots, we follow the following rules:
If the discriminant is
The quadratic equation has 2 distict real solutions and its graph has 2 x-intercepts
The quadratic equation has one repeated real solution and its graph has 1 x-intercpet
The quadratic equation has no real solution and its graph has no x-intercpets. Its square root is imaginary and yields 2 complex solutions
Test out the discriminant rule in assignment 3 to find out the behaviour of the roots.