Transformations of quadratics in the form y=ax(b-x)

Question

If I have a quadratic graph in the form y=ax(b-x), what effect (transformation) will it have on my graph if I change the values of the constants 'a' and 'b'?

Answer 1

When you have a quadratic of the form ax(b - x) you know its zeroes are going to be 0 and b.

 

If a is positive, the graph will face down (the bowl will be upside down). If a is negative, the graph will face up. 

 

You know that the vertex is always halfway between the zeroes of a parabola, because parabolas are symmetrical.

 

So the vertex of the parabola will be where x = b/2.

 

The coordinates of the vertex will be (b/2, ab²/4) because the y point of the vertex is what you get when you plug b/2 into your equation.

 

Looking at this vertex, we can find what happens to its placement when we consider different values of a and b. If you increase b it will move right and up. If you increase a it will move up. If you decrease b (as it becomes negative) it will move left and up. If you decrease a it will move down. 

Answer 2

a affects the steepness of the sides of the parabola.  

 

If:

     a > 1 then the parabola is steep, opening upwards

     a = 1 standard parabola opening upwards

     0 < a < 1 the parabola is flattened, opening upwards

     -1 < a < 0 flattened parabola opening downwards

     a = -1  standard parabola, opening downwards

     a < -1  steep parabola, opening downwards

 

 

b affects the x-intercepts of the parabola, this formula will produce a graph that intercepts at 0 and b

 

This equation will always have one x-intercept at the origin, because

if x = 0,  y = a(0)(b-x) = 0

    

As you increase b, the second intercept will move to the right.  As you decrease b, the second intercept will move to the left.

 

The axis of symmetry will move in the same direction as the second intercept moves: x = b/2

 

As b moves further from the origin (increases if b>0, decreases if b<0) the vertex will get more distant from the x-axis, to the positive if a<0, to the the negative if a>0.  If b=0, the vertex is the origin.