Quadratic functions and Parabolas

by Hwa Young Lee

In assignment 1 we investigated sine functions and sinusoids. This time we are going to explore parabolas. The graph of a quadratic function is called a parabola.

Let's examine graphs for the parabola

for different values of a, b, and c (a, b, c can be any rational number).

First, we start with a. What does the parameter ‘a’ do?

To find out, we fix the other values of b and c both equal to 1. So, we have

.

Let’s try varying a from -3 to 3. Then we get the following graphs.

 

We can see that when a=0, we get a line. If we go back to our equation, we know that if a=0, our function is no longer a quadratic function. Thus, in this assignment, we eliminate the case when a=0, to focus on parabolas.

You might be guessing what the parameter a does to the graph, but to get a better look, we will compare the graphs when a varies from 1 to 10:

and when it varies from -10 to -1:

We can see that the parabolas are both opening up or down more rapidly as the absolute value of a increases. Also, when a is positive, the parabola opens up while the parabola opens down when a is negative.

Now, let’s leave ‘b’ alone for a while and go on to ‘c’.

To find out what c does to the graph, we fix the other values of a and b both equal to 1. So, we have

.

Let’s try varying c from -3 to 3. Then we get the following graphs.

You might have figured out what the parameter c does already, but let’s just see what the graph looks as c varies from -10 to 10 to get a closer look.

We can see that the parabolas are shifting vertically. When c is positive, the parabola shifts upward; when c is negative, the parabola shifts downward.

Finally, let’s go on to ‘b’. To find out what b does to the parabola, we fix the other values of a and c both equal to 1. So, we have

.

Let’s try varying b from -3 to 3. Then we get the following graphs.

Is there a common point to all of the graphs? We can see that the parabolas always pass through (0,1) but the axis of symmetry shifts horizontally as the value of b changes. Also, the vertex of the graph shifts vertically. We rewrite the quadratic to get the idea of what’s happening:

.

We now see that b influences both the x-value and the y-value of the vertex, which is .

Since the x-value of the vertex determines the axis of symmetry, we know that when b is positive, the axis of symmetry shifts to the left and when b is negative, the axis of symmetry shifts to the right.

Since the y-value of the vertex is determined by both b and the value of c (in this case c was equal to 1), the direction of the vertical shift is determined by the sign of .

Since the value of b has influence on the vertex, we can also notice that the value of b has something to do with the zeros of the graph. Also, try to guess the locus of the vertices of the different parabolas as b varies. This is going to be examined in assignment 3.