Let's take a closer look at that infamous U-shaped curve that shows up throughout mathematics, the parabola. Specifically, how does changing coefficients and constants in the general equation of a parabola affect the graph.
The equation for a parabola can take on a lot of forms. One that is commonly used is the "general equation" of a parabola. It looks something like this . . .
where a, b, and c can be any real number (for example 2, 1/9, pi).
So the question is, what do these numbers have to do with how the graph of our function looks? Open this file and take a moment to pick your own values for a, b, and c (try several different combinations) by changing them to numbers and examine the graphs of the resulting parabolas.
What features were similar in the graphs of your functions? What looked different?
To get a better sense of exactly what's going on when we change the coefficients of the terms in our general equation or the constant term (c), let's fix two of them and vary the third.
We'll begin by letting both 'a' and 'b' equal 1 and letting 'c' be any real number. So the equation for our parabola will be as follows:
Check out the graphs of parabolas with different values for c.
What do these graphs have in common? How are they different?
Check out a variety of parabolas that result from changing the value of the constant 'c.' Click on the play arrow at the bottom of the screen to see how the graph of the parabola changes as 'c' varies between -10 and 10.
All of our parabolas open upward. They don't seem to get wider or thinner. Hmmm . . . it looks like changing 'c' doesn't really change the shape of our parabola but just moves the graph up and down. Interesting . . .
These parabolas appear to have a common shape and face the same direction. Is there anything all these parabolas have in common? Perhaps a common point? What do you see about the graphs of the following parabolas?
From what we can see of these two parabolas, they don't appear to have a common point. If they did intersect (have a common point) it would have to be out of view, more toward y equals positive infinity. Move the graph window toward y equals positive infinity in this file and look for a point of intersection.
It sure doesn't look like these two ever meet. How can we be sure? Let's prove it!
First we'll pick two arbitrary values for 'c.' We'll call them 'k' and 's.' So the equations for our parabolas look like this.
To find where they intersect, we'll set them equal to each other and solve for 'x.'
And we see that the only time two parabolas like this will intersect is when 'k' equals 's' which is to say that they are the same parabola! So, as we suspected, these parabolas with 'a' and 'b' fixed and 'c' varying can't intersect.
Let's try it out.
The graph that results from letting 'c' equal 0 doesn't look drastically different from parabolas with other c-values.
Take another look at parabolas with various values of 'c.' Do you notice any connection between the value of 'c' and where the parabola crosses the y-axis?
It seems like each parabola crosses the y-axis at the point (0,c). Let's investigate the arbitrary parabola by letting the x-value be 0 to see if this is always true.
When we solve for y, we indeed find that y must equal c when x is 0. Therefore every parabola of the form crosses the y-axis at (0,c).
Do you think this will be true for parabolas having any values of 'a' and 'b' (not just a=1 and b=1)? Why or why not? Hmmm . . .check it out.
We can see that some of our parabolas cross the x-axis and some don't. What does it mean when a function crosses the x-axis? It means that at that point, the function's y-value equals 0. By inserting 0 for y and solving for x one can find out if and where the function crosses the x-axis. If we solve for the roots of our general equation (without inserting a numerical value for 'c') we can find out for what values of 'c' our parabola will have a real root, that is, cross the x-axis. When we substitute 0 for y we see that
To solve this equation, we'll have to use the quadratic formula.
In order to have real solutions to the quadratic formula, we can't have a negative number under the radiacal sign. That is to say the following:
And we discover that whenever 'c' is less than or equal to 1/4, our parabola will cross the x-axis and have real roots.
Now we'll let 'b' and 'c' equal 1 and let 'a' be any real number. Thus, the equation for our parabola is now
Check out the graphs of these parabolas with different values for 'a.'
When we varied 'c' we found that the shape of the parabolas didn't change, just their vertical position in the plane. Does this also seem to be true when we fix 'b' and 'c' and vary 'a'? How do these parabolas seem the same. What aspects are different?
Check out all of the parabolas that result as 'a' moves between -10 to 10. Press the play button at the bottom of the screen to begin the animation.
The shape of our parabolas certainly seems to change as 'a' changes. Some open upward, some downward, some are skinny, others more broad. But they do all seem to go through one same point. What is this point?
All of the parabolas we've looked at seem to go through the point (0,1) which is to say that they have the same y-intercept. How can we be sure that any 'a' value we pick results in a parabola that passes through (0,1)? Looks like it's time for another proof!
To start we'll let 'a' be an arbitrary real number. To find out the y-intercept of this arbitrary parabola we should let the x-value of our equation be 0 and then solve for y.
Ah hah! And we see that our arbitrary parabola goes through the point (0,1). This means that all parabolas of the form have a common point, namely (0,1).
Well, let's look at what happens to the equation for our function and make a prediction.
And we're left with the equation of a line. Let 'a' be 0 in this file to see if the graph is, indeed, a line.
Again we see situations where some values of 'a' create parabolas that intersect the x-axis while others do not. How can we tell what values of 'a' will produces real roots (or x-axis crosses) and which won't. Again we'll use our arbitrary parabola where b=c=1 and 'a' is any real number.
Like before, we know that the y-value of a point where a parabola crosses the x-axis must be 0, so let's let the y-value in our equation be 0 and solve for the x-coordinate. Be aware that the quadratic formula might pop up again . . .
For this equation to have real roots, it must be true that the number under the radical sign is not negative. That is to say the following must be true:
Therefore, for a parabola of the form to cross the x-axis, the value we pick for 'a' must be greater than or equal to 1/4. You can use this file to pick some of your own values for 'a' and see if this seems true.
So far we've seen that varying 'c' seems to change the vertical position of the graph and varying 'a' alters width and the direction a parabola opens. Now the question remains, what does changing 'b' do? Again we'll fix 'a' and 'c' to be 1 and let 'b' vary. Let's look at a few examples first.
How do the graphs look the same? What aspects appear to change as 'b' changes?
Press play to see how the graph of a parabola changes as 'b' moves between -10 and 10.
When 'b' moves between -10 and 10, the graph of the parabola seems to hop forward, that is it's vertical and horizontal position in the plane changes. However, the shape of the parabolas appears pretty constant. Additionally, all these parabolas seem to go through a common point somewhere near (0,1). Let's do some algebra o see if this is true for any value of 'b.' We'll start by letting 'b' be any real number and 'x' be zero.
And we see that when x=0, y=1. Since 'b' was arbitrary, we can conclude this will be true for any value of 'b'.
Like before we'll solve for the roots of our function
by letting y=0 and solving for 'x'. Here goes . . .
For our parabola to have real roots we know the discriminant (part under the radical) must not be negative. That means the following must be true:
So it looks like whenever 'b' is greater than 2 or less than -2, the resulting parabola will have real roots. Pick several different values for 'b' in this file to see if this seems true.
Changing 'c', the constant term changes the vertical position (y-intercept) of a parabola.
Changes 'a', the coefficient of the squared term changes the width of a parabola and the direction it opens (up or down). If a=o, the graph ends up being a line.
Chaning 'b', the coefficient of the linear term changes the vertical and horizontal position of the parabola. However, the general shape of the parabola doesn't change.
What about coefficients and constants of a cubic function . . .?
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