Beware the Double Crosses

by Arielle Alford

Below we find the graph of the two lines

We see that these fairly common lines intersect once.

So, what does the product of these two functions look like?

Let's define a new function which is the product of our two linear functions. So we have

Below is a graph of our new function h(x).

Hmmm . . . this looks a lot like a parabola. Is the hypothesis that h(x) is a parabola supported algebraically? Let's simplify h(x) and find out.

Our guess that h(x) is a parabola is supported since our simplified equation for h(x) is a second degree equation. Will our h(x) always be a parabola? Let's find out by making our linear functions f(x) and g(x) be as general as possible by using variables for the slope and y-intercept of each function.

Since f(x) and g(x) are both linear, then m and n can not equal zero. This means that the coefficient of our squared term can't be zero. This means that h(x) is, in fact, always parabolic when f(x) and g(x) are linear.

How does h(x) relate to its underlying linear functions f(x) and g(x)?

Let's put the graphs of f(x), g(x), and h(x) on the same axes and investigate.

In this case the parabola - our h(x) - crosses each linear function - f(x) and g(x) - two times. In other words, there are two points, or roots, where h(x) = f(x) and also two points where h(x) = g(x).

Will the parabola always cross each line twice (have two roots)?

Let's explore some situations where f(x) and g(x) are perpendicular. This time we'll let f(x) and g(x) be the following:

This means that

Below is the graph of our three functions f(x), g(x), and h(x) where n ranges from -5 to 5. Press play to see how the relationship between the three graphs changes when n changes.

How does the relationship between the graphs of the three functions change as n changes? Do you notice any situations where h(x) doesn't appear to cross f(x) or g(x) two times each? If so, what does the value of n appear to be when this occurs?

So, it looks like h(x) touches, but does not cross g(x), when n=0 but how can we be sure?

Let's prove it by calculating where g(x) and h(x) are equal (that is, where they intersect).

Remember that we want to find the n-value where h(x) touches but does not cross g(x). This means we are looking for a double root. Recall that double roots can be factored the following way:

When this is expanded we find that, in the case of double roots

So, for the solution to the equation we found to be a double root, it must look like the expanded equation of a double root. That is, the following relationships must be true:

And this confirms our hypothesis that when n=0, h(x) touches but does not cross g(x)! Let's look back at the graph of this specific case (when n=0). So we have the following functions:

Hmm, it also looks like h(x) touches (but does not cross) f(x) when n=0.

Let's check by finding out where h(x) and f(x) intersect.

And our suspicions are confirmed.

Was that just a fluke or are there other situations where h(x) is tangent to both f(x) and g(x)?

We'll again examine perpendicular lines, but let's let the y-intercept vary for both f(x) and g(x). So we'll have the following equations for our functions f(x), g(x), and h(x):

In this Graphing Calculator application we've chose 3 for 'a' and 7 for 'b'. How are the graph of f(x), g(x), and h(x) related? Now try out different values for 'a' and 'b' by simply changing 3 and 7 in our equations to other numbers. Do you find that the graph of h(x) usually crosses the graph of f(x) and g(x) once, twice, no times?

Now let's use some algebra to figure out what values of 'a' and 'b' will result in situations where h(x) is tangent to f(x) and g(x).

To see where h(x) and f(x) intersect, we'll set the equations for each function equal to each other and solve for the x-value where this is true.

Now we set these two functions equal to each other.

Since we want the solution to this equation to be a double root, it must follow the double root "pattern."

This means that the following relationships must be true:

Let's use these two equations to solve for our unknowns 'a' and 'b.'

Now we have to solve this equation to find out information about 'a' and 'b.' Let's make the following substitution to simplify matters.

Now our equation looks like this.

If we insert b+a for z in our equation we see that

So, any time the value of b = -a+1 in our equations for f(x) and g(x), f(x) will be tangent to h(x) at one point. But whether h(x) is also tangent to g(x) when b=-a+1 still remains. To see if this is true, we'll let b=-a+1 in our equation for g(x) and then find out where h(x) and g(x) intersect.

Ah hah! Just the the kind of root we were looking for - a double root!

So, the criterion that b=-a+1 produces a situation where h(x) is tangent to both f(x) and g(x). Use this file to insert your own values of 'a' into the formulas for f(x), g(x) and h(x) and visually check to see if this is true.