Explain why the x-coordinates of the points where the graphs of the equations y = 4^x and y = 2^(x + 2) intersect are the solutions of the equation 4^x = 2^(x + 2)

Question

anonymous · Answer

This is a good thinking problem. What does it mean when two lines intersect?

anonymous · Answer

Uh, I don't know how to answer that other than where two lines meet on a coordinate plain

anonymous · Answer

That's good enough, and what happens at the intersection that is unique?

anonymous · Answer

I don't know what you mean, but I suppose it's unique because its the only place they meet?

anonymous · Answer

It's unique because that is the only place where the coordinates of the first line and the coordinates of the second line are exactly the same

anonymous · Answer

So x1=x2, and y1=y2

anonymous · Answer

Now look at the example they gave you. We will label the first as y1 and the second as y2. y1=4^x     y2=2^(x+2)

anonymous · Answer

If we set 4^x and 2^(x+2) equal to each other, then that must also mean that y1=y2 right?

anonymous · Answer

So there is the first half of the intersection, now x1 needs to equal x2.

anonymous · Answer

Okay

anonymous · Answer

Hmm I'm having a tough time explaining this part, gimme a sec

anonymous · Answer

Hah, okay. :)

anonymous · Answer

this is because its the only point for which y have the same value for both the function... for a particular value of x

anonymous · Answer

What..?

anonymous · Answer

Oh, is that the answer?

anonymous · Answer

Oh okay I got it now. You want to find the x value that makes both sides equal, since that automatically means that all x's in both sides are equal.

anonymous · Answer

its the only point ... for which you will get value for both the functions same.... for a given particular value of x... this particular value is called the solution of x in the equation

anonymous · Answer

So that will determine what y is, and when y1 is also equal to y2, 4^x=y1=2^(x+2)=y2

anonymous · Answer

The explanation is a bit lengthy, so I'll try to summarize it. Both of those equations are equal to some y. When those y's are the same along with the same x's, you have an intersection.

anonymous · Answer

Okay

anonymous · Answer

Imagine plugging in random x's into both equations. Whichever yields matching y's is the answer

anonymous · Answer

Okay, thank you.