- [Instructor] We're told, this is the graph
of y is equal to 3/2 to the x.
And that's it right over there.
Use the graph to find an approximate solution
to 3/2 to the x is equal to five.
So pause this video and try to do this on your own
before we work on this together.
All right, now let's work on this.
So they already give us a hint of how to solve it.
They have the graph of y is equal to 3/2 to the x.
They graph it right over here.
And this gives us a hint, and especially because it's,
they want us to find an approximate solution,
that maybe we can solve this equation
or approximate a solution to this equation through graphing.
And the way we could do that is we could take
each side of this equation and set them up as a function.
We could set y equals to each side of it.
So if we set y equals to the left-hand side,
we get y is equal to 3/2 to the x power,
which is what they originally give us,
the graph of that.
And if we set y equal to the right-hand side,
we get y is equal to five.
And we can graph that.
And what's interesting here is if we can find
the x-value that gives us the same y-value
on both of these equations,
well that means that those graphs are going to intersect.
And if I'm getting the same y-value for that x-value
in both of these, well then that means
that 3/2 to the x
is going to be equal to five.
And so we could look at where they intersect
and get an approximate sense of what x-value that is.
And we can see it, at least over here,
it looks like x is roughly equal to four.
So x is approximately equal to four.
And if we wanted to, and we'd be done at that point.
If you wanted to, you could try to test it out.
You could say, "Hey does that actually work out?
"3/2 to the fourth power, is that equal to five?"
Let's see, three to the fourth is 81.
Two to the fourth is 16.
It gets us, it gets us pretty close to five.
16 times five is 80.
So it's not exact, but it gets us pretty close.
And if you had a graphing calculator
that could really zoom in and zoom in and zoom in,
you would get a value, you would see that x is
slightly different than x equals four.
But let's do another example.
The key here is that we can approximate solutions
to equations through graphing.
So here we are told, this is the graph
of y is equal to, so we have this third-degree polynomial
right over here.
Use the graph to answer the following questions.
How many solutions does the equation
x to the third minus two x squared minus x plus one
equals negative one have?
Pause this video and try to think about that.
Well when we think about solutions to this,
we could say, all right, well let's imagine two functions,
one is y is equal to x to the third minus two x squared
minus x plus one, which we already have graphed here.
And let's say that the other equation
or the other function is y is equal to negative one.
And then how many times do these intersect?
That would tell us how many solutions we have.
So that is y is equal to negative one.
And so every time they intersect,
that means we have a solution to our original equation.
And they intersect one, two, and three times.
So this has three solutions.
What about the second situation?
How many solutions does the equation all of this business
equal two have?
Well same drill, we could set y equals to
x to the third minus two x squared minus x plus one.
And then we could think about another function,
what if y is equal to two?
Well y equals two would be up over there,
y equals two.
And we could see it only intersects y equals
all of this business once.
So this is only going to have one solution.
So the key here, and I'll just write it out,
and these are screenshots from the exercise
on Khan Academy where you'd have to type in one,
or in the previous example, you would type in four.
But these are examples where you can take an equation
of one variable, set both sides of them
independently equal to y, graph them,
and then think about where they intersect.
Because the x-values where they intersect
will be solutions to your original equation.
And a graph is a useful way of approximating
what a solution will be, especially if you have
a graphing calculator or Desmos or something like that.