Approximating solutions with graphing calculator
We're told this is the graph of ( y ) is equal to ( \frac{3}{2} ) to the ( x ) and that's it right over there. Use the graph to find an approximate solution to ( \frac{3}{2} ) to the ( x ) is equal to ( 5 ). 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 ( \frac{3}{2} ) to the ( x ); they graph it right over here. This gives us a hint, and especially because they want us to find an approximate solution, then maybe we can solve this equation, or approximate a solution to this equation through graphing.
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 ( \frac{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 ( 5 ), and we can graph that.
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. If I'm getting the same ( y ) value for that ( x ) value in both of these, well then that means that ( \frac{3}{2} ) to the ( x ) is going to be equal to ( 5 ). We can look at where they intersect and get an approximate sense of what ( x ) value that is. We can see, at least over here, it looks like ( x ) is roughly equal to ( 4 ).
So, ( x ) is approximately equal to ( 4 ), 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? ( \frac{3}{2} ) to the fourth power, is that equal to ( 5 )? Let's see. ( 3 ) to the fourth is ( 81 ); ( 2 ) to the fourth is ( 16 ). It gets us pretty close to ( 5 ).
Sixteen times five is eighty, so it's not exact, but it gets us pretty close. 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 ( 4 ). 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—we have this third degree polynomial right over here. Use the graph to answer the following questions: How many solutions does the equation ( x^{3} - 2x^{2} - x + 1 = -1 ) have? Pause this video and try to think about that.
When we think about solutions to this, we can say, "All right, well let's imagine two functions." One is ( y ) is equal to ( x^{3} - 2x^{2} - x + 1 ), which we already have graphed here, and let's say that the other equation or the other function is ( y ) is equal to ( -1 ). Then how many times do these intersect? That would tell us how many solutions we have. So that is ( y ) is equal to ( -1 ), and so every time they intersect, that means we have a solution to our original equation.
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 ( 2 ) have? Well, same drill: we could set ( y ) equals to ( x^{3} - 2x^{2} - x + 1 ), and then we could think about another function. What if ( y ) is equal to ( 2 )? Well, ( y = 2 ) would be up over there, ( y = 2 ), 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.