All right, so just a little basic algebra review on zeros and roots and x-intercepts. So what are all of these things? So I wanted to just use a basic example, something that we can easily graph to make sure that we understand this.
So I'm going to go ahead and let f of x be this quadratic function x squared minus 1. And let's say I want to find the zeros of this function, okay? What that means is I want f of x to be x squared minus 1. be zero. Okay.
So the zeros of this function are the same thing as saying the roots of the equation where I've let f of x be equal to zero. So the zeros of the function are the same thing as saying the roots of this equation. All right.
So what we're going to do is solve this equation to figure out what those x values are that make f of x equal to zero. All right. We can do this in a couple of ways. first, and I'm actually going to do this two different ways.
I felt like this was going to be a good review idea here. The first technique that I'm going to use is if I were to add one to both sides of the equation. Okay. So then X squared would be equal to one. And then if I want to take the square root of both sides, I need to remember that I could square either a positive or a negative.
And still get the same result. So if I square positive 1, I get 1. If I square negative 1, I also get 1. All right, the other method for this would be to factor. I'm looking at difference of perfect squares.
So I can factor x squared minus 1 into x minus 1 times x plus 1. And then I can set each of these factors equal to 0. And so for the first equation here, I would add one to both sides. For the second equation, I would subtract one from both sides. Either way, I look at this, my solution to this equation, my solutions to this equation would be positive and negative one, positive and negative one.
Okay, so the zeros of this function are plus and minus one. The roots of this equation are plus and minus one. And if I graph this function, which by the way, don't forget is just the parabola x squared shifted down one unit. So it's going to have a vertex at zero comma negative one, and it's going to open upward.
I'm going to do my best to make this symmetric, but that was bad. Over here to the right, I'll have an x-intercept of positive 1, 0. And over here to the left, I'll have an x-intercept of negative 1, 0. All right, and so we can find zeros of a function that's the same thing as finding roots of an equation in x. And then I want to find, I want to be able to graph the x-intercepts of this function.
y equals x squared minus 1. All right, so now let's go into a questioning type that will help you with your prerequisites. This will be quick. So if I hand you an equation in x or y or z, if I hand you an equation in a single variable like what you see here, all right, and I ask you to figure out what function do we need to find the zeros of if we're trying to find the roots of these equations. Let's come back. The root is of an equation set where you've got an expression set equal to zero.
Okay, so we're finding roots of an equation where we've got an expression in x set equal to zero. So that should be our first goal here. We want to set this equal to zero.
So for this first equation, it really doesn't matter how you want to do it. We could subtract x squared to the left and add 2x to the left, or I could do maybe a little less work and subtract the sine of x over to the right, which is what I'm going to choose to do. So if I subtract sine of x from both sides of the equation, I'm left with 0 equals x squared minus 2x minus sine x.
So this expression that I'm setting equal to 0 is the function. that I'm setting equal to zero. Remember, that's the whole point of zeros, right? We want to know when is f of x equal to zero. Well, the thing that I have equal to zero is going to be my f of x.
So the answer here, the function I want to find the zeros for is x squared minus 2x minus sine x. If you had done the alternative and subtracted x squared and added 2x, the zeros for that equation or the zeros for that function would be the same. So it doesn't matter.
Now let's talk about why that is. If I did the opposite and subtracted x squared and added 2x, let me write that down for you so you can just see it. So we would have then sine of x minus x squared plus 2x.
This function, okay, actually I'm going to give it a different name to be more... precise here because they are not the same function. They do have the same zeros, but they're not the same function. I want you to realize that this is the negative, g is the negative of f.
So essentially what you're looking at is a situation where, and this isn't the graph of this function necessarily, but what you want to be thinking about is one of these functions will be negative. you know, containing some number of zeros, right? Zero, zero, zero, zero, x-intercepts, right? And the other function, the other option, right? So if, you know, Joe writes this, but Brianne writes this, they're both correct because Brianne's option is just the reflection of Joe's.
But the question was, what are the zeros, right? And the zeros are the same regardless, right? So either answer would be correct here. All right, so now over here, same idea. We want to set this equal to zero, and that would be the function.
Then the expression that is equal to zero would be the function we're looking for. So I'll go ahead and subtract 3x and add 5 from both sides. So zero equals e to the x.
plus 4x squared minus 3x plus 5. All right. And so again, this is the function, this expression in x over here is the function that I am setting equal to zero. So the function I'm trying to find zeros for is e to the x plus 4x squared minus 3x. Plus 5. So maybe that's Stephen's choice. And then Jessica goes with 3x minus 5. And then she subtracts e to the x and 4x squared from both sides.
Either one of those will be correct answers. And I want you to be careful because when doing the prerequisite, we don't actually ask you to find the zeros. We just want to know if you can figure out what the function is, whose zeros we are trying to find.
OK, so it's not really that long of a question. It's not that much work to do. It's just making sure that you understand what's being asked.