# MGF1107 unit 3 spring 2017

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### MGF1107 unit 3 spring 2017

Good morning, can you hear me? okay well this week is a busy week because we have three sections to cover so I’m going to go ahead and get started and this is the unit on linear and exponential functions. We’re going to start with section 6.6 and this is really just a review of graphing linear equations. I know you took MAT 1100 so this is all going to be pretty familiar to you The Cartesian coordinate system looks like this and we have two axes x and y They intersect at the origin Horizontal is x and vertical is y. When we plot numbers, points on the graph, we use ordered pairs of numbers. They’re always given in the form of a comma b For instance here if we want to plot the point 5, 3 We start at the origin positive numbers go to the right on the x and negative numbers to the left We go over 5 and then vertically positive numbers go up, negative numbers go down A quick review here I’m sorry I did it too quick what are the coordinates of the points? So C here You’re going to start at the origin you’re going negative direction on x and then a positive direction on y Point D you’re going to positive direction on x and negative direction on y So that point is 3. Also, this is in Quadrant 2 This one here is going to be 3 and negative 3 and is located in quadrant 4 Remember the quadrants are numbered counterclockwise starting here: 1, 2, 3, 4 Okay now we’re going to talk about graphing linear equations, first by plotting points. So the graph is an illustration of all the points whose coordinates satisfy the equation We can put equations in a couple different forms. One form is ax plus by equals C where a and b are not 0. These are going to be straight lines when they’re graphed They’re called linear equations in two variables x and y Since we only need two points to draw a line that’s really all we need to graph the linear equation. But we always suggest finding a third point just to make sure that you didn’t make a mistake on one of the first two points Again all three points on the same line means the points are collinear The first thing you want to do when you’re graphing is solve the equation for y and then pick at least three values for x and find the corresponding values of y Then plot those points An example; if we wanted to graph the line y equals 2x plus 2, first we’re going to pick some points Let’s see we’re going to find I’ve got the points already picked for us. We’re going to do x equals 0, 1 and negative 1 When we put 0 into the equation, 2 times 0 is 0, plus 2 gives us 2. When we put x equals 1 into the equation, 2 times 1 is 2, plus 2 gives me 4. When I put negative 1 in, 2 times negative 1 plus 2 gives me 0. So I’ve got 3 ordered pairs: 0, 2 1,4 negative 1, 0 and I can graph those points. Here they are on the line It turns out actually this first point is the y-intercept and the last point is the x intercept Now, you can also graph just using the intercepts. The point where the line crosses the x axis is the x intercept [ back to slide ] Example, here’s the x intercept, the point where the line crosses the x axis The y intercept is the point where the line crosses the y axis To do to the intercepts it’s easier to leave it in this standard form for a linear equation It’s better to leave it in this form than to solve for y like we did in the prior example. You’ll see when I go ahead to find the intercepts to find the x-intercept we set y equal to 0 and solve for x When it’s in the standard form the other term just drops out So we’ve got 2x equal to 6 so X is 3 The x intercept for this is 3, 0 Then to find the y intercept we’re going to let x equal 0 and again this term is going to drop out. We come up with y equals 2. So the intercept here, the x and y intercept rather, is 0, 2 Once I have the two intercepts I can graph the equation. We’ll do a checkpoint here just to make sure we have a third point so we’ll let X equal negative 3. So I put negative 3 into the original equation, that was 2x plus 3y equals 6 I get negative 6 plus 3 y is 6, add the 6, 3y is 12, y is 4 So the third point I have is

negative 3, 4 Here’s my graph, here’s my x-intercept here’s my y-intercept and there’s my checkpoint and I see that they fall right on a nice straight line. Now the slope of the line, which we studied in MAT 1100 The slope of the line is a measure of the steepness of the line. It’s the ratio of the vertical change. Remember the Cartesian coordinate system and the slope is the ratio of the vertical change, the change in y, over the change in x. We can use any two points on the line to find the slope vertical change over horizontal; y2 minus y1, divided by x2 minus x1, using any two points on the line. It’s also sometimes written like this the change in y, the Greek letter Delta means change, so the change in y divided by the change in x A couple of examples Positive slope: as x increases y increases so the line will look like this Negative slope: as x increases y decreases A horizontal line: has 0 slope. Everywhere along the line y equals some number A vertical line has undefined slope Example: find a slope of the line that connects two points We have negative 1, negative 3 and 1, 5 I’ll specify this point is point two and this is point one. So, y2 minus y1 over x2 minus x1 then I go ahead and simplify the numerator and denominator. I’ve got 5 plus 3 is 8 over 2, or 4. So the slope of the line that connects those two points is 4. The slope of 4 means there is a vertical change of 4 units for each each horizontal change of 1 unit This slope is positive so the line rises going from left to right. Here’s what it looks like. Here’s the line and those are the points that we used Remember the slope was 4 so if I move from this point here, the y-intercept, x increases by 1, y increases by 4 So I go from this point to the next point x increases by 1, y increases by 4 and that’s what they’re showing you over here on this graph You can read the slope right off of the line. Again, this was covered in MAT 1100 A nice skill to have is to be able to read the slope from the line You can also use one of the intercepts in the slope to graph the line The other form of the line, if you remember we had the form up here of ax plus by equals c, and we call that standard form This is what we call the slope intercept form You can always take the standard form and rewrite it In slope intercept we see y equals mx plus b where m is the slope and 0, b is the y-intercept. This is a nice form to have the linear equation in because we can simply read the slope and the y-intercept right from the line and we can graph using simply the slope and the y-intercept To do that you want to solve for y. Put it in slope intercept form. Find the slope and the y intercept from the equation and then plot the y-intercept and then use the slope to find a second point. Let’s take an example We want to write 4x minus 3y equals 9 first in slope intercept and then we’re going to graph it Here’s the solution. 4x minus 3y The first thing we did was we want to get y by itself. I subtracted 4x from both sides and so these are going to cross out cancel and then I’ve got negative 3y over negative 3. Divide both sides by negative 3 and so I end up with y Negative 3s cancel. Then I end up with negative 4 over negative 3 gives me positive 4/3x minus 3 So here’s my slope and my y-intercept is 0 negative 3 To graph it, y-intercept 0 negative 3 and then remember my slope is 4 over 3 so it means x goes up by 4, y increases by 3 Here’s the next point on the line and I can actually check that point. The coordinates of that are 3, 1 and I could put that back into my equation and check and make sure that it’s true. So if I put

in x equals 3 let’s go ahead and do it y equals … so I have 4/3 and I’m going to substitute in a 3 for x, minus 3 What happens here, I’ve got the 3s will cancel. So 4 minus 3 equals 1 and that’s the point that I had right here, and so that satisfies the equation It’s nice to check especially when you’re using this method to graph that you didn’t make a mistake when you applied the slope to find another point on the line; confirm that point makes a true statement in the original equation Now we’re going to go the other way we’re going to give you a graph and we’re going to say what’s the equation of the line. By looking at the line I can tell the y-intercept, it’s 0, 1 That part’s easy Next we want to find a slope Notice; what is the form of the line? Will it have a positive slope or a negative slope? That gives us some hint of the kind of answer It’s going to have a negative slope if I think about moving from this point to this point, x increases by 3 but y decreases by 1. So the slope is going to be negative 1/3. So to go from this point to this point, x is going up by 3 but y is going down by negative 1 So the slope is negative 1/3. The equation of the line is going to be y equals negative 1/3x plus 1 There’s my equation of the line so you can go both ways now you can take an equation like this and graph the line using just that equation and then you can also look at a graph and give us the equation of line now this is only true with a with a linear function obviously not with anything else So two horizontals and verticals We want to graph the line y equals 2 and x equals negative 3 Remember, the y equals were horizontal lines. That’s what that would look like and then the x equals are the vertical lines Remember… let’s go back a minute on this one the slope is 0 on the horizontals and on the verticals the slope is undefined Let’s finish up with a little definition on this section, the dependent and independent variables. This was in MAT 1100 When we’re graphing equations, especially true when we get to the regression section, we label the horizontal as x and the vertical is y. For each equation we determine the values for y by substituting for x, which we just went over. Since the value of y depends on the value of x we call the y the dependent variable and x the independent variable Always label the vertical axis with dependent and the horizontal with the independent. Remember, in some applications we don’t have x and y. We may variable labels that match the problem. So x and y are kind of the generics but a lot of times we’ll use other labels for the two variables On the horizontal it’s always going to be the independent and on the vertical is going to be the dependent variable Let’s skip up to Section 6.10 We just did the linear In 6.10 we’re going to expand this a little bit more and we’re going to look at linear and also exponential functions We’re going to talk here about relations and functions. A relation is any ordered pair of numbers or some kind of an association between the x’s and the y’s. A function is a special kind of a relation where every value of the independent variable corresponds to a unique value of the dependent variable The domain is a set of values used for the independent. The range is a set of values for the dependent There’s an example in your book I’m not going to go over here, but they’re looking at a function like this The equation says the cost of some product. Each product is \$1.29 so we multiply the product price times the number we sold to get the total cost. So the domain here are the n values. What values can I put in here for the number of items That’s what we call the input Then the output here is going to be the cost in this example. So if I buy no product I pay nothing If buy one product I pay \$1.29 and so forth The mapping occurs so that from the domain it maps on to only one and only one value in the range. Now, in the range multiple values from the domain

can be mapped but not the other way Let’s take a look at a way to determine from a graph if we have a function. If a vertical line can be drawn so that it crosses the graph at more than one point then x does not have a unique y and it wouldn’t be a function Notice here, this is a parabola which we’re not going to go over in this section. If I drop a vertical line here it only crosses the graph at one place so that tells me that that’s a function. We call this the vertical line test A couple examples Determine if the graph represents a function and if it does what’s the domain and the range We use that vertical line test. We drop the vertical line down and if you notice here when x is negative 2, y might be 1 or it might be negative 1 So it’s not a function. [Student question] Yeah, basically the vertical line test, let me show you the next one so you get an idea If you look at this one when I drop a vertical line here this is a function because the vertical line only crosses in one place That tells me in this case when x is around 3 my y value is here on the graph There’s only one output value On this graph here, if I go to x equals negative 2, I can either go to positive 1 or I could go to negative 1 So this is not a function. I don’t get a unique y for each value of x Notice on the y side, you don’t do it horizontally because you can use the y values more than once. It’s the x uniqueness that defines the function Let’s do the domain and the range on this one. Remember domain is x. What values of x could we use or does this graph define? Look along the horizontal axis and we see that X is going from about 0 up to about 8 Then the range goes this way You look to see what’s the smallest value that Y takes on in this graph and it goes from about negative 1 to positive 1. Does that make sense? Do you see the difference between the domain and the range? Domain means you look this way. So what values of x can I put into this function to get an output? Then Y goes this way [Student typing] You’re right. When we get to the exponential’s in a minute we have a lot of domain and ranges that go to infinity This one stops, notice the graph? We don’t actually do a domain and range if it’s not a function you look here in the y-direction on this graph here, if there was a little arrow here it would say we keep going and going so then you would have an infinity positive and negative infinity. But you’re right, a lot of the times they’re going to have infinity in the ranges. Okay, we just did the linear so this should just be a review here. The other thing we can do when we’re talking about functions is we change the y to say f of x because we’re talking about a function of x. We’re saying that the y is actually the function evaluated at that particular value of x. So f of 1, f of 2, whatever we happen to be doing. It’s nothing different in terms of what we graph The same thing: to evaluate the function we substitute the given value and determine f of that value for x For instance, if I have this function f of x equals negative 4 plus 7. Evaluate it when x equals 3 and also when x equals negative 2. So I’m going to get two answers What’s nice here is the terminology shows you what you’re doing so everywhere I see x I’m going to put in 3 I’m going to get negative 12 plus 7 is going to give me negative 5. Then I’m going to evaluate f of negative 2 Put in negative 2 for x. I get 8 plus 7 and I get 15 This is just like evaluating that we did in the 1100 class but we’re just using the functional terminology We already did the graph so this should be pretty much of a review The graphs of the linear functions are straight lines and these will pass the vertical line test except for a vertical line They’re graphed by plotting points using intercepts or the slope and the y-intercept. We just went through that in the first section

Graph this function using the slope and the y-intercept. From the equation we can see f of x is y. We can write this like this The slope is going to be negative 1. The y-intercept is 0, negative 1 which I can tell from the equation. We can plot the intercept and then use the slope to find a second point Because the slope is negative we know the line is going to be decreasing as x increases. So I plot the y intercept which was 0, negative 1, and then I’m going to use the slope of negative 1 to get some more points so I could go x increases by 1, y decreases by 1. x increases by 1 y decreases by 1 So there’s another point on the line and there’s my line. This is what we just did in section 6.6 so this should be a review Here’s another one: graph using the slope and the y-intercept The slope here is negative 2, y-intercept is 0, 3 I plot 0, 3 and then I’m going to use my slope As x goes up by 1 y goes down by 2. [Repeat] and I’ve actually now got three points on the line and I can graph it. You may want to check these points like this is 2 negative 1 make sure that makes sense in the original equation. We’re going to do an evaluation Find f of x when x is 0. So I substitute in 0 and then I get my y-intercept 0, 3 Then let’s do f of negative 1 That would be right here, f of negative 1 and I get positive 5. Notice that’s on the graph; 1 negative 1 positive 5 Let’s try an application or two in here Marta is part owner of a newly opened candy company The annual profit is given by this function; p of x equals 3.5x minus 15,000, where x is the number of pounds of candy sold per year If Marta sells 20,000 pounds of candy this year, determine her profit We’re going to evaluate the function when x is 20,000 The profit of selling 20,000 units is equal to 3.5 times the 20,000 minus the 15,000 and so her profits going to be \$55,000 So if she sells 20,000 pounds of candy she will make a profit of \$55,000 There’s something similar in the homework but they’ll give you the function and then they’ll ask you to evaluate Now let’s go to exponential functions Remember, we did cover this tin the 1100 class Exponential function. Many real-world problems can be modeled using an exponential function. It means things are increasing or decreasing rapidly In a linear, for every unit of change the relationship changes by a fixed amount which is the slope Exponential means we’ve got a rapid rate so we’ve got exponential growth or we’ve got exponential decay growth means increasing, decay means decreasing. One form of an exponential could be written like this: y equals a to the x. Or we can use the functional notation, f of x equals a to the x In both cases a is going to be positive and a is not going to be equal to 1 Then the value of x is going to determine whether it’s increasing or decreasing. Here is an exponential function on population. As the years go by the population is increasing exponentially, that curved line Here is the basic form We went over this in MAT 1100 so let’s go through the definitions here The basic form: P naught is the original amount present. If you think of a population, it’s the initial number of people, or if you think of radioactivity it’s the initial amount of the compound Then P of T is the amount present after T years, and a and k are constants We just said a is greater than 0 and it’s not equal to 1 Depending on what the exponent is, is going to tell us if it’s increasing or decreasing If k is greater than 0, a is greater than 1, we have an increasing function If k is less than 0 and a is still greater than 1, it’s going to be a decreasing function One special kind of exponential is

I pick some x values and I went ahead and did them on paper You can do them in your calculator When you put in a negative 3 here, when x is negative 3, it is 1/8. Put in negative 2 you get 1/4 and so forth. Notice here as x is increasing y is increasing. So we have a general idea of what the graph is going to look like Look at the points. Negative 3, 1/8 is pretty close to the x-axis and then negative 1, 1/2 negative 2 was 1/4 the y-intercept here is 0, 1 and then we had 1, 2 a 2, 4 , a 3, 8 and then we just connect the dots like that In your homework you’re not going to be doing it by hand you’re going to be picking the correct graph The first thing I would do is look at whether or not as x increases y increases or decreases because generally of the four choices you’re going to be able to eliminate one of them. Then take a look at your points Let’s look at the domain and range I’ll keep the graph up here. Domain means the x values. What x values can this function take on? Notice on the graph here there’s a little arrow on either end of the curve so that means it keeps going and going and going. Yeah, any, and this is when they’re going to write it as all real numbers or…. I don’t remember the answers in MyMathLab so it could be written as all real numbers or it might be written as negative negative infinity less than x… less than positive infinity So these are two ways to say it could be any number I’m not sure in MyMathLab which one they use but these are saying the same thing Let’s look at y now Y is in this direction. Notice the graph is entirely above the x-axis so what does that tell me about y? It’s never going to be negative And, it’s not ever going to be 0 because if you notice this is never ever going to go across that x-axis When we talk about the range here, range is going to be y is going to be strictly greater than 0 So does that make sense? That was where the domain and range came in I’m thinking on almost everything you’re going to do in Exponentials the domain is always going to be all real numbers The range may change depending on where the graph falls because sometimes they’re above or below the x-axis Does that makes sense? Okay, good Now the last section A lot of stuff this week a busy week, I know. after this will be two sections a week In spring semester the midterm comes a week later than it does in the fall and so we end up having to push more stuff in after the midterm it’s not a good thing I know but and it’s more logical to keep material together that goes together. Yeah I know I wish you would tell the higher-ups at the college we’ve been arguing this for a long time. We would prefer you to have the midterm before the spring break I think it would make a lot more sense but it falls on deaf ears. Anyway, yeah faculty have been lobbying for that for a long time but it’s never changed Anyway this will be the worst week I promise you but the three sections go together so that’s why I think it makes sense to put these three together Now, we’re going to do linear correlation and regression. This again goes along with and again it’s linear okay so what kind of all goes together here Correlation is used to determine whether there is a linear relationship between two variables and how strong it is Regressions is how we determine the line that best fits the data {laughing about long descriptions] okay Linear correlation coefficient We use letter r. It’s a unit-less measure that describes the strength of the relationship between the two variables and the relationship has to be linear It’s really important that you make a graph before you calculate the correlation coefficient. If the value is positive then one variable increases and