WEBVTT mathematics/algebra-1/smith
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Welcome to www.educator.com.
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In this lesson, we are going to work more with graphing functions now that we know a little bit more about them.
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What you will see is that when it comes to graphing functions, we use a lot of the tools that we use with graphing just a normal equation.
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We will see first how you can use a chart to plot a whole bunch of different points and graph out an entire function.
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The good news is if you have a linear function, we can often use our tools for lines to shortcut that process.
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Once we can look at the graph of the function, we will be able to determine whether it is truly a function using something like the vertical line test.
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More importantly, once we have the graph of the function we can test out what its domain and range is just from looking at its graph.
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Let us go ahead and take a look.
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Graphing a function is the same process as graphing an equation.
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We want to look at the relationship between the variables.
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In order to do that, we can simply develop a table of values.
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Remember doing this in our graphing of linear equation section, we may pick a lot of different values for x and see what the corresponding y value is.
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The only difference that we might make with function is simply use different notation.
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I would still pick a lot of different values for x but then I would simply see what the corresponding output is for my y values.
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Always keep in mind that when we are dealing with our inputs, those are going to be our x values.
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When we are dealing with our outputs, those are going to be our y values.
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We will still be able to plot them on a coordinate axis.
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Our x and y will correspond to the inputs and outputs of that function.
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Now if a function represents a line then I have some good news for you.
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You can use a lot of your techniques, especially about slope intercept form.
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Here is a function written in slope intercept form, you can see that it still has the n being slope and it still has b being the y intercept.
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The only change that I have done here is instead of writing out the y, I'm using my function notation.
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This just gives us the name of the function and tells us that our independent variable here is x.
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Since it is in slope intercept form, I could simply graph something like this by first using the y intercept as a starting point
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and then using the slope to identify another point on that graph.
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I have 2 points that I could connect them and then I'm good to go.
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To determine if the graph is a function or not we can use what is known as the vertical line test.
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The way the vertical line test works is you imagine a vertical line on the graph.
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As long as it only crosses the graph in one spot, then you can consider it a function.
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If the vertical line crosses the graph in only one spot and then we can consider it a function.
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If it crosses in more than one spot that is where we will get into trouble.
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Here I have two little diagrams, this one crosses here and here, we would say that this is not a function.
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This one, it only crosses in one spot and if I was to move that dotted line into a different spot over here, it still crosses only one spot.
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In that situation, I would say that this is a function.
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To determine the domain and range of a function when you are looking at its graph,
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think of tracing back all of the values that we used back to the x values on the x axis and back to the y values on the y axis.
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This looks a little difficult to do at first but it is not that bad.
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I imagine picking out some point out on the graph but if I am looking for the domain, I will trace it back to figure out what value on the x axis it came from.
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If I pick another point, trace that back where did it come from on the x axis.
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I do this for many, many different points I’m always tracing it back
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What I'm looking to do is trace back essentially every single point on that graph.
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Now what this would end up doing is I will end up plugging back many different points
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and they would end up shading in the domain of all the x values that we have used in the function.
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This one, if I trace this back trace it back, you can see that it creates that entire line.
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In this part of the line out here comes from tracing back values on this side.
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Even the ones that it can not see it, sure enough they traced all the way back.
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In a similar fashion, you can figure out the range by taking these points and going to the y axis.
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It is because the y’s represents the outputs, shading the axis, so you knew what was in your range.
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That way we get a little bit better sense of how to graph functions and things about them.
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Let us practice.
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Let us use the vertical line test to see whether these following relations are functions or not.
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The way a vertical line test works is we imagine a vertical line or test it to see if it crosses in only one spot.
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On this first one over here, let us go ahead and put down a vertical line.
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No matter where we move that vertical line, it looks like it will only end up crossing in one spot.
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Since it only crosses once, we will say that it is a function.
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With this one over here, when I put down a vertical line it is easy to see that it crosses in 2 spots.
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It crosses in two spots, not a function.
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The vertical line test says it must only cross in one spot at the most.
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Let us go ahead and see if we can graph one of our linear functions.
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In this one, it is a special type since it is linear.
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We want to look at the form to see if that will help us out.
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This one is written in slope intercept, I know that the y intercept is the 3 and I have a slope of -1/4.
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Let us start with our first being right at 3 and from that point I will go down 1 to the right, 1, 2, 3, 4.
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I have a second point, so I will connect the two.
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There is my line.
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I can also graph out this line by simply choosing a whole bunch of different values for x and evaluating them one at a time.
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I simply use the slope intercept form because it will be a lot quicker.
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I do want to point out that either way would be fine, just pick out some different things for x, plug them in and see what you will get for y.
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You use the graph to determine the domain and range of the function.
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This is unusual in that with last time we actually looked at the equation and tried to pick out what the natural domain was.
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In here, I just have the graph and I have no idea what is being used in here but I can see the inputs and outputs.
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Remember that is every single point on this graph here.
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To first figure out the domain, I will imagine all the points and trace them back to this x axis.
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What I'm doing is I’m figuring out what points we get shaded in on that x axis when I start tracing them all back.
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What it looks like it is doing is it is tracing out a lot of different values here.
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In fact, even my little point way out here we get trace back.
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I will end up shading quite a bit of the x axis.
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One thing to note is nothing is over on the side.
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The reason why I have nothing over there, is there is no graph to trace back to the x axis.
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Our domain looks like it would start here at -3 and it will go on and on forever from there.
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We can use the same process to figure out what the range is.
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We will simply take all our points now and trace them back to the y axis.
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We will see what this shades in as we do this, bring that one back and you can see I’m shading a whole bunch of different values here.
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In some places you might have more than one spot it traces back, but that is okay.
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Let us see, keep shading going to the y axis, this guy will go back to -2.
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Notice how below that I'm not going to shade anything on that part because there is no graph to trace back to the y-axis.
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What do we have for our range? Well, the lowest value I have here is that -2.
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We will start there and then it keeps going on from there since the rest of it is shaded.
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Let us do one more domain and range.
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Graph the relation and determine if it is a function then state its domain and range.
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We got a little bit to do with this one, let us first just develop a graph in it.
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This one is not a linear equation so I do not have too many shortcuts at my disposal.
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I’m just going to end up creating a table of values to help me out.
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Let us choose some different values like 4, 5, 8, and 13.
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These values will make it a little bit easier to evaluate this.
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If I was to use 4, I would end up with 2 × √4 - 4 or 2 × √0 which is 0.
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That is one point I know is on my graph, at 4, 0.
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Let us go ahead and put in our next value and that would be 5.
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2 × √5-4 that would be 1, √1 = 1 so I have 2 as this value.
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I will plug in 5 and I have 2 as my output.
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Looking good, let us try some more.
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Let us go ahead and put in the 8.
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8 - 4 would be 4 and the √4 is 2, 2 × 2 =4.
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This shows that when I put in 8 my output is 4.
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Let us put in 1, 2, 3, 4, 5, 6, 7, 8, 4, there you go.
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It looks like 13 is going to be off my-
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13 - 4 would be 9, 2 × √9, 2 × 3 or 6.
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That is another point on our graph.
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Looking at our points, I might as well start putting them together so we can have a nice little curve right here.
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Onto our first question, I was able to graph the relation, but is it a function or not?
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Does it pass the vertical line test?
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That is our question that we should be asking.
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If we imagine a vertical line on here, does it cross once, more than once? What is happening?
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What was this vertical line? I can see that no matter where I put it, it is only going to cross this graph in one spot.
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I will say yes it is a function since it passes the vertical line test.
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Now we have to figure out what is its domain and range.
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What are all the inputs we could use and what are all the outputs?
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In making this chart, I can already see some of the inputs that I used that you could potentially use even more than that.
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If we trace back all these values, it also includes all the numbers between the ones we used.
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I’m tracing things back to the x axis and it looks like I would shade in all of this.
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This starts at 4 and continues on from there.
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The domain would start at 4 and just go on from there, 4 to infinity.
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If I take all of the same values then I start to trace them back to the y axis it will shade in a lot of other values but it looks like nothing less than 0.
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We would shade in all that and now we have our range from 0 up to infinity.
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You can see that graphing functions are the same process as just graphing any type of relation.
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Keep track of your inputs and outputs.
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If it is a special type of function like a linear function, then use your tools for graphing lines.
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When it comes to the domain and range, look at your inputs and outputs by tracing all of the values back
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and then show the intervals of all the numbers that should be included.
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