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Aztec ASE

Added Mar 23, 2022 •

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Construct and Compare Models and Functions

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are you guys welcome back to another hiset math lesson our final math lesson actually we finally made it through the entire course here we are today and we're talking about at how to construct and compare models and functions it's quite a lengthy lesson will split it up into two days today and tomorrow start off by going over our objectives you want to do is identify the key features of functions never going to want to construct and compare linear quadratic and exponential square root cube root logarithmic piecewise absolute value step polynomial rational trigonometry trigonometric functions wow that is a lot Justin one second piece they're going to build new functions from existing functions by transmission fourth solve problems using various types of all right let's start by talking about some of the key features of functions looking at a graph of specific functions is important because General key features of functions all right and intercepts are places where the graph is going cross over any access whether it's the x-axis or the y-axis that's what they're calling us here in the second one does in the scrap the Y intercept occurs at the point is 0 1 + 30s right where the line crosses the y-axis and the blue. Is showing where line crosses access a process into place this we also talked about yesterday talk about a minimum and the max you know with positive graphs having minimums and a negative graphs having maximums positive because it opens up and down words all right and sometimes you can have a situation in which we have both that's called having a relative maximum and a relative minimum so that means both our presents so we're just looking High Point and the low point on our function of thinking of it is mountains and valleys enough to help us locate these points on the function all right there thing we already talked about is the axis of symmetry talked about the axis of symmetry that's basically where we can draw a line down the middle of the function and have the same thing on both sides how's the temperature okay and then I'll do we have a positive function here right now it's positive because it opens up words values in that function that are negative right those are all the values that are going to be you know if we're looking at the Y axes are all below the zero if you look at the x axis there between the one and for this you shape part of the function that is below the x axis a are located okay so take a look at this graph and you guys tell me where does this line cross the x axis at what number do you see across from the Xbox what number is it awesome. it crosses from 654 + 3 + 2 + 1 just one number there's only one number where it crosses the x-axis look at the x-axis and see 3 number 3 what is the correct coordinate pair for the number three is it 03 or 30 3030 perfect the x value always comes first alright we also you know looked at a function like this where the different end behavior right left hand side of the function of words and so is the right hand side from where do we see the function increasing I'm sorry I can barely dear Daniel can you repeat the question again for you where do we see the function increasing you can see what that means eat. They don't have to have any measurement here so don't have any feet or inches are not measuring so from what interval do we see is increasing 3 -3 - 3 okay so this one you know you know it's it's kind of hard to say because usually you know when we have a function we have it in a week might have a linear function right or we may have a problem but in this case since the arrows are both pointing upwards I would say negative Infinity to positive Infinity so either direction it's it's increasing up right that's my my interpretation of the right for sure I know for a fact that this side let me a highlight I know for a fact that this side definitely is increasing right that for sure we know because that's that's typical line graph Behavior where from left to right going up the problem is with this one on the right you know we have no we normally look at a line from left to right but this is beginning at the origin point in moving the left branch of the moon in the opposite direction but since they're both moving up words I'm going to say negative or positive all right no good yeah they only selected the right hand side that one for sure we knew was was going on positive it's hard to say you know reason being is that you know both of these lines are facing upwards right so they both heading to the positive numbers on the y-axis right on the x-axis that's a different story right the x-axis you only have positive numbers to the right but on the y-axis you have positive numbers upwards right so go ahead and take a look at this skills check and then in the chat message me your response and then we'll review together as it pops all right let's go so let's let's discuss what number is so what number is asking so fat the ascent bi-weekly are you why oh no not the but that would be the y-intercept but the isotope is the number you know that the line approaches but never touches so now at least it's Cy equals to meaning. approaches you know the two marker you seem like the art of the line here if I put it under then it approaches at the Never all right what does influence mean all right to look at all the diff variations of functions that exist Hobby Lobby have a lot of flights all right linear function because Russians so let's look at some of the characteristics we can clearly see what it looks like when scrap write a linear function is in the shape of a line as expected by the name it has one white intercept that only crosses the y-axis one time and sometimes you process at the origin for the origin is at 0 0 you won't see any exponents on this function right so I'm going to have to all right. We have no exponents visible and then the end behavior means how does the line act as well if you're going to have one side that is decreasing or increasing depending when I go to where you look at it okay on to the next type of function this is what is called a quadratic function the quadratic function is what we're looking at a quadratic functions going to look either like letter U or an upside-down U and some of the features that it's going to have is just going to have its intercepts right so you know not necessarily but and I won't always have regularly. right one always have exit intercept could be none could be 1 or could be 20. Depends on how many times the function crosses the x-axis you could also have a y-intercept example right here it's at 4 in a quadratic function is always going to be raised to a power of two so I can receive the exponents going to the power of 2 you know that everything is that if it's a positive order to function it's going to have a minimum it's a negative quadratic function it's going to have a Maxima with the axis of symmetry we talked about ice capades Michigan where you draw a line down the middle and have the same thing on both sides so take a look at this this is called an exponential function right this one we didn't learn about right the only thing we learned about in regards to this is the how to put an asymptote where a portion of the graph approaches a number and gets closer and closer and closer but never touches it right you know what a task last is what is the Behavior right on this line to the naked eye it looks like it's constant right because it's usually a really what's happening is it's decrease right it's decreasing ever-so-slightly if we were to hypothetically drain all the water of the ocean but we were to drain it one drop at a time imagine that you would say oh is the water level remaining constant or is decreasing to the naked eye would look like it's you know constant right and maybe for years and years and years I probably would look on you know if we drain the ocean one drop by the time but in reality you know the water level would be decreasing ever so slightly so miniscule number is it looks in a one of those you know after beats of an exponential function is that you have slowed growth and then rapid growth for the huge Spike right if you want to look up an example of an exponential function human population over you know exponential function because we're starting off consistent and I'm spiking up dramatic so yeah so quickly or Shrink depending on what's Happening one of the sides of a square and they are the inverse of logarithmic functions don't worry about that we did not cover that but do you have any lessons on logarithmic functions exponential function about the attributes of the graphs all right because I compare in a quadratic and exponential function increase or decrease on both ends of the graph I have to either both or both signs are pointing down that's different from the exponential function right which levels off becomes nearly horizontal because you know it's not actually remaining five it's actually decreasing more and more more but never touched. so how do we describe this you have to be covered exponentially covered linear quadratic and so we should be familiar with how each of those what type of function does this look like I think I'm just seeing a or c one of those I don't know which one so you're right it is one of those so remember what shape does a quadratic function look like what letter does it look like are you so does that look like a u it's either going to grow quickly or Shrink but started a high number and then quickly decrease okay listen to befriend simple kind of function is this linear okay now we're looking at an equation so remember we add the properties of of the equations write equation is this notice the actual what do we have quadrant we have a power raised to two let's talk about the attributes of a graph of a square root function again don't be alarmed we've never talked about this before right we didn't do any exercises on this we're only talking about the attributes of it right we have them square roots like that we've done the square root graph any square root function before or learned about the property so this is the first time we're doing looks like a look you're the expert evil is within the radical so that means it's always going to be located square roots it grows slowly but continues to increase in One Direction as X approaches positive or negative Infinity Infinity could be aiming towards the left and it says the domain and range are restricted to positive real numbers since the value inside the radical and not be negative all right but does anyone I know why we can't have a negative number under the square root sign why wouldn't we be allowed to do and I don't think of why the reason why we can negative under the square root sign is because there's no number that when multiplied by itself will give you a negative right you would need a number with opposing science so takes a positive and and in order to create a negative number or you can't multiply the same number twice and get a negative answer all right another function that we did not discuss this is what it's called cube root function so this would be like asking oh what is the cube root eat right answer to that would be two meaning that number if you multiply that times itself 3 * 282 * 2 * 2 cube root of 8 let's look at the function does the acts variable is what the radical all right that's the same thing as a square root function has an index of 3 okay we expect Rose slowly but continues to increase in One Direction and decrease in the other direction alright so we can think of it as an extension right looks very similar to the square root function except we have the inverse going on on the left hand side you're the tell us that right that has opposite and behavior the domain and range or not restricted since the value inside the radical me positive or negative alright the reason why when we have to group we can put negative number is because if we multiply a negative number times itself three times you will end up with a negative right negative times negative I * R * a negative becomes so you can do it all right another function we didn't talk about logarithmic function looks a lot like the other function we just looked at Wright's looks take a look at some of the differences the X-ray able is within the logarithm slowly but continues to increase in One Direction and decrease in the other game similar shapes of exponential right just in a different 90 degrees the demesne is restricted sense of value inside the logarithmic and not be negative one side of the function purchase the extra never reaches it this is a vertical asymptote and they are the inverse of the exponential function you look very similar to each other all right just wanted this one YouTube vlog I swear all right what about this one Vlog Foursquare logarithmic all right so and a function is showing yes Peru all right so even though we don't have the two written here it's always assumed but it's square root night was cube root or 4th through 5th week whatever we would have a number that number written that the square root symbol open another type of function do you know we didn't go over before this is called voice function so you'll notice that it is broken down into different pieces so let's take a look at its attributes combination of two or more functions each function price two different portion of the domain so notice your we have you know one function which is a black line with a circle filled in and then we have a red line with a circle I thought I should be familiar for my lesson a inequalities we'll talk about it talk about we have a line under the inequalities symbol if so that means it's a closed Circle and if we don't open remember piecewise is going to be the ones that are broken apart alright absolute value we discuss absolute value in the past but we never talked about the function absolute value function that was one of the first ones that we looked at in this lesson where are the positive numbers well depends which value you're looking for positive are you looking for X values being positive or you looking for a y value being positive are all of the X values all the positive values are going to be to the right of the y-axis on the right hand side that's where all the positive X values are all the positive y values are going to be above the x-axis so where were you looking for all right look at its attributes all piecewise functions they are all pieces so you'll notice it's a piecewise function because you know the arrows are two separate functions that are basically connecting at at that one vertex axis of symmetry occurs at the vertex so I can that's that's where the two rays meet and you know it's also you know where to cut down does it have a minimum of a maximum or minimum at the bridge right between if it opens upward or downward trend and the end behavior is the same at both ends of the function so we both go upwards towards upwards for the positive numbers on the y-axis or you could say 1 heads towards the left negative Infinity the right with positive you know the arrow on the left when you should be heading up North West and the one on the right is heading Northeast okay system so if you find something that looks something that looks on or doesn't seem right take a screenshot or write down what lesson it's from when send it to me so I can take a look at it okay what is this one here look at the two vertical lines between the x - 5 absolutely all right I'll talk about the graph of a polynomial function the polynomial function means more than one that's the first point that's what they talked about right has more than one term separated has variables that cannot have negative exponents so we don't even need to worry about negative Xbox even seen that the entire class I mentioned briefly are you sure you're going to have a relative maximum and relative minimum RedTube do they have the relative minimum you know the lowest points on the function of the road maximum the highest point and then look at the zeros right the X intercepts you also call that as zeros or the solutions to the function of these are all the places where the graph crosses the x-axis and then let's talk about the end behavior so that's a pens so we have negative Infinity all right look at a rational function a lot of these look very similar you notice a lot of the same Trends with the graphs another type of function that we didn't discuss so this is this is a rational function or the display Vision Kia of X over G of x equals 0 second Point says have an x-ray able with a negative exponent 4 have an x-ray able in the denominator I should say not I'm have to ask him to on vertical or horizontal and have a domain restrictions because the denominator cannot equal 20 if never be 0 what's one of the main the main is all of the X values from left to right and range is all the way up and down okay look at this wavy function here I got another thing we didn't discuss this is a trigonometric represented by sine cosine tangent functions and it repeats with Peaks and throws the. 1 Peaks the next or one maximum or minimum another maximum amplitude is the height from the central line of the functions to the peak and then also kids are a toots are the same this okay so this one quite literally looks like waves maximum maximum okay so which one is this one I quite recognizable what is this see and what is this remember the polynomial one the graphic look at didn't look like this but it's not taking very bright as minimums relative minimum the maximum and then when I looked at the rational you know we saw a mirrored across the stagno all right graphs and tables of linear functions From Any Given linear function a table points can be created take the function f of x 2x + 3y substitute values for x and simplify to find the corresponding so we'll look at this page and I'll pass on 25 and do art do we have a linear function right you know we're told it's a linear function but we would also be able to tell just by looking at it because it doesn't have any exponents so here they do the first input for us the first input is the number zero so they replace x 10 and Saul ran so now we can do the same thing for one so what do I get when I input one what is my outfit what is my sign 703 9 here we are we have all of our inputs and outputs right so then we could go to a graph and put all these points 0 3 1 5 2 7 3 9 and again to graph a linear function you only need two points so once you have few points are all sets all you need you're the end of cracking it again another use for points you only need to write if they didn't put the last two there they would still the line would still go over those two points all right so we'll do this final skills check and I will pass there for part 1 so what am I going to get when I input one where do I get my info what about when I put two 7 10 cool already there

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