Functions Notes 1.5

Added: 2026-05-11

[00:00:06]notes 1.5 geometric properties of linear functions let's go ahead and read example number one you need to rent a car for one day to compare the charges of three different companies company one charges twenty dollars per day worth an additional charge of 20 cents per mile so company 1 initial charge is 20 and 20 cents for every mile company 2 charges thirty dollars per day with an additional charge of ten cents per mile and then company three company three has a flat rate of seventy dollars this is for each company find a formula for the cost of driving a car m miles in one day well i just did that c in terms of m then graph the cost functions for each company from 0 to 500.

[00:01:23]okay so let's go ahead and draw an x-y-axis and in this case we're not using x and y our independent variable is m and c of m is our dependent variable okay so they want us to go from zero to five hundred zero to five hundred um you can just estimate i'm gonna go by hundreds 100 200 300 400 500.

[00:02:14]okay so 100 500. all right and then um our y-axis here i think i want to go by a hundreds um let's go by tens okay let's go by so let's say i call this let's start here 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80. so we're going by 10.

[00:03:05]they don't have to be the same on the x and the y as long as the equal increments here equal increments here okay so let's start by plotting the first fret our y-intercept or our initial condition is 20.

[00:03:22]and you really only need to plot um two points to determine a unique line so if we want to do say you know 500 if we want to plug in 500 into the equation it'd be 20 or 500 or 500 times 0.2 which is a hundred and then you add 20 so we plot this as 120. but that's going to run off our graph so let's go with something smaller let's go with say 300.

[00:03:58]if you plug 300 into this equation multiply that by 20 or 0.2 you get 60 plus 20 you get 80. so 300 80 is the point on the first graph so we're going to estimate that to be about right here and again it is just an estimate so let me connect those with a line and there's our c1 let's do the same thing for c2 c2 starts at 30.

[00:04:40]okay and how about if we plugged in 500 for that one or 400 it's completely up to you let's do 400.

[00:04:49]10 of 400 is 40 plus 3 is 70. so if we're plugging in this one we would get approximately here which is 70. okay and there's our second line and then the third one our third company is always 70 dollars always 70 dollars okay i missed that a little bit but we're okay there you get the idea okay so there is the graph it says how many miles would you have to drive in order for company two to be cheaper than company one company two could be cheaper than company one well let's see let's see when they're equal to each other let's find this intersection right here what does it look like about 100 let's see if that's right if we do company two equaling company one all right subtract 20 from both sides you get 10.

[00:06:19]subtract.1 m from both sides and you get.10 m divide by a tenth dividing by a tenth is the same as multiplying by ten so you're going to get m is equal to a hundred so that's when they're equal to each other okay so as you can see the pink after that the pink the second company is going to be less than the first company so you would have to travel more than 100 miles for company two to be less than company one and you can see that on the graph okay all right so using slopes and y-intercepts see if you can match each one of these formulas or equations to one of these graphs here you can stop the video and try it now all right let's see how you did b a c d and e y intercept slope y intercept slope so how do we know which one is slope 2 and which one is slope of 4 well let's take a look at these facts here what are the facts about the line that's in y equals mx plus b form well if you didn't get those right review these facts the y-intercept the b is also called the vertical intercept tells us where the line crosses what the y-intercept is where it crosses the y-axis when your slope is positive the line rises left and right so as you read the line is the line rising or as you read this way is the line falling when it's the slope is less than zero the line falls left to right now how do you know which one is steeper how do you know if this is 2 or this is 4 well when you take the absolute value of the slope the larger the value the larger the absolute value the steeper the graph okay next is a review of parallel and perpendicular from geometry so we have two lines we'll call this y sub one and y sub two slope one slope two okay what do you know when two lines are parallel well when they're parallel they have the same slope but they would have different y-intercepts same slope same y-intercept same line how about perpendicular what do you know about perpendicular lines well their slopes are opposite reciprocals of each other so basically change the sign and flip it over okay let's look at a couple examples here it says consider the lines at the right we know that this line right here is negative two it says one of the lines is negative two well i know that this is a negative slope and this is a positive slope so that slope is negative 2. how about the slope of the other line can you find that well they give us two points here this point right here is 0 negative 2 and this point right here is 2 0. you could use the slope formula or you could just count rise over run 2 over 2.

[00:10:58]so we know the slope of this one is equal to one now we also know the y-intercept here is negative two so if you know a y-intercept and a slope you would use slope-intercept form and again a review of slope-intercept form is y equals mx plus b that's slope intercept form so for line one we call this line one right here the positive one for line one our slope is one so it's one x plus the y-intercept which is negative two so plus a negative two you're just going to write x minus two there's your equation 4y sub 1.

[00:11:58]let's do the equation for this one we'll call this line y sub 2. y sub 2 we're going to use the slope intercept because we know the slope is negative 2 and we know the intercept is positive three there's your equation of the two lines now the question is find the exact coordinates of their point of intersection well as you can see here it's not very obvious we know it's less than two and below zero but we don't know what it is exact means leave your answers in fractional form okay so to find the solution or the point of intersection you're finding the solution of this system and because they're both solved for y we're going to go ahead and we are going to use substitution or basically sub this into here or just set them equal to each other so when are the equations or the lines equal to each other so solve for x if you want to stop the video and try it yourself you can we are going to add 2x to both sides we are going to add 2 to both sides and we are going to divide by 3.

[00:13:17]so that is not the answer that's the x coordinate that's the x value of this point right here does that is five thirds just a little bit less than two sure is but now we need the y value so the y value you can substitute x into here or here it doesn't matter i'm going to use the first one y is equal to five thirds minus two for y equals five thirds minus six thirds so i can get an exact answer no decimals no calculators gotta get used to that for calculus so that would give me a negative one third so my solution is the x value comes first then the y value my solution is five thirds negative one third okay so a little bit less than two a little bit below the x-axis that is your point of intersection or the solution to the system okay take a look at the last two here this one wants you to find the equation of the line that's parallel to 2x minus 3y equals 2.

[00:14:37]so that means they have the same slope that means they have the same slope so in the last uh section i showed you a shortcut if you have an equation in standard form ax plus b y so your b is negative plus b y equals c you could find the slope by taking the opposite of the a value divided by the b value so the opposite of your a value divided by your b value gives you two thirds so in this problem i have a point and i have a slope but i don't have the y-intercept i have a point and a slope so on this one you should use the point slope formula which just as a reminder y minus your y value equals your slope times x minus the given x value so in this case y minus our given y value is one the slope is two thirds x minus our given x value is one and that is the point slope form of the line which is acceptable if that's what they ask you for sometimes they may ask you for the slope intercept form so all you would do is you would take this and you would solve for y so we will do that quickly here multiply by two thirds we're going to multiply this by two thirds and we're going to add the one from the other side so slope-intercept form would be two-thirds x that's going to be three-thirds three-thirds plus a negative two-thirds gives you one-third that is the slope form of the line but when you're given a point and you're given a slope plug in the point here plug in the slope here and that is the equation of the line so let's see what we have on the last one yep same thing we know a point and we know the line is perpendicular not parallel but perpendicular so over here when it was parallel the slope was two-thirds but on this one the perpendicular slope is going to be the opposite reciprocal of two-thirds and we have a point and we have a slope so we are going to plug that into point slope just like we did on the last example so there's your point-slope form of the equation solve for the y a negative times a negative is a positive three halves i'm going to add this one from the other side so i get y equals a negative three halves x plus five plus five halves almost wrote five thirds plus five halves three over or two over two you get five halves now what if they asked you for standard form you go from point slope form you go to slope-intercept form and then from here put it in standard form standard form the x's and the y's are on the same side the most important thing though is that you can't have a fraction for your a in your b so we can't have this here so what do you think you would do to get rid of a fraction multiply both sides by the common denominator which is 2.

[00:18:50]so that's going to give you 2y equals a negative three x plus five add three x move the x and the y to the same side and that is standard form or 3x plus 2y minus 5 is sometimes also referred to as standard form which i think your book uses this but most books use this um either one is acceptable for standard form if it's multiple choice they will have one or the other and that is it