Mastering the Desmos Calculator for Efficient SAT Math Success

[Music] Nothing in life is certain except death and the uh Desmos calculator on the SAT

this calculator is actually a godsend but most people don't know how to use it including KH Academy themselves which

still tells you to solve every single Problem by hand I love you s but this is stupid Desmos is fast and Desmos is

accurate and in this video I'm going to tell you exactly how to use it and so do not let your Zoomer brain get distracted

unlike my fellow sat YouTubers I'm not going to put you to sleep this video is fast it's efficient and it's going to

help you improve your score so Now's the Time to lock in the first thing you want to know is that the Desmos desmos.com

calulator is pretty much the exact same version that you'll get on the SAT the only differences are you can't upload

images sure and you can't uh Play Sounds which I'm sure is disheartening for most of you okay now on to single variable

equations most people are sheep and they're going to solve this problem by moving X plus 6 to the other side

subtracting by 55 Distributing factoring splitting up rearranging and picking the positive solution but Sigma males like

myself are just going to copy and paste it into Desmos and look at where the vertical line is so even if you have no

idea how to do any of this as long as you know how to use a keyboard you should be able to solve any single

variable equation on the SAT whether it's a square root equation or an absolute value equation or a rational

equation or a quadratic equation or another quadratic wait why is there a t okay hold on guys remember I'm a sigma

male I've got this uh this question is asking for when there are no real solutions so I can just replace t with

all of the possible answer choices and then see which graph doesn't show any solutions and that's the correct answer

I know guys I know hold your applause systems of equations you're never going to guess what you do for

these plug them in asmos that's right baby we're going to Harvard now if you want to get technical you can think of

the red line as every single point that satisfies the first equation and the blue line is every single point that

satisfies the second equation so the point of intersection is the only point that satisfies both equations this works

for quadratics too as long as there are two equations and two variables you can find their point of intersection

sometimes there might be multiple points of intersection like in this goofy system now you may realize that every

single variable equation can be Rewritten as a system of equations and vice versa for example let's say we have

this single variable equation right here with Solutions 2 and 7 if we wanted to express this as a system you can set y

equal to the left side of the equation and y equal to the right side of the equation notice that the quadratic and

the line intersect at x = 2 and x = 7 which were our two solutions and so for the SAT you should have a conceptual

understanding of solutions you should understand Solutions in the context of systems of equations like this and you

should understand Solutions in the context of single variable equations like this and you might be thinking well

yeah I see how they relate but when would I ever want to convert one into the other well look at this problem if

we plug it into Desmos it seems like there's no solution right this is just like that one problem from 2 minutes ago

with the constant but if you split this up into a system of equations you'll see that these are actually just the same

line and a line intersects itself infinite number of times therefore there are actually infinite solutions not no

Solutions so be warned when doing what I did 2 minutes ago because no Solutions and infinite solutions of a single

variable equation look the exact same okay side tangent that has literally nothing to do with the SAT Math I bought

this green t-shirt off Amazon to try to use as a green screen but my editing software doesn't like it so it's just

this comically bad green screen but at least I only bought one t-shirt am I right like it could have been

worse just like systems of equations you can solve systems of inequalities just by plugging them straight into Desmos

Desmos will shade the regions that satisfy each inequality so for a problem like this where you're trying to find a

solution you don't need to look for the point of intersection but rather any point where the red and blue regions

overlap in that case this would be 140 now the SAT might ask questions that give you an in quality in context so in

this scenario we're buying apples and bananas but we have a price constraint and a quantity constraint and price and

quantity constraints are a very common problem type you'll see on the SAT and this question is not just asking what is

a valid point for this inequality it's specifically asking for the maximum number of bananas so the way we would do

this is first we' want to graph our inequality we can't use A's and B's cuz Desmos thinks those are constants so we

have to use x's and y's so you can think of every single x coordinate as the number of apples we're buying and the

y-coordinate as the number of bananas we're buying so for example this point right here means we're buying four

apples and two bananas and you can think of the green region as every combination within our budget and the black region

as every combination where we're buying at least six fruits and we're interested in the overlapping region where we're

simultaneously staying within our budget but also buying enough fruit and we're trying to find the most amount of

bananas we can get while being Within this overlapping region in this case bananas are y so we want to take the

greatest y within our region which is going to be 10 and you might be thinking well aren't there a lot of numbers

greater than 10 like we're still in the Shaded region when we're at y = 12 or Y = 14 or yal 16 but look at what happens

to X our X has to be negative for us to buy more than 10 bananas that means we're buying -3 apples to pay for our 16

bananas you can't do that you can't just just walk into the store and be like I'll take -3 Apples please um to finance

the rest of your purchase so just because the graph seems to indicate something doesn't mean that you can

ignore the context of the original problem but students do this all the time the question will ask for a

positive solution and they'll pick the negative one the question will ask for y and they'll pick X don't get lost in the

sauce and remember what you're actually solving for this is just general sat advice by this point of the video if you

are not already sold on how fast and easy Desmos is let me prove it to you I'm going to show you how many problems

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subscription okay next up quadratics if you don't know what a quadratic is it's a function where the highest powers to

and graphically it makes a parabola like the St Louis Gateway [Music]

Arch actually I'm mistaken this guy wrote a 10-page paper on why the St Louis Arch is actually not a parabola

Jesus Christ the point is quadratics are incredibly common on the SAT and Desmos is very much your friend anytime you're

asked to find the x or y intercepts of a quadratic just plug it into Desmos and click on the intercepts there you go and

by the way you can do this for other functions too it's just that those are usually easier to do by hand then

there's the vertex of a quadratic the vertex is basically the end of a quadratic it's the minimum or the

maximum and it's very easy to find if you plug the quadratic into Desmos and then click on it that's it now what if

they try to Spook you this problem isn't just asking for the minimum of f ofx it's asking for the minimum of f ofx +

5 people said I use the vine boom too much so I've replaced it with gong Sound effect. MP3 I think it's an improvement

okay back to the problem finding the minimum of f ofx + 5 is easy watch this [Music]

that's right it grabs the translation for you this is crazy you can also do this for any other sort of

Transformations so take advantage of this okay buckle up this next question is important in the XY plane the graph

of the equation Y = x^2 + 9x - 100 intersects the line yal C at exactly one point what is the value of C so we graph

our quadrant and our line into Desmos and for some reason it treats c as a variable rather than as a constant so

rewrite c as 0x + C and then you can add the slider but wait our C slider doesn't bring our line all the way down to the

quadratic so we need to click on our lower bound over here and set it to something much lower like 100 so now we

can bring our line all the way down to the quadratic but if we zoom in we'll see it's not actually touching the

quadratic when it's at -80 it's one too low and when it's at -79 it's one too high so instead let's manually input all

four answer choices into desos -41 over4 um nope that's not touching -00 nope that's not touching NE - 319 over 4

there we go but this seems much longer and more tedious than it should be if we acknowledge that the point of

intersection is the vertex of the quadratic then we don't even need this line we can just graph our quadratic

click on the vertex and that's our answer - 79.7 is -39 over4 the reason I went

through this whole laborious process is that this is what most of my students do and I want you to avoid this mistake

when there is one solution or one point of intersection between a quadratic and a horizontal line it's at the vertex

don't forget that now you might get unlucky and college board just gives you a problem where the line is not

horizontal at which point you kind of have to plug in every single answer choice but most of the time the line's

going to be horizontal and all you're doing is finding the vertex in this problem we're given a

quadratic equal to zero and told the equation has no real solutions when C is greater than n the easiest way to think

about no Solutions is with a system of equations so we want to do what we did earlier in the video and split each side

into its own equation now the question becomes simpler it's asking what does c need to be greater than for our

quadratic to be above the x-axis so we drag our C until the quadratic reaches the x-axis and see that any value above

289 is above the x-axis therefore 289 is our answer if it doesn't make sense what I'm doing go back through this section

and try it yourself I'm going quickly so I don't have to make 19 videos about this but if you actually want to

internalize this stuff you need to practice and think through these problems on your own now here's the

catch with problems that ask about no Solutions if there are no solutions for a system with a quadratic use Desmos if

there are no solutions for a system of linear equations do not use Desmos for example take this problem you're given

two lines and asked if the system of equations has no solution what is the value for a well if two lines have no

solution that means they never intersect if two lines never intersect that means they're parallel if two lines are

parallel they share the same slope I like to think of this as two cars traveling one behind the other if

they're going the exact same speed they're never going to crash just like how two lines that increase at the same

rate are never going to intersect so this question is very simple we just want the slope of the second equation to

be the same as the slope of the first equation so a equal 2 even if you get a harder variation of this type of problem

you can rearrange it into slope intercept form and then set the slopes equal to each other and yes you can go

into Desmos and move your slider around until the lines are parallel but this is very imprecise and you're too likely to

make an error so again use Desmos for no solutions to a quadratic use paper and pencil for no solutions to a system of

two [Music] lines okay this one's just funny to me

you can literally type median followed by your data set in parentheses and Desmos will give you your median it's

the same thing with mean it's as simple as that the thing is college board knows this so for every stupid easy mean and

median problem they'll give you one of the hardest problems on the entire test so if you're trying to get a say 700 or

higher you still need a strong understanding of mean and median but for the rest of you you kind of just plug it

in so for the rest of this video I'm going to be covering topics that are a little more ambiguous on whether or not

you should be using Desmos some tutors will say you should some tutors will say you shouldn't I'm going to share my

thoughts functions are weird in theory you shouldn't need Desmos for a problem like this if you're decent at algebra

and arithmetic you can do the math pretty much in your head but if you're the kind of person to mess up 56 divid 7

then Desmos is probably more reliable you can treat functions like single variable equations simply replace n with

X set the function equal to 56 and then you get your value for n as a vertical line same thing with this problem where

you're turning a function into a table you shouldn't need Desmos but it doesn't hurt you can go to this plus sign and

add a table and then for your table set your first column to X and your second column to x uh 2x^2 + 9 so then we can

input our values - 1 0 and 1 because that's what our answer choices have and it'll automatically populate the other

side of the table so we can select answer Choice a I've seen plenty of people use Desmos

to make circles but most Circle problems on the SAT are really just testing if you know the circle formula like this

problem if you know the circle formula you can clearly identify your Center and radius in less than 5 Seconds so

learning your circle formula should be the priority however there are problems like this one below where you're told AB

lies on the circle and asked which of the following is a possible value for a this is basically asking you which X

values on the circle and if you graph the circle in Desmos you can clearly see that the circle goes from -15 to 7 which

means -14 is the only possible x value so some circle problems are better with Desmos some are worse

you know it's an equivalent expressions problem when it asks you which expression is equivalent to this and

this is definitely not a problem type you should use Desmos for is it technically possible yes is it

optimal no please for the love of God just learn how to factor and do Algebra it comes in handy on so many different

parts of the SAT so just learn it and get used to it the greatest common factor of this expression is X so the

answer is the one that factors out X it's that simple you don't need to plug in every single answer choice to figure

that out you just need to give it 5 Seconds of thought and so to summarize everything from this video you should be

using Desmos 4 single variable equations systems of equations systems of inequalities number of solutions x and y

intercepts vertices of quadratics quadratics that intersect the line at one point quadratics that intersect the

line at zero points Computing mean and median directly and finding valid points on a circle you should not use Desmos

for linear systems of equations with no solution mean and median problems that aren't literally just finding mean and

median finding the center or radius of a circle equivalent expressions or literally anything else on the SAT if

you found this video helpful please like And subscribe and if you're interested in private tutoring from me personally I

know crazy you can go over to learn sat math.com in book of session I've been at this for 3 years now I really know how

to teach this test and I've helped countless students improve their scores thanks for watching and good luck

studying