Part 1: Use Desmos to Ace the SAT

Summary

This video tutorial explores the use of Desmos, a powerful graphing calculator, for solving various questions on the digital SAT. The presenter begins with a basic introduction to Desmos, demonstrating how to graph equations, adjust settings, and utilize features like sliders for dynamic visualization. Through a series of SAT problem examples, including linear and quadratic equations, the video showcases how Desmos can simplify the process of finding solutions. It emphasizes the importance of understanding Desmos' functionality to efficiently tackle SAT questions, providing viewers with practical tips and strategies to enhance their test preparation.

Takeaways

• 💻 Desmos is a powerful graphing calculator accessible online at desmos.com, offering a user-friendly platform for graphing various equations.
• 🔢 You can easily graph basic functions and equations by typing them into Desmos, such as linear equations and quadratic functions.
• ✍️ For exponentiation, use the shift key and the number 6 to enter exponents in your equations, facilitating the graphing of polynomial functions.
• 🖋️ Desmos allows the addition of multiple functions in one graph, with each function automatically assigned a different color for distinction.
• 📚 The settings menu (gear icon) in Desmos offers customization options, including changing graph colors and adjusting grid visibility.
• 🔍 The 'Home' button quickly resets the graph to its default zoom level, making it easy to return to the original view after exploring different parts of the graph.
• ⚙️ Advanced settings enable manual adjustment of axis ranges and steps, enhancing the visibility of specific graph features according to user preference.
• 🛠 When inputting equations from the SAT into Desmos, removing '=0' might be necessary for the graph to display properly.
• 🔄 Desmos' slider feature allows dynamic exploration of how changing parameters affect the graph, aiding in solving equations with variables.
• 📝 The tutorial emphasizes the versatility of Desmos in solving a variety of SAT problems, including systems of equations and quadratic equations, by visualizing and manipulating graphs.

Q & A

What is Desmos and how can it be used for the digital SAT?

Desmos is a graphing calculator accessible online at desmos.com. It can be used to solve mathematical problems, such as those found on the digital SAT, by allowing users to input equations and visually see the graphs generated from those equations.

How do you graph a basic function, like 3x minus 5, on Desmos?

To graph a basic function like 3x minus 5 on Desmos, simply type the equation into the input field. Desmos will automatically graph the line corresponding to the equation.

How can you adjust the exponent in an equation on Desmos?

To adjust the exponent in an equation on Desmos, press the shift key and then the six key (which has the caret symbol '^') to enter the exponent mode (superscript), then type the desired exponent number.

What is the process to change the color of a graph on Desmos?

To change the color of a graph on Desmos, click the settings button (gear icon) next to the equation you wish to change, then click on the color option and select the desired color.

How can you reset the zoom to the default setting on the Desmos graph?

To reset the zoom to the default setting on the Desmos graph, click the home button located near the graph area. This will return the graph to its original zoom level.

What should you do if typing an equation exactly as it appears on the SAT into Desmos does not generate a graph?

If typing an equation exactly as it appears on the SAT into Desmos does not generate a graph, try removing the '=0' part of the equation. This often resolves the issue and allows the graph to be displayed.

How can you add a slider for a variable in Desmos and what is its benefit?

To add a slider for a variable in Desmos, input an equation with a variable (e.g., c) and Desmos will prompt you to add a slider. This feature allows you to dynamically adjust the value of the variable and see how the graph changes in real time, which is useful for visualizing the impact of different variable values on the graph.

What is a common workaround for graphing equations that don't directly support implicit equations of x and y on Desmos?

A common workaround for graphing equations that don't support implicit equations of x and y on Desmos is to add a term involving x that doesn't change the equation, such as adding '+0x' to the equation. This allows Desmos to process the equation.

How can Desmos be used to find the intersection point of two graphs?

Desmos can be used to find the intersection point of two graphs by graphing both equations and using sliders to adjust variables, if necessary. This visual approach helps identify the point(s) where the graphs intersect, which is particularly useful for solving problems involving systems of equations.

Can Desmos solve simple algebraic equations, and how?

Yes, Desmos can solve simple algebraic equations by graphing the equation. The point where the graph crosses the x-axis represents the solution to the equation, allowing users to find the answer without manually solving it.

Outlines

00:00

📊 Introduction to Desmos for the Digital SAT

The video begins with an introduction to Desmos, a graphing calculator accessible at desmos.com, highlighting its use for answering questions on the digital SAT. It covers basic functionalities, such as graphing linear and quadratic equations by typing in expressions directly and adjusting their appearance on the graph. The presenter explains how to use the shift key for exponents, add new functions, change graph colors, reset zoom with the home button, and modify graph settings like axis steps and grid lines for clearer visualization. The segment emphasizes the importance of familiarizing oneself with Desmos to efficiently solve SAT questions.

05:03

🔍 Troubleshooting Graphs and Leveraging Desmos Features

This section deals with troubleshooting issues when graphing equations directly from the SAT on Desmos, such as removing the '=0' from equations to display the graph. It demonstrates using Desmos to solve a specific SAT question about finding the value of a constant 'C' where a line intersects a parabola at exactly one point, showcasing the utility of sliders for changing variable values dynamically. The presenter shows how to adjust the slider range and step to find the precise value of 'C' for the intersection, emphasizing Desmos's capability to visually solve equations and the strategy to include necessary terms like '0x' for equations lacking an 'x' term.

10:04

🛠 Advanced Graphing Techniques for SAT Problems

The video progresses to more advanced uses of Desmos for solving SAT problems, including adjusting slider steps for more accurate solutions and demonstrating how to graph implicit equations by adding terms. It solves another SAT question involving the intersection of a parabola and a line to find a specific value, highlighting the need for precision in adjusting the slider for accurate intersection points. The explanation covers how to interpret Desmos's error messages and modify equations to meet its requirements, illustrating the platform's versatility in handling various mathematical problems.

15:06

🎓 Practical Examples and SAT Strategy with Desmos

In the final part, the presenter uses Desmos to solve additional SAT questions, focusing on equations involving quadratics and systems of equations. This includes tips on using sliders for variables and interpreting the graphical results to find solutions. The video also touches on solving equations directly in Desmos without algebraic manipulation, showing how the graph intersects the x-axis to reveal solutions. The conclusion offers study tips, encourages viewers to explore further Desmos resources, and invites feedback for future video topics, underscoring the tool's value in preparing for the SAT.

Keywords

💡Desmos

Desmos is introduced as an online graphing calculator that allows users to visualize and manipulate mathematical functions and equations. It plays a central role in the video, serving as a tool to help viewers understand and solve questions from the digital SAT. The script demonstrates various features of Desmos, such as graphing linear and quadratic equations, adjusting graph settings, and using sliders for dynamic visualization of how changing variables affects graphs.

💡Graphing

Graphing is a fundamental concept in mathematics and in the video, where it refers to the process of representing mathematical functions and equations visually on a coordinate plane. The script walks through how to input equations into Desmos and view their corresponding graphs, highlighting the importance of graphing in understanding mathematical relationships and solving problems, especially in the context of the SAT.

💡Digital SAT

The digital SAT is mentioned as the context for the video's content, implying a standardized test that evaluates mathematical proficiency among other skills. The video aims to equip viewers with strategies and tools, specifically through Desmos, to tackle the math portion of the digital SAT, demonstrating the application of graphing calculators in answering exam questions effectively.

💡Functions

Functions are a key concept in the video, representing mathematical relationships between sets of numbers. The script provides examples of entering functions into Desmos, such as linear functions (3x - 5) and quadratic functions (3x^2 - 4), and shows how these are graphically represented. Functions are central to the video's theme as they form the basis of many SAT questions and their graphical representations in Desmos aid in solving these questions.

💡Sliders

Sliders are a feature of Desmos highlighted in the video for dynamically adjusting values of variables in equations and observing the impact on graphs. This feature is particularly useful for visualizing how changes in parameters affect the shape and position of graphs, such as in the examples where the value of 'C' is adjusted to find where two graphs intersect at exactly one point.

💡Quadratic Equations

Quadratic equations are a specific type of polynomial equation featured prominently in the video. The script explains how to graph quadratic equations in Desmos and use the platform's features to solve SAT problems involving quadratics. Quadratic equations are used to demonstrate the application of Desmos in solving real SAT questions, such as finding the intersection points of parabolas and lines.

💡Intersections

Intersections refer to the points at which graphs of two or more functions meet. The video uses Desmos to find intersections between different types of functions, such as lines and parabolas, to solve SAT questions. Understanding intersections is crucial for solving problems related to systems of equations and analyzing graphical solutions.

💡Coordinate Plane

The coordinate plane is a two-dimensional plane formed by the intersection of a vertical (y-axis) and a horizontal (x-axis) line. In the video, Desmos is used to graph equations on the coordinate plane, demonstrating how mathematical relationships are visually represented and analyzed. The coordinate plane is foundational for graphing and solving equations in the context of the SAT.

💡Zoom and Settings

Zoom and settings in Desmos are discussed in the video as tools for adjusting the view and appearance of graphs. These features allow users to focus on relevant parts of a graph, change axis scales, and toggle grid lines for better visualization. These adjustments are essential for closely examining the characteristics of functions and their graphs, especially when solving complex SAT problems.

💡System of Equations

A system of equations is a set of two or more equations with the same variables. The video demonstrates using Desmos to graph systems of equations and find their solutions, particularly where the equations intersect. Solving systems of equations is a common SAT problem type, and the ability to visualize these systems graphically on Desmos is presented as a valuable strategy for test-takers.

Highlights

Introduction to using Desmos for the digital SAT

Overview of Desmos' default screen and basic functionalities

How to graph basic functions and create exponents in Desmos

Adding multiple functions and changing their colors for differentiation

Resetting the graph to default zoom and modifying grid settings

Adjusting axis scales and steps for detailed graph views

Troubleshooting graphing issues with equations equal to zero

Solving an SAT question on graph intersection points using Desmos

Utilizing sliders in Desmos to find specific values in equations

Exploring Desmos' keypad functionality for equation input

Demonstrating how to solve quadratic equation problems with Desmos

Using Desmos to find the number of real solutions in quadratic equations

Applying Desmos for system of equations to find specific intersection points

Clarification on interpreting Desmos results for SAT question answers

Showcasing the versatility of Desmos in solving a variety of SAT problems

Conclusion and encouragement to use Desmos as a study tool for the SAT

Transcripts

hey guys in this video we're going to

look at Desmos and how we can use it to

help us answer some questions on the new

digital sat

[Music]

[Applause]

so on the right here on my screen you

could see if you've ever used Desmos

this is the graphing calculator

and you could access it if you want to

just play around with it on your own you

could go to desmos.com calculator of

course you could just Google it you'll

find it easily but this is the kind of

the default screen that you come to when

you're on Desmos and before we get into

solving some questions from the SAT

let's just talk about some of the

functionality

components to Desmos that we're going to

need to know so of course you can just

kind of type in whatever you want 3x

minus 5 and you can see that that line

3x minus 5 is graphed there

if you want to do an exponent like if I

want to make it 3x squared I can hit the

shift key on my keyboard and then I can

hit the the six key which is like the

little up arrow thing which I'm sure you

know how to do an exponent but that's

how you do it so shift and then six and

it goes to that what's called a

superscript and you could put a 2. so 3x

squared and then if you just like hit

the right arrow from here it'll

go down back to the normal level so 3x

minus 4 maybe is that okay so that's how

you just kind of start with graphing

some basic functions in Desmos if you

want to add a new function you could

just click down here on the second line

down

and you could put in another function

like y equals 4X minus 1. and there's

that and you can see that the the graph

automatically changes to a different

color typically

um you could change it if you want I

don't know why you would care to do that

but you could hit the settings button or

that little

um what do you call that yeah the gear

and then you could click the color and

you could change it right and it changes

from green to Orange to whatever the

heck you want it to be okay but that's

not that that important if you scroll

you could scroll by dragging obviously

that's easy but if you want to get back

to the default Zoom of this graph you

could just always hit this home button

here and it'll go to the default which

was right there okay so that's helpful

if you're zooming out really far to see

a graph that's not anywhere close to the

origin and if you're maybe scrolling

like this and you need to just quickly

get back to the start you just hit the

home and it resets everything which is

really really nice

now another thing that you can do is if

you hit the the wrench there you can

change the settings so if you you know

just play around with this if you want

but if you don't want those little grids

grid lines you could uncheck that and

you could kind of see in the background

there it makes those disappear

um you could change the x-axis and the

y-axis so maybe instead of zooming it

with the zoomed in plus sign and zoom

out minus sign if you want to manually

change how far the x-axis goes or the

y-axis you can change that here you

could even change the step of the axis

so watch this is maybe important enough

to to really talk about so if I zoom in

notice that you can't see you know one

two three four all you see is zero and

then five is the next number but that

would be changing the steps so if you

check the wrench here click the wrench

you could change the step to one and you

could change the y-axis step to one and

that way you could see all the numbers

and if you don't want those little grids

grid lines you could Click Of course

like we already said uncheck the minor

grid lines and this is kind of a nice

way to see

the graph so I prefer that of course you

could do whatever the heck you want if

you zoom it out it makes them disappear

but you get the idea so that's kind of

the the basics of Desmos the other thing

that I should mention is sometimes when

we see equations on a on the SAT like if

we look at this second one down here 64x

squared plus BX plus 25 equals zero

don't worry about the B for now but if

we were going to type that into Desmos

and let's maybe even put a a number in

in place of that b so if it was

something like 64 x squared whoops I

totally messed that up 64x

whoa what's going on x

there we go squared

and then if it was just like Plus

5X

plus 25

. now if I hit equals zero like they

have there I'm wondering if this is

going to do what I want it to do

is there a graph there

I don't see anything graphed maybe I'm

missing it

but as far as I zoom out I don't see

anything and so what I'm going to do is

I'm going to take away the equal zero

and notice that there's now a graph

there so sometimes if you type in the

problem from the SAT exactly into Desmos

for whatever reason it might not like it

and so my suggestion would be to either

take away the equal zero at the end

which is usually what causes that to

happen or

um well that's that's a good way to do

it and if that works then of course you

know go for that and now yeah we can see

that it's a it's a parabola it's kind of

hard to see it but it's still like a

U-shaped graph and you can see the

vertex there the minimum point at the

bottom 0 25

that was just an example I made up but

yeah that would be that problem and why

don't we use Desmos now in order to

solve maybe this question here the first

one here so it says in the X Y plane and

that's a good hint to use Desmos because

we're talking about a graph a line with

equation 2y equals c for some constant C

intersects a parabola at exactly one

point if the parabola has equation y

equals negative 2x squared plus nine x

what is the value of C so really quickly

we've got two equations they're

intersecting but we want them to

intersect in exactly one point and so

we're going to be able to figure out the

value of C such that these graphs

intersect like they said in just one

point so why don't we start by just

typing in I'll do the parabola first

that quadratic equation and it would be

y equals negative 2 x squared

plus 9x and another thing that I should

say is in Desmos if you want you can

pull open this keypad and I think on the

SAT that might be the default setting so

that you have this here and I could even

move myself so that you could see

some of the the functionality of this

you have numbers you have exponents you

have square roots things like that and

so you could always use that keypad

there instead of manually typing it out

on your own keyboard so that's helpful

too

so anyway let's go back to

this problem so we've got our first one

we see the parabola there I can go back

to the home default Zoom if I want and

then I'll type in the other equation

which is a line 2y equals c so 2y equals

c and uh-oh notice this little

exclamation point it's like an error and

this is an example of what Desmos

doesn't like so we typed in the equation

just as we saw it but for some reason

it's not really working and notice what

it says there it says we only support

implicit equations of X and Y so I take

that to mean there's no X in this

equation

so I need to add an X in you might say

well I can't just add an X in and that's

true but think about it there's not any

x's in it right now so we could actually

put in there C plus 0x

right because

there's no x's in the equation and so if

I put 0x in there 0 times x is just zero

it doesn't change anything right so

sometimes you're going to need to think

like that and put in plus 0x

but that's a rare problem or it doesn't

happen too often but that's what you

would do there so now notice that desmo

says that we can add a slider for C so

I'm going to tap that blue button on C

and look what it does

it gives you this slider and this is so

cool this is one of the coolest features

of Desmos you can move it yourself to

change the value of C to see what the

graph would look like if C was for

example 5 or 7 and you could see it

moving up and up and up now what are we

going to look for remember we want these

graphs to intersect at exactly one point

and so if I keep dragging this bigger

and bigger notice that that line is

getting closer to the top of the

parabola at the vertex there and it

seems like that's where it's going to be

able to intersect it in just one point

but notice that I've gotten as big as

though let me get for C and I'm not

there yet so what I'm going to do is I'm

going to tap on this number here 10 and

that's going to allow me to change the

range of c and clearly I need it to be

bigger than 10 so maybe I'll do

something like up to 25 I don't know I'm

just guessing and maybe the minimum I

can make it just 10 because I know it

has to be bigger than that so this means

C is between 10 and 25 five and I'm

going to assume it's probably going to

be a whole number answer sometimes it's

not but usually it is so I'm going to

put a step of one that means it's just

going to go up by 1 every time instead

of decimals like it was last time I was

sliding it and now I can hit enter and

you could see now that the slider is

going to go from 10 to 25 so let me drag

it and we're going to look for it to

intersect in just one point and uh oh

it doesn't seem like it's intersecting

it in just that one point notice that

it's a little bit off still

and so what I need to do is actually

seems like it's going to be between 20

and 21. so what I can do is I can go

back and I can change that step so that

it's not a whole number anymore maybe

it'll be like a half step so 0.5 and

maybe that'll help so 20 doesn't work 20

and a half also doesn't work but it

looks like our graph the vertex that

we're looking to to line up with is just

right between those two points right or

those two numbers so if I go down from a

half of a step to maybe a quarter point

25

that's probably going to work and so

there we have it 20.25 and you can see

that they intersect right there in that

one point and if I even zoom in more

drag this up you can see that it's

definitely just intersecting it in that

one point now there's of course other

ways you could do this without Desmos

but the point of this video and me

showing you this example is just to show

you that an example like this doesn't

need to be solved in Desmos but it

certainly could be and so you just need

to know what to look for when you're

using Desmos for these types of problems

so that first one we got it's 20.25 now

why don't we clear all these out you can

just hit these x's and it'll get every

get rid of everything and we'll go back

to the the home default zoom and why

don't we go over here and look for a new

problem that we can solve in Desmos so

we'll do this one next

64x squared we already kind of looked at

this one 64 x squared

plus BX

and you can see that it wants a slider

which is good equals zero now remember

the way I typed it in would that equal 0

before it didn't really seem to work and

you could see now that it's still not

graphing it so this is where again we're

going to take out the equal zero and now

it's okay it just needs that slider

which is great so let's go and add that

slider and it says in the question for

which of the following values of B will

the equation have more than one real

solution now in the context of a

quadratic like this a solution is an

x-intercept so if it has more than one

solution a real solution it just means

it's going to have two x-intercepts it's

going to cross this x-axis twice the

question is which B value does that now

you have two options you can take each

answer and plug it in in place of B so

you could just take out B here and put

negative 91.

and you could see what the graph does

now we actually got lucky here because

that first one we tried negative 91 we

don't even need this slider anymore

because look that graph does cross the

x-axis twice that would be more than one

real solution and therefore the answer

is just a negative 91. now if you didn't

plug in the answer like we did what you

could have done of course is like we

were going to do before put that slider

in and if you add the slider and change

it

um let me zoom out so that we could kind

of see the graph see it's way up there

if you drag it it's not really moving

much you might even not even be able to

really see it moving but that's because

these values for B that we need are big

numbers so I'm going to change the range

again and I need it to be

the smallest number is negative 91. so

why don't I put negative 91 and then the

biggest B value is 40 so I could type 40

and that would be the range of B and I

could put a step of 1 here and that

would allow us to to see what happens

and you could see clearly that when we

drag it to negative 91 it crosses the

x-axis two times like we already saw and

all the other numbers negative 80

that crosses it just one time

and positive five I mean you get the

idea this isn't going to work right it's

way up high it's off of our picture and

40 is as well okay so

either way we still get the same answer

it's a negative 91. hopefully you can

see just another good example of a good

way to use Desmos to get an answer very

very quickly if you did this problem

using algebraic methods without a

calculator it would just take you a

little bit longer unless you're really

really confident in using maybe the

quadratic formula the discriminant

things like that now why don't we just

take a look at maybe one more question

this is just another kind of similar

type of problem but it's another one

that we can use Desmos for so we'll get

rid of these and go back to the home

zoom and here's another example where it

says that they intersected exactly one

point so we're going to do essentially

what we did in the the one a little bit

ago I'll type in 2x squared and I don't

know if typing in the Y in front of it

is going to make a difference I think

it'll be fine so why don't I do y equals

2x squared minus 21x

plus 64. whoops what the heck

Plus

64. okay there it is graphed no trouble

with putting the Y there and then y

equals 3x plus a and here's our chance

to put a slider which is what we want

and we want these to intersect in just

one point but notice that that line has

a certain angle this time it's got a

slope of three so it's not just a

horizontal line like we saw in that last

example so this one might be a little

bit trickier to see but we know that the

a value

oh this one's actually different so

first of all we'll slide a so that that

line intersects the parabola just once

and you can see it's getting closer and

closer and it seems like negative eight

first of all it's a whole number so

that's a likely candidate for the answer

and of course now we can go up here and

we can see that if we zoom in and go

this way that these guys intersect in

yeah just one point which is right there

it looks like 6 10 okay so that is the

answer for where they intersect that's

good we're on the right track but do not

just pick negative eight and move on

because a is negative eight but they

don't ask for a they ask for the value

of x so be very very careful this means

that at the intersection point which is

what they say here they intersect at

this point x y that would be the point

that we just found here 610 and I can

just click that and it'll show me that

point so 610 and so the value of x at

that intersection point would be 6 and

our answer would be C okay so there's a

lot of different types of problems we

can use to use Desmos to find on the SAT

a lot of them have to do with quadratics

or systems of equations where we have

two lines or graphs intersecting I just

thought of one more that I want to show

really really quickly and I'm just going

to make it up off the top of my head to

show you that even a a simple problem

like solving equations can be used in

Desmos to find the answer so an example

might be if they gave you 4X minus and

let's try to make it really complicated

because

we want to see the power of what Desmos

can do x minus 7 equals let's say X Plus

9. okay so that might look pretty

complicated and certainly you can solve

it and I'm sure most of you know how to

solve that you would distribute the

negative two you would get all the X's

to one side get all the constants to the

other and solve for x but guess what I

put it in Desmos and I didn't even have

to think look this vertical line forget

about the fact that it's vertical the

line crosses the x-axis here at negative

five that means the answer for this

problem the value of x would just be

negative 5.

so easy so even on simple equation

problems like this if you don't want to

waste the time to solve it out and risk

making a mistake as long as you just

type it in correctly Desmos will show

you the answer because it's going to be

where that line crosses the x-axis so

um yeah we'll stop here there's plenty

of question types that you can use

Desmos for hopefully this was a good

summary of that and um yeah best of luck

to you when you're studying feel free to

check out our blog where we have good

resources on the SAT and on Desmos you

can check out our program methodize

which maybe you're already using right

now and you can also

take a lot of practice tests from the

sat's official website so good luck in

your studying and reach out to us if you

have any questions

hey thanks for watching let us know in

the comments if there's any topics that

you want to see a video on going forward

be sure to like And subscribe and check

out our channel for more content