20 SAT Math Questions You Can Annihilate Using DESMOS

Cornerstone Test Prep and Tutoring

6 Mar 202452:01

EducationalLearning

32 Likes 10 Comments

TLDRThe video script is a comprehensive guide on utilizing Desmos, a powerful online graphing calculator, to solve a variety of mathematical problems that students might encounter on standardized tests like the SAT. The speaker enthusiastically walks viewers through 20 different questions, ranging from basic to advanced levels, demonstrating how Desmos can simplify the process of solving equations, finding intersections, calculating areas, and even dealing with word problems involving exponential growth and statistics. The guide emphasizes the importance of active participation while watching, the use of a mouse for efficient navigation, and the capability of Desmos to handle both basic calculations and complex graphing. It also provides strategies for solving problems, such as focusing on details to avoid mistakes, using Desmos's built-in tools to find equations of best fit, and manipulating sliders to find exact values for variables. The script is not only informative but also motivational, encouraging students to practice with Desmos to build confidence and efficiency in tackling test questions.

Takeaways

📈 **Desmos Utilization:** The video emphasizes the importance of using Desmos for solving a variety of mathematical problems, including those that may be encountered on the SAT.
🖱️ **Mouse Usage:** It is recommended to use a mouse with a wheel for quick navigation and graph manipulation within Desmos, which can enhance problem-solving efficiency.
🧮 **Calculator Functions:** Desmos is not only for graphing but also for basic calculations, reducing the need to switch between tools during exams.
📚 **Reference Sheets:** The SAT provides a reference sheet with useful formulas, which can be leveraged alongside Desmos for a comprehensive approach to problem-solving.
✅ **Active Participation:** Encourages viewers to actively engage with the video and Desmos simultaneously to build muscle memory for efficient problem-solving.
🔍 **Zoom and Pan Features:** Using Desmos's zoom and pan features can help in accurately identifying solutions and understanding graph behavior.
📉 **Graph Interpretation:** The video illustrates how to interpret graphs in Desmos to find solutions to equations, such as using vertical lines to denote solutions and the vertex for finding minimum values.
🔢 **Direct Equation Input:** Desmos allows direct input of equations and will automatically solve them, which can be particularly useful for complex problems.
🔧 **Equation Manipulation:** Desmos can manipulate and find equivalent expressions, which is showcased through the process of testing multiple-choice answers by overlaying graphs.
📊 **Statistics Features:** Desmos can calculate descriptive statistics like mean and median, which can be useful for quickly solving statistics-based problems.
⏱️ **Time Efficiency:** The video script highlights that using Desmos can save time, allowing for quick solutions to problems that might otherwise be time-consuming.

Q & A

What is the main purpose of using Desmos in the context of the video?
-The main purpose of using Desmos in the video is to demonstrate how it can be used to solve a variety of mathematical problems efficiently, particularly those that are expected to appear on the SAT. It is used to illustrate how Desmos can handle equations, graphing, and even basic calculations to assist students in their test preparation.
What is the significance of using a mouse with Desmos?
-Using a mouse with Desmos, especially one with a wheel, allows for faster navigation and manipulation of the graph. It enables users to zoom in and out and pan around the graph quickly, which is particularly useful when trying to identify specific features or values on the graph.
What is the advantage of using Desmos for basic calculations?
-Desmos can handle basic calculations quickly and efficiently, which means students can avoid switching between their calculator and computer. It can provide immediate results for simple operations, saving time and effort during problem-solving.
How does Desmos help in solving the problem involving the triangle with an area of 104 square cm?
-Desmos helps by allowing the user to input the area formula for a triangle directly into the software. By setting the area to 104 and defining the base and height in terms of x, Desmos can solve for x, giving the value of the height when the base length is x - 3 cm.
What is the strategy suggested in the video for finding the Y-intercept of a function?
-The strategy suggested in the video is to input the function directly into Desmos and then pan up to find where the graph intersects the y-axis. Desmos will highlight the intercept point, and the user can zoom in to identify the exact value.
How does Desmos assist in determining the equation of an exponential relationship from a given table of values?
-Desmos can automatically generate the equation of an exponential relationship by inputting the table of values into the software. It uses the given points to form-fit an exponential equation that passes through all the points, providing the user with the equation in terms of parameters A and B.
What is the recommended approach for finding the intersection points of two lines in the XY plane using Desmos?
-The recommended approach is to input the equations of both lines into Desmos. The software will then graph both lines, and the user can visually identify the points of intersection. Desmos can also provide the exact coordinates of the intersection points.
How can Desmos be used to find the minimum value of a quadratic function?
-Desmos can be used to find the minimum value of a quadratic function by entering the function's equation into the software. The user can then click on the vertex of the parabola, which corresponds to either the minimum or maximum value, depending on the direction of the parabola's opening.
What is the strategy for finding the value of 'n' for line M that is parallel to the line represented by the equation Y = 7x + 5 using Desmos?
-The strategy involves entering the equation of the line to which line M is parallel and then creating a second line that passes through the origin (0,0) and another point with an adjustable 'n' value. By adjusting 'n' and using Desmos's graphing capabilities, one can visually determine when the second line is parallel to the first by matching their slopes.
How does Desmos help in solving a system of inequalities?
-Desmos allows the user to graph each inequality separately, showing the regions that satisfy each inequality. By overlaying the graphs, the user can identify the region that satisfies both inequalities simultaneously, which is the solution to the system.
What is the advantage of using Desmos for statistical problems, such as finding the mean or median of a data set?
-Desmos can quickly calculate the mean or median of a data set by inputting the data into the software. It automatically sorts the data and provides the correct statistical values, making it easier and less prone to manual calculation errors.

Outlines

00:00

📚 Introduction to Desmos Tutorial

The video begins with an introduction to a Desmos tutorial, emphasizing the importance of active participation and having Desmos ready for real-time practice. The presenter encourages the use of a mouse for efficient graph manipulation and highlights Desmos' capabilities in both advanced and basic calculations. A brief mention of the SAT test and the utility of a reference sheet is also made before jumping into the first question.

05:01

🔍 Finding Roots of an Equation

The presenter demonstrates how to use Desmos to find the roots of the equation x = 70x + 3, specifically focusing on the negative solution. By entering the equation directly into Desmos, the roots are found, and the presenter shows how to identify and input the correct answer.

10:02

📈 Identifying the Y-intercept of a Function

The video moves on to finding the Y-intercept of a given function F of x = 27*(2/3)^x. The presenter explains the process of entering the function into Desmos and adjusting the graph to find the Y-intercept, which is identified by the point where the graph intersects the Y-axis.

15:04

🧩 Solving a Triangle's Area Problem

The presenter tackles a word problem involving a triangle with a height of x cm and a base length of x - 3 cm, aiming to find the value of x given the triangle's area is 104 square cm. By setting up the area equation and using Desmos to solve it, the presenter shows how to find the positive solution, which is the length of the base.

20:04

📊 Fitting an Exponential Equation to a Table

The presenter creates a table in Desmos corresponding to an exponential relationship between time (Y) and total amount (D) in a savings account. Using Desmos' equation creator, the presenter finds the exponential equation that best fits the given data points, showcasing how Desmos can derive the equation from the table.

25:05

🤝 Intersection Points of Two Lines

The presenter discusses finding the number of intersection points between two lines represented by y = 7x and y = x + 18. By entering both equations into Desmos, the presenter shows that the lines intersect at only one point and guides on how to input the correct answer.

30:07

🔢 Solving a System of Linear Equations

The presenter solves a system of linear equations, 9x + 6y = 30 and 4x + 3y = 18, to find the value of x. The presenter emphasizes the importance of understanding the question's goal and using Desmos to find the intersection point of the equations, which directly gives the required value of x.

35:07

📉 Finding the Minimum Value of a Function

The presenter finds the minimum value of the function f(x) = 12x^2 - 30x - 162. By entering the function into Desmos and using the vertex feature, the presenter identifies the minimum value and its corresponding x-coordinate, demonstrating how to input the correct answer.

40:08

🏞️ Parallel Line and Point Intersection

The presenter solves a problem involving line M, which passes through (0,0) and is parallel to the line represented by y = 7x + 5, and also passes through the point (4, n). By using a slider for n and adjusting it until the line appears parallel to y = 7x + 5, the presenter finds the correct value for n.

45:09

🔍 Intersection of a Line and a Parabola

The presenter finds the value of P for a line with equation 4y = P, which intersects a given parabola Y = 5x^2 - 7x at exactly one point. Using Desmos, the presenter adjusts the value of P until the line intersects the parabola at a single point, demonstrating how to use the software to solve the problem.

50:10

📊 Desmos for Statistics and Data Analysis

The presenter shows how Desmos can be used to find the median and mean of a given data set. By entering the data into Desmos and using the median and mean functions, the presenter calculates and identifies the correct answers for both statistical measures.

🔧 Solving an Expression Equivalence Problem

The presenter tackles an expression equivalence problem, where given 28/(4x + 32), the task is to find the value of R such that the expression is equivalent to 7/(x + r). The presenter uses Desmos to test each answer choice until finding the one that matches the original graph, thus solving for R.

🚀 Modeling a Projectile Motion Problem

The presenter uses Desmos to solve a word problem involving an object launched from a platform, where the height of the object above the ground is given by the function h(t) = -16t^2 + 96t + 100. By graphing the function and applying restrictions, the presenter finds the height from which the object was launched.

📈 Finding the Height of a Cone

The presenter solves a geometry problem involving finding the height of a cone with a volume of exactly 2π cubic inches. By entering the formula for the volume of a cone into Desmos and solving for the height, the presenter finds the correct height of the cone.

🔗 Solving a System of Inequalities

The presenter demonstrates how to solve a system of inequalities in the XY plane to find which point is a solution to the system. By graphing the inequalities and identifying the overlap region, the presenter tests the given answer choices to find the correct solution point.

🔢 Finding a Value that Affects Mean and Median Equally

The presenter solves a problem where an integer X is added to a data set, and the goal is to find a value of X that makes the mean and median of the data set equal. Using Desmos, the presenter creates a list for the data set and adjusts a slider until the mean and median are the same, finding the correct value for X.

📈 Growth of an Online Newspaper's Subscribers

The presenter solves a word problem involving the growth of an online newspaper's subscribers, which triple every 6 months. By entering the growth formula into Desmos and adjusting a slider to find the correct multiplier, the presenter determines the value of n that represents the number of 6-month periods.

🔢 Finding the X-Intercept of a Shifted Line

The presenter finds the x-intercept of line H, which is a shifted version of line G that passes through specific points. By using Desmos to find the equation of line G and then shifting it up by five units, the presenter identifies the x-intercept of line H.

🍹 Solving a Beverage Mixing Problem

The presenter solves a classic word problem involving a beverage dispenser containing a mix of grape and apple juice. By setting up a system of equations based on the total volume of juice and the relationship between the quantities of grape and apple juice, the presenter uses Desmos to find the amount of apple juice in the mixture.

Mindmap

Keywords

💡Desmos

Desmos is an advanced online graphing calculator and mathematical tool used for visualizing and solving mathematical problems. In the video, it is the central tool used to demonstrate how to approach a variety of math questions, from basic to advanced levels. Desmos is shown to be instrumental in solving equations, understanding functions, and graphing various mathematical relationships.

💡Equation

An equation is a mathematical statement that asserts the equality of two expressions, often containing one or more variables. In the context of the video, equations are used to represent different mathematical problems, such as finding the solution to x in 'x = 70x + 3' or determining the relationship between two variables in an exponential function.

💡Graph

A graph is a visual representation of data in which certain quantities are represented by the coordinates of points. The video emphasizes the use of graphing to understand the behavior of functions, find intercepts, and solve systems of equations. Desmos allows users to graph functions and see their properties, such as vertices for finding minimum or maximum values.

💡Slope

Slope is a measure of the steepness of a line, indicating how much the y-coordinates change with respect to the x-coordinates. In the video, the concept of slope is used to determine when two lines are parallel, as parallel lines have the same slope. For instance, finding the value of 'n' for the line 'y = nx + b' that is parallel to 'y = 7x + 5' involves setting their slopes equal to each other.

💡Exponential Function

An exponential function is a mathematical function of the form y = a * b^x, where b is a constant. The video demonstrates how to use Desmos to find the equation of an exponential function that passes through given points, which is useful for modeling growth or decay processes.

💡Statistics

Statistics refers to the branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In the video, Desmos is used to calculate statistical measures such as the mean and median of a data set, which are important for understanding the central tendency of the data.

💡Word Problem

A word problem is a problem that uses a verbal, or written, description to present a real-world scenario that requires a mathematical solution. The video shows how to translate word problems into mathematical equations or functions that can be solved using Desmos, such as finding the number of ounces of apple juice in a punch based on given conditions.

💡System of Equations

A system of equations is a set of two or more equations that need to be solved simultaneously. The video illustrates how to use Desmos to solve systems of equations graphically by finding the point of intersection of the equations, which gives the solution to the system.

💡Inequalities

Inequalities are mathematical statements that compare expressions that are not necessarily equal, using symbols such as 'less than' or 'greater than'. The video demonstrates solving systems of inequalities using Desmos to find the region that satisfies all the given inequalities, which is useful for optimization problems and modeling constraints.

💡Volume

Volume refers to the quantity of space occupied by a solid object. In the context of the video, the volume of a cone is used as an example to show how to use Desmos to solve for unknown dimensions, such as finding the height of a cone given its volume.

💡Shift

In geometry, a shift or translation refers to moving a shape a certain distance in a given direction without rotating, resizing, or otherwise changing it. The video discusses how to find the equation of a line that has been shifted vertically (up five units) in the Cartesian plane using Desmos.

Highlights

Desmos can solve a wide range of mathematical problems, from basic calculations to complex equations and graphing.

The presenter emphasizes the importance of active engagement with Desmos while watching instructional videos.

Using a mouse with a wheel facilitates faster navigation and manipulation within Desmos.

Desmos is not only for graphing but also excels at performing basic calculations, making it a comprehensive tool for various mathematical tasks.

The SAT provides a reference sheet with geometry formulas, which can be accessed within Desmos.

Desmos automatically solves equations, such as the one involving x = 70x + 3, and identifies negative solutions.

For finding the Y-intercept of a function, Desmos allows you to pan and zoom to locate the point where the graph intersects the y-axis.

Desmos can handle area calculation problems, such as finding the base or height of a triangle, by inputting the area formula and solving for the variable.

The presenter demonstrates how Desmos can derive the equation of an exponential relationship from given data points.

Desmos simplifies finding the intersection points of two lines by inputting their equations and observing their common solution.

To find the minimum value of a function, Desmos can identify the vertex of a parabola, which corresponds to either a minimum or maximum value.

Desmos helps visualize and calculate the value of 'n' for a line parallel to another line and passing through a given point.

The presenter shows how to use Desmos to find the value of a constant 'P' in a line equation that intersects a parabola at exactly one point.

Desmos can calculate the median and mean of a data set, which is useful for statistical problems.

The tool can evaluate expressions and functions, such as finding G(3) for the function G(x) = 6 + x^3.

Desmos assists in solving word problems, such as calculating the height of an object launched from a platform given its height function.

The presenter illustrates how to use Desmos to find the height of a cone given its volume, using the cone's volume formula.

Desmos can solve systems of equations, such as finding the value of 'D' for which an equation has no solution, by graphing the involved functions.

The tool can solve inequalities and determine the solution set for a system of inequalities in the coordinate plane.

Desmos can be used to solve for variables in word problems involving growth rates, such as the number of subscribers to an online newspaper over time.

The presenter demonstrates solving a classic word problem involving a mixture using Desmos, emphasizing the importance of entering the correct variable as the solution.

Transcripts

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Desmos Tutorial SAT Math Problem Solving Graphing Calculator Algebra Geometry Test Preparation Educational Content Exponential Functions Word Problems Statistics