Cochino’s math
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This document provides an overview of key concepts in functions and their graphs including: - Properties of lines like slope and equations of lines - Basic functions including domain, range, and the vertical line test - Transformations including translations, reflections, stretches, and combinations of functions - Polynomials and rational functions including solving quadratics and graphing rational functions - Exponential and logarithmic functions including their properties and equations - Conic sections including ellipses and hyperbolas with equations for different center points.
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My talk about computational geometry in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
4.2 notes
This document discusses how to graph linear equations in a coordinate plane. It defines a linear equation as one whose graph is a line and provides the standard form of a linear equation as ax+by=c. It explains that solutions to linear equations are ordered pairs that make statements true when substituted into the equation. To graph a linear equation, you create a domain of at least three convenient values, solve the equation for those values, and plot the points. Examples are given of graphing the equations y=b, x=a, y=3x-2, y=2.5, and x=-4. The vertical line test is introduced to determine if an equation represents a function.
Yocco41
(1) The document discusses cubic and quartic polynomial functions. Cubic polynomials are of degree 3, while quartic polynomials are of degree 4.
(2) The graph of a cubic function is a continuous smooth curve that can have up to two turning points, where the curve changes from increasing to decreasing. These turning points can be local maxima or minima.
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Investigation 8;06
1. The document discusses different families of lines graphed on a number plane. The red lines are parallel and have negative x terms in their equations. The blue lines are also parallel and have positive 2x terms.
2. Three lines intersect at the origin - the green line y=x, the blue line y=2x, and the red line y=-x. These lines all lack a constant term.
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1) Different forms of equations for lines (slope-intercept, point-slope, etc.) and quadratics (general, vertex).
2) How to determine if lines are parallel or perpendicular.
3) Methods for graphing and solving linear and quadratic equations (finding intercepts, using a graphing calculator, extracting roots, using the quadratic formula).
4) An example word problem on investing money with different interest rates and determining the amount to invest to earn a target return.
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11.6 graphing linear inequalities in two variables
This document discusses how to graph linear inequalities in two variables. It provides examples of graphing various inequalities, including 2x + 3y ≤ 6, 2x + y > 3, and y < 3x. The process involves graphing the boundary line defined by the equal case of the inequality and then shading the appropriate region based on checking if a test point satisfies the inequality. Key steps are to draw the boundary line as solid or dashed based on the symbols (< or > versus ≤ or ≥) and then shade the region containing points that satisfy the inequality.
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1. The document discusses different families of lines graphed on a number plane. The red lines are parallel and have negative x terms in their equations. The blue lines are also parallel and have positive 2x terms.
2. Three lines intersect at the origin - the green line y=x, the blue line y=2x, and the red line y=-x. These lines all lack a constant term.
3. Investigating the graphs shows that lines with a negative x term slope down as x increases, while lines with a positive x term slope up. The constant b relates to where the line intercepts the y-axis, with higher b above the axis and lower b below it.
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- A linear inequality describes a region of the coordinate plane bounded by a line. Any point in the shaded region is a solution to the inequality.
- To graph a linear inequality, first solve it for y and graph the resulting equation as a line. Then, test a point not on the line to determine which side of the line to shade based on whether it satisfies the inequality.
- The line is drawn solid for ≤ or ≥ and dashed for < or >. Shading the correct side of the line indicates the full solution set of the inequality.
5.2 solve quadratic equations by graphing.vertex and intercept form
Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where: a = 3
h = 0
k = 8
So the equation in vertex form is:
y = 3(x - 0)^2 + 8
Or simply:
y = 3x^2 + 8
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1. Absolute-value functions can be transformed through translations, reflections, stretches, and compressions. Translations move the graph up/down or left/right and change the vertex. Reflections flip the graph across an axis. Stretches and compressions make the graph wider/narrower or taller/shorter.
2. Examples show how to write the rule for a transformed absolute-value function g(x) based on an original function f(x). The transformations can be vertical/horizontal shifts, reflections, or vertical/horizontal stretches/compressions.
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The document discusses various functions including:
1. Coordinate planes and plotting points.
2. Linear functions defined as f(x)=mx+b and their properties like slope and intercepts.
3. Quadratic functions defined as f(x)=ax^2+bx+c and their properties like intercepts, vertex, and parabolic graph shape.
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Graphing Linear Inequalities
- A linear inequality describes a region of the coordinate plane bounded by a line. Any point in the shaded region is a solution to the inequality.
- To graph a linear inequality, first solve it for y and graph the resulting equation as a line. Then, test a point not on the line to determine which side of the line to shade based on whether it satisfies the inequality.
- The line is drawn solid for ≤ or ≥ and dashed for < or >. Shading the correct side of the line indicates the full solution set of the inequality.
5.2 solve quadratic equations by graphing.vertex and intercept form
Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where: a = 3
h = 0
k = 8
So the equation in vertex form is:
y = 3(x - 0)^2 + 8
Or simply:
y = 3x^2 + 8
Absolute value functions
1. Absolute-value functions can be transformed through translations, reflections, stretches, and compressions. Translations move the graph up/down or left/right and change the vertex. Reflections flip the graph across an axis. Stretches and compressions make the graph wider/narrower or taller/shorter.
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3. Graphs confirm that the transformations are applied as expected to f(x), resulting in the graph of g(x) with the
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The document discusses different types of functions including:
- Even functions which are symmetrical about the y-axis and have domains of all real numbers and ranges of zero to infinity.
- Quadratic and quartic functions in the form f(x)=a(x-b)n+c, where a controls dilation from the x-axis, (b,c) is the turning point, and b and c represent translations along the x and y axes.
- Odd functions where f(-x)=-f(x) and have domains and ranges of all real numbers.
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Absolute value functions have a V-shape and model situations involving distance or edges. The graph can be transformed by changing the slope (a), shifting the vertex horizontally (h), or shifting the vertex vertically (k). To graph, identify the vertex and axis of symmetry, then use the slope to sketch the right side and symmetry to complete the left. Writing the equation involves identifying the vertex (h, k) and slope (a).
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A polygon is a closed two-dimensional shape with straight or curved sides. It can be defined by an ordered sequence of vertices and edges connecting consecutive vertices. The scan line polygon fill algorithm uses an odd-even rule to determine if a point is inside or outside the polygon by counting edge crossings along a scan line from that point to infinity. Boundary fill and flood fill are two area filling algorithms that color the interior of a polygon or region by recursively filling neighboring pixels of the same color.
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This document discusses graphing linear inequalities in two variables. It provides definitions of key terms like half-plane and boundary. It also gives helpful hints for graphing different types of inequalities based on whether the sign is >, <, ≥, or ≤ and whether it involves just x and y or a slanted line. Examples are provided to demonstrate how to graph inequalities like y > 3, x - 2, y - 3x + 2, and an applied problem involving nickels and dimes. The key steps for graphing a linear inequality on the coordinate plane are outlined.
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The document discusses linear equations and graphing lines. It covers plotting points, calculating slope, writing equations in slope-intercept form, and graphing lines by making a table or using the slope and y-intercept. Methods are presented for finding the equation of a line given two points, the slope and a point, or from a graph.
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The document discusses absolute value functions. It defines absolute value and notes that the graph of an absolute value function is V-shaped and symmetric about the y-axis. It provides details on how to graph an absolute value function given its equation, including identifying the vertex and using the slope and symmetry to plot additional points and complete the graph.
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1. The document discusses various properties of graphs including symmetry, even and odd functions, translations, reflections, and stretching or compressing graphs.
2. It provides examples of applying these properties to determine if a graph is symmetric, even or odd, or to sketch graphs based on translations, reflections, or vertical/horizontal stretching and compressing of a original graph.
3. Several exercises are provided to apply these graph properties and transformations to specific functions and points on graphs.
2.8 a absolute value functions
1. The document discusses absolute value functions and how to graph and write equations for them.
2. It explains that the graph of an absolute value function is V-shaped and defines the vertex and line of symmetry.
3. Examples are provided to demonstrate how to graph an absolute value function by finding the vertex and slope and plotting points, and how to write the equation of an absolute value function given its graph.
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Computational geometry is a subfield of algorithms that deals with efficient structures and algorithms for geometric problems. It focuses on problems in computer graphics, CAD, modeling, and other applications. Key topics include surface reconstruction, collision detection, smallest enclosing balls, boolean operations on shapes, and arrangements of lines. Computational geometry aims to solve geometric problems without roundoff errors through techniques like CGAL, an open source C++ library.
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5.2 solve quadratic equations by graphing.vertex and intercept form
5.2 solve quadratic equations by graphing.vertex and intercept form
Notes on 3.2 properties of linear frunction graphs
Notes on 3.2 properties of linear frunction graphs
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- Content topics covered include limits of functions, continuity, derivatives and their applications (related rates, max/min problems), integrals, and differential equations.
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The domain is the set of all possible square footages, which is 1450 to 2100 square feet. The range is the set of all possible costs, which would be $75 * 1450 = $108,750 to $75 * 2100 = $157,500.
So the domain is [1450, 2100] and the range is [108750, 157500].
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Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where:
a = 3
h = -2
k = 8
So the equation in vertex form is:
y = 3(x - (-2))^2 + 8
y = 3(x + 2)^2 - 4
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This document provides an overview of different types of functions that will be investigated in an Algebra II course. It discusses linear, quadratic, cubic, exponential, and absolute value functions. For each function type, it describes their graphical representation and symbolic notation. Students are instructed to graph examples of each function type using graph paper and calculators to check their work. The document emphasizes that students will explore these and new functions in more depth during the Algebra II course.
Linear_and_NonLinear_Functions_a (1).pptx
This document provides an overview of linear and non-linear functions for an 8th grade math class. It defines linear functions as those that form a straight line when graphed, while non-linear functions do not. Linear functions can be represented by graphs, equations in slope-intercept form (y=mx+b), or tables with constant rates of change for both x- and y-values. Non-linear functions have curved graphs or tables without constant rates of change. Examples of linear and non-linear equations and tables are provided to illustrate the differences.
Lecture 01-2 (Functions).pptx
This document provides information about an Applied Calculus course taught by Imran Qasim at Mehran University of Engineering and Technology. The key points are:
1) The course covers topics in differential and integral calculus, including functions, limits, derivatives, integrals, and their applications.
2) Students are expected to have prior knowledge of functions, limits, and differentiation before taking the course.
3) The course contents will help students develop expertise in techniques for differentiation and integration, as well as apply calculus to solve real-world problems.
Ap calculus warm up 8.27.13
To find the limit of a function as x approaches positive or negative infinity, examine the degrees of the numerator and denominator polynomials. If the degrees are equal, the limit is the ratio of the leading coefficients; if the numerator degree is greater, the limit does not exist; if the denominator degree is greater, the limit is 0. Rational functions can have horizontal or vertical asymptotes. Examples show evaluating limits of rational functions by comparing degrees and leading coefficients as x approaches infinity. Practice problems apply these techniques to additional rational function limits.
Chap7 2 Ecc Intro
Elliptic curve cryptography uses elliptic curves over finite fields for public-key encryption. It offers the same security as other public-key cryptosystems using smaller key sizes. The points on an elliptic curve over a finite field form a finite abelian group which can be used for cryptographic operations like point addition. Point addition involves calculating the slope between two points and using it to find the x-coordinate of the sum point, while point doubling uses the tangent line to find the double of a point.
Linear Regression.pptx
Linear regression is a supervised learning technique used to predict continuous valued outputs. It fits a linear equation to the training data to find the relationship between independent variables (x) and the dependent variable (y).
Gradient descent is an algorithm used to minimize the cost function in linear regression. It works by iteratively updating the parameters (θ values) in the direction of the steepest descent as calculated by the partial derivatives of the cost function with respect to each parameter. The learning rate (α) controls the step size in each iteration.
Multivariate linear regression extends the technique to problems with multiple independent variables by representing the hypothesis and parameters as vectors and matrices, allowing gradient descent to optimize the parameters to fit the linear model
4.2 vertex and intercept form
This document discusses graphing quadratic functions in vertex and intercept forms. It defines key terms like vertex, axis of symmetry, and parabola. It explains how to graph quadratic functions written in vertex form and intercept form. Examples are provided for graphing functions in each form. The document also discusses how to change between the vertex, intercept, and standard forms of a quadratic equation.
PP1 Presentation
This document provides an overview of functions and their representations. It discusses the domain and range of a function, and how functions can be represented verbally, visually, numerically, and algebraically. It also describes the vertical line test used to determine if a graph represents a function. Inverse functions and the horizontal line test are introduced. Common types of functions covered include linear, polynomial, cubic, rational, and trigonometric functions. Mathematical modeling using functions is also briefly discussed.
Similar to Cochino’s math (20)
General Mathematics - Representation and Types of Functions
General Mathematics - Representation and Types of Functions
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UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
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What will you get from this session?
1. Insights into integrating generative AI.
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
ここ3000字までしか入らないけどタイトルの方がたくさん文字入ると思います。
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Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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Cochino’s math
- 1. Cochino’s Math By Jocelyn Hernandez, Leslie Zamudio, Horacio Sevilla & Yesenia Gonzalez
- 2. Functions and Their Graphs • Properties of Lines • Slope m=y2-y1/x2-x1, • Point slope y-y1=m(x-x1), Slope Intercept y=mx+b • Vertical X=Y Horizontal Y=X
- 3. Basic Functions, Functions and Graphs • Function: A function has a value in the domain with exactly one value in the range • Domain: The input or (x) • Range: The output or (y) • Vertical line test: if it passes it’s a function • Solving for a function: Algebraic, Numeric or Graphically • Continuity of a graph: Contiguous • Increasing and decreasing graphs. • Asymptotes: a line that a graph approaches but never reaches
- 4. Transformations (Shifts, Stretches, and Reflections) • Horizontal Translations: • Y=f(X-C) Translation to the right by C units • Y=f(X+C) Translation to the left by C units • Vertical Translations: • Y=F(X)+C Translation up by C units • Y=F(X)-C Translation down by C units • Reflections across the X-axis: y=-f(x) • Reflections across the Y-axis= f(-x)
- 5. Transformations (Shifts, Stretches, and Reflections) • Stretches and Shrinks: • Horizontal: Y=F(X/C) {A stretch by a factor of C if C>1} • {a shrink by a factor of C if C<1} • Vertical: Y=C*F(X) {A stretch by a factor of C if C>1} • {A shrink by a factor of C if C<1}
- 6. Combination of Functions • Sum: (F+G)=F(X)+G(X) • Difference: (F-G)(X)=F(X)-G(X) • Product: (FG)(X)=F(X)*G(X) • Quotient: (F/G)(X)=F(X)/G(X), Provided G(X) cannot equal 0
- 7. Polynomials and Rational Functions • Ways to solve a Quadratic Equation: Factoring, Using the Quadratic Formula, Completing the Square ax2 + bx = c • Polynomial Function • One-To-One Functions • Horizontal Line Test -If some horizontal line intersects the graph of the function more than once,then the function is not one-to-one. -If no horizontal line intersects the graph of the function more than once,then the function is one-to-one.
- 8. Polynomials and Rational Functions • Synthetic Division: • Real zeros and complex numbers -complex numbers are numbers such as : 4+3i , 5i+i etc... • real zeros are the intercepts of a quadratic equation
- 9. Polynomials and Rational Functions • Rational Functions :To graph a rational function, you find the asymptotes and the intercepts, plot a few points, and then sketch in the graph example equation:
- 10. Polynomials and Rational Functions • Fundamental Theorem of Algebra • Any polynomial of degree n ... has n roots, but you may need to use complex numbers example of a polynomial • this one has 3 terms • The Degree of a Polynomial with one variable is the largest exponent of that variable.
- 13. Exponential and Logarithmic Functions • Exponential Function : f(x)=a. bx • Logarithmic Function: y= Logbx • Properties of Logs: • Product: logbMN=logbM+logbN • Quotient: LogbMN=logbM-LogbN • Power: LogbNy=ylogbN
- 14. Exponential and Logarithmic Functions • Basic Common Logarithms Functions: • -Log101=0 because 100=1 • -Log1010=1 because 101=10 • -Log1010y=y because 10y=10y • -Loglogx=x because Logx=logx • Log always finds the exponent!!
- 16. Ellipses with center (h,k)
- 17. Hyperbolas