Lesson 21: Curve Sketching (Section 021 handout)

The document provides an overview of polynomials, including definitions, examples, and key concepts. Some key points covered include: - A polynomial is an expression with multiple terms and powers that can be simplified. - Polynomials have coefficients, degrees, roots, and can be written in nested form using brackets. - Methods for evaluating, dividing, and factorizing polynomials are discussed, including synthetic division and finding factors. - Graphs can be used to determine the equation of a polynomial function based on intercepts. - Approximate roots can be found between values where the polynomial changes signs.

The document provides examples of writing linear equations in standard form (Ax + By = C) and point-slope form (y - y1 = m(x - x1)) by converting between the two forms. It also gives examples of finding the slope and y-intercept of a line from two points to write the equation in point-slope and slope-intercept forms. Finally, it demonstrates finding the x- and y-intercepts of lines from their equations and using them to graph the lines.

This document discusses how to graph linear equations using slope-intercept form. It identifies the slope and y-intercept of several equations in slope-intercept form, such as y=3x+4 and 3x-3y=12. It then shows how to graph these equations by plotting the y-intercept and running the line with the given slope. As an example, it graphs the equation y=5x-3 on an xy-plane. It also demonstrates graphing parallel lines, such as y=3x+4 and y=3x+6, which have the same slope but different y-intercepts.

1. The document is from a Calculus I class at New York University and covers optimization problems. 2. It provides examples of optimization problems involving finding the maximum area of a rectangle with a fixed perimeter and the largest area that can be enclosed by a fixed length of fencing. 3. The document reviews strategies for solving optimization problems, such as understanding the problem, drawing diagrams, introducing variables, expressing the objective function, and finding extrema using calculus techniques like taking derivatives.

The document provides an assessment review with multiple choice questions about math concepts like algebra, geometry, and coordinate planes. It includes 15 questions testing skills like simplifying expressions, solving equations, factoring polynomials, and graphing lines. The questions are formatted with explanations of steps required to arrive at the answers.

This document provides an overview of integration and how it relates to calculating areas under curves and between functions. Some key points covered include: - Integration uses the concept of reverse differentiation to calculate the area under a function. - The area under a function between two x-values a and b is calculated as the definite integral from a to b. - Integrals may be used to find the area between two curves by subtracting their integrals between the intersection points. - Integrals can represent either positive or negative areas depending on whether the area is above or below the x-axis.

The document discusses techniques for clipping and rasterization in computer graphics. It covers line segment clipping algorithms like Cohen-Sutherland and Liang-Barsky. It also discusses polygon clipping, including brute force, triangulation, and a black box pipeline approach. Finally, it covers rasterization techniques for points, lines, and polygons, including inside-outside testing methods, fill algorithms like flood fill and scanline fill.

This document consists of an 11 page mathematics exam with multiple choice and free response questions. The exam covers topics including algebra, geometry, trigonometry, statistics, and financial mathematics. Students are asked to show working for credit and are provided graph paper, calculators, and other tools to aid in solving problems. The exam is designed to test skills and knowledge expected of secondary students in an International General Certificate of Secondary Education (IGCSE) level mathematics course.

The document provides instructions on graphing parabolas using vertex form and translations. It defines the vertex form of a parabola as y = a(x - h)2 + k, where (h, k) are the coordinates of the vertex. Examples show how to find the image of a parabola under a translation Th,k and graph parabolas by hand by determining the vertex and symmetrical y-values. Steps are given to graph a parabola as finding the vertex, symmetrical values, and filling in the graph.

This document contains 6 math problems involving graphing functions. Each problem has parts that involve: 1) Completing a table of values for a function. 2) Graphing the function on graph paper using given scales. 3) Finding specific values from the graph. 4) Drawing and finding values from a linear function related to the original. The problems provide practice graphing and extracting information from graphs of quadratic, cubic, rational, and other polynomial functions. The document demonstrates how to set up and solve multi-step math word problems involving graphing functions.

This document provides homework questions on solving equations and graphing equations. It includes: 1) Solving equations with one variable like 3x + 2 = 5x + 3 and determining the number of solutions. 2) Solving a two variable equation 2x + 3y = 12 and finding points that satisfy the equation. 3) Graphing equations like y = x^2 - 3x + 2 on a graphing calculator and determining features of the graph like where it crosses the x-axis. 4) Applying a transformation like (x,y) -> (x + 5, y) to the points on a graph. 5) Identifying points that do or do not

The document discusses graphing horizontal and vertical lines. It defines horizontal lines as having an equation of the form y=k, and vertical lines as having an equation of the form x=k. Examples of graphing specific horizontal and vertical lines are provided, as well as finding the equations of lines given points and finding intercepts of lines.