2.8 graphs of factorable polynomials t
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This document discusses how to graph factorable polynomials by identifying the roots and their orders from the polynomial expression, making a sign chart, sketching the graph around each root based on the order, and connecting the pieces to obtain the full graph. As an example, it identifies the roots x=0, x=-2, and x=3 of order 1, 2, and 2 respectively from the polynomial P(x)=-x(x+2)2(x-3)2, makes the sign chart, and sketches the graph around each root to ultimately connect them into the full graph of P(x).
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The document discusses quadratic functions. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c where a ≠ 0. It explains that the graph of a quadratic function is a parabola. If a>0, the parabola opens up and there is a minimum point. If a<0, the parabola opens down and there is a maximum point. The roots of the quadratic function correspond to the x-intercepts of the graph. If b^2 - 4ac > 0 there are two distinct roots, if b^2 - 4ac = 0 there is one repeated root, and if b^2 - 4ac < 0 there are no real roots
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Quadraticsportfoliopowerpoint 100325142401-phpapp02
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Coordinategeometry1 1
The document discusses coordinate geometry and defines key terms like axes of reference, coordinates of a point, distance formula, section formula, and area of a triangle. It provides examples of using these concepts and formulas to solve problems like finding the coordinates of a point dividing a line segment in a given ratio, or the area of a triangle given the coordinates of its vertices.
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Function evaluation, termination, vertical line test etc
The document discusses functions and their inverses. It defines key terms like domain, range, and inverse functions. It provides examples of determining if a relationship is a function using the vertical line test or one-to-one mapping. The document also covers finding inverses of straight lines, parabolas, exponential graphs and logarithmic graphs. It shows how to sketch these graphs and determine their domains and ranges. Finally, it discusses solving cubic expressions through finding factors and remainders.
Mathematics 9 Quadratic Functions (Module 2)
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
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1. This module covers quadratic functions and how to graph them. Students will learn to identify the vertex, axis of symmetry, and direction of opening of quadratic graphs.
2. Key aspects covered include how changing coefficients a, h, and k affect the width, translation, and vertical shift of the graph. When a increases, the graph narrows; changing h translates the graph left or right; and changing k shifts the graph up or down.
3. The steps for graphing a quadratic function are finding the vertex, axis of symmetry, direction of opening, and making a table of values to plot points.
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2.8 graphs of factorable polynomials t
- 1. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph. Example C. Given P(x) = -x(x + 2)2(x – 3)2, identify the roots and their orders. Make the sign-chart. Sketch the graph about each root. Connect them to complete the graph. The roots are x = 0 of order 1, x = -2 of order 2, and x = 3 of order 2. + order=2 order=3
- 2. Graphs of Factorable Polynomials The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is x = -2 order 2 ++ x = 0 order 1 x = 3 order 2 By the sign-chart and the order of each root, we draw the graph about each root. (Note for x = 0 of order 1, the graph approximates a line going through the point.) Connect all the pieces to get the graph of P(x).