Algebra lesson 4.2 zeroes of quadratic functions
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This document provides information about quadratic functions and solving for their zeroes (x-intercepts). It discusses factoring quadratic expressions, using the zero product property to set each factor equal to zero. It also introduces the quadratic formula as a way to solve quadratic equations that are not factorable. There is an example of using the quadratic formula to find the zeroes of the function f(x)=x^2 - 3x - 1. The document concludes with practice problems for students to solve for the zeroes of various quadratic functions.
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Module 2 exponential functions
This module introduces exponential functions and covers:
- Finding the roots of exponential equations using the property of equality for exponential equations.
- Simplifying expressions using laws of exponents.
- Determining the zeros of exponential functions by setting the function equal to 0 and solving for x.
The document provides examples and practice problems for students to learn skills in solving exponential equations and finding zeros of exponential functions.
Module 3 quadratic functions
This module covers quadratic functions and equations. Students will learn to determine the zeros of quadratic functions by relating them to the roots of quadratic equations. They will also learn to find the roots of quadratic equations using factoring, completing the square, and the quadratic formula. The module aims to help students derive quadratic functions given certain conditions like the zeros, a table of values, or a graph.
Quadraticfunctionpresentation 100127142417-phpapp02
The document defines quadratic functions and discusses their various forms, including general, vertex, and factored forms. It also covers solving quadratic equations using methods like the quadratic formula, factoring, and completing the square. Additionally, it discusses key features of quadratic graphs like x-intercepts, y-intercepts, the vertex, and concavity. Examples are provided to illustrate finding these features and graphing parabolas.
Solving quadratic equations[1]
The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations arising from applications like finding the dimensions of a rectangle given its area. Sample problems are worked through step-by-step to find the solutions or roots of quadratic equations.
Solving quadratic equations
The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations as well as word problems involving quadratic equations. Factoring involves writing the quadratic as a product of two binomials and setting each factor equal to 0. Using square roots involves taking the square root of both sides of the equation to isolate the variable. Completing the square rewrites the equation by adding or subtracting terms to put it in the form (x+a)2=b and then taking the square root of both sides.
10.4
Here are the steps to solve each equation by graphing:
1) x2 + 5x + 6 = 0
Write in standard form: x2 + 5x + 6 = 0
Graph the related function y = x2 + 5x + 6. The x-intercepts are the solutions, which are -3 and -2.
2) x2 + 8x + 16 = 0
Write in standard form: x2 + 8x + 16 = 0
Graph the related function y = x2 + 8x + 16. The x-intercepts are the solutions, which are -4 and -4.
3) x2 - 2x + 3 = 0
Write in standard
Function
FUNCTION
A function is a set of ordered pairs which relates two sets such that to each element of one set there is only one element of the second set
Algebra slideshow
The document provides instructions for graphing and solving various types of quadratic equations. It defines standard form, vertex form, and intercept form of quadratics. It explains how to graph quadratics by finding the vertex and intercepts. Methods covered include factoring, taking square roots, completing the square, and using the quadratic formula. Examples are included to demonstrate each process.
Quadratic equations
This document discusses quadratic equations and methods for solving them. It begins by defining quadratic equations as second degree polynomial equations of the form ax^2 + bx + c = 0, where a is not equal to 0. It then presents several methods for finding the roots or solutions of quadratic equations: factoring, completing the square, and using the quadratic formula. Examples are provided to illustrate each method. The document also discusses graphing quadratic functions and key features of parabolas such as vertex, axis of symmetry, and direction of opening.
Modeling with Quadratics
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
Quadraticequation
A quadratic equation is a second-order polynomial with terms up to x^2. It has two roots, or solutions, which may be real or complex. The quadratic formula can be used to find the exact numeric values of the roots. A root is a value that makes the quadratic equation equal to zero. A double root occurs when the two roots are equal, making the quadratic a perfect square trinomial.
Grph quad fncts
The document discusses graphing quadratic functions. It defines a quadratic function as f(x) = ax^2 + bx + c where a, b, and c are real numbers and a is not equal to 0. The graph of a quadratic function is a parabola that is symmetrical about an axis. When the leading coefficient a is positive, the parabola opens upward and the vertex is a minimum. When a is negative, the parabola opens downward and the vertex is a maximum. Standard forms for quadratic functions and methods for finding characteristics like the vertex, axis of symmetry, and x-intercepts from the equation are also presented.
First Quarter - Chapter 2 - Quadratic Equation
Okay, let's think through this with the new information:
* The equation modeling the height is: h = -16t^2 + vt + c
* The initial height (c) is still 2 feet
* The initial velocity (v) is now 20 feet/second
* The target height (h) is still 20 feet
So the equation is:
20 = -16t^2 + 20t + 2
0 = -16t^2 + 20t + 18 (subtract 20 from both sides)
Evaluating the discriminant:
(20)^2 - 4(-16)(-18) = 400 - 288 = 112
Since the discriminant is positive
Assignment5
This document outlines 6 questions for a math assignment on root-finding algorithms and optimization techniques:
1. Find the zeros of f(x) = sin(10x) - x.
2. Use iterative methods to find the 1000000th positive solution to x = tan(x) and explain the method.
3. Analyze the convergence of fixed-point iterations for 4 functions related to solving x^2 - 3x + 2 = 0 and implement the schemes to verify the analysis.
4. Derive and apply Newton's method to approximate 1/y without division and investigate cases where it converges/diverges for the equation 1/x - y = 0.
5.
Module 1 quadratic functions
This module introduces quadratic functions. It discusses identifying quadratic functions as those with the highest exponent of 2, rewriting quadratic functions in general form f(x) = ax^2 + bx + c and standard form f(x) = a(x-h)^2 + k, and the key properties of quadratic graphs including the vertex and axis of symmetry. The module provides examples of identifying quadratic functions from equations and ordered pairs/tables and rewriting quadratic functions between general and standard form using completing the square.
CalculusStudyGuide
The document provides examples and explanations of concepts related to calculus including continuity, differentiability, limits, derivatives, and the mean value theorem. Some key points:
- It gives examples of determining if functions are continuous and differentiable at various points, including functions with absolute value.
- The mean value theorem is explained and examples are worked through showing a function satisfies the mean value theorem and finding the value of c.
- Numerous examples demonstrate calculating derivatives using rules like product, quotient, chain and implicit differentiation. Examples include derivatives of trigonometric, exponential and logarithmic functions.
- Implicit differentiation is used to find the equation of a tangent line to a curve at a given point.
C2 st lecture 2 handout
This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.
Quadratic equations
This document defines and explains how to solve quadratic equations. A quadratic equation is an equation that can be written in the standard form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to 0. There are several methods for solving quadratic equations covered in the document: factoring, taking the square root, completing the square, and using the quadratic formula. The discriminant, b^2 - 4ac, determines the number and type of solutions.
April 19
The document provides an agenda for a math class that includes reviewing solving quadratic equations by factoring and completing the square, and introducing the quadratic formula. The class will start with a warm-up reviewing solving quadratics, including finding zeros, vertices, and axes of symmetry. New material will then be presented on the quadratic formula and worked examples will be provided. Students will finish with class work problems solving quadratics using various methods.
Mathematics 9 Quadratic Functions (Module 1)
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
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Algebra lesson 4.2 zeroes of quadratic functions
- 1. QUADRATIC FUNCTIONS MIDTERM EXAM: FEB 1, W, 10-11:30 AM BAR AND DINING LAB (Bring Calculator) Lesson 4.2 Real Zeroes Factoring Quadratic Formula
- 2. Zeroes • The values of x so that f(x) = 0. • Real (as opposed to imaginary) zeroes are also the x-intercepts of the function.
- 3. Zeroes of Quadratic Functions CASE 1: FACTORABLE QEs • The Zero Product Theorem For a product to be equal to zero, at least one of its factors must be equal to zero. i.e. If abcd = 0, then either a=0 or b=0 or c=0 or d=0. • To solve (x+2)(x-3)(2x+1)=0, find values of x that will make each factor equal to zero. • Hence, the solutions are x = -2, 3 and -1/2.
- 4. Zeroes of Quadratic Functions Example 1: Find the x-intercept/s of f(x) = x2 – 9. Solution: 0 = x2 – 9 0 = (x – 3)(x + 3) Hence, x = 3 and -3
- 5. Answers to the SW: 1. 0 = 4x2 – 9 0 = (2x + 3)(2x – 3) Solution: x = -3/2 and 3/2 2. 0 = x2 - 5x – 24 0 = (x – 8)(x + 3) Solution: x = 8, -3 3. 20 = 9x - x2 0 = - x2 + 9x – 20 = (-x + 4)(x – 5) Solution: x = 4, 5 4. 0 = 3x2 + 4x – 4 Solution: x = 2/3, -2 5. 3 = 4x2 + 4x Solution: x = -3/2, 1/2
- 6. Quadratic Formula • Not all quadratic equations are factorable but that does not mean that it does not have zeroes. • The following formula solves for the value of x that satisfies the equation 0 = ax2 + bx + c: • Notice that this is actually two formulas, one using + and the other using -
- 7. Quadratic Formula Example: Find the x-intercepts of f(x)=x2 – 3x – 1 Solution: a = 1, b = -3, c = -1 -(-3) ± (-3)2 - 4(1)(-1) x= 2(1) 3± 9 + 4 x= 3+ 13 3- 13 2 x= and 2 2 3± 13 x= 2
- 8. Seatwork 1. Solve for x: a) 12 = x2 + x c) 2x2 + 6 = 7x b) 0 = 2x2 – 2x – 3 d) 0 = -x2 + 3x + 5 2. Given the function f(x) = x2 + 8x + 10, do the ff: a) Find the vertex, y-intercept and x-intercepts b) Graph f(x) showing the points in (a). c) Identify the domain, range, axis of symmetry and interval/s where f(x) is positive.