This is called the vertex form of the quadratic equation. To graph the equation
|
|
- Cleopatra Ellis
- 5 years ago
- Views:
Transcription
1 Name Period Date: Topic: 7-5 Graphing ( ) Essential Question: What is the vertex of a parabola, and what is its axis of symmetry? Standard: F-IF.7a Objective: Graph linear and quadratic functions and show intercepts, maxima, and minima. To graph parabolas whose equations have the form ( ) and to find the vertices and axes of symmetry. This is called the vertex form of the quadratic equation. To graph the equation first make a table of pairs of values of x and v that satisfy the equation. Then plot the graph of each ordered pair of coordinates, as shown in the figure below. x y Summary
2 If you plotted more points, you would see that they all lie on the smooth curve shown in the figure below. This curve, called a parabola, is the graph of. In the figure notice that if the point (x, y) is on the parabola, then ( x, y), its "mirror image" across the y-axis, is also on the parabola. This property can also be seen in the table where the coordinate pairs are mirror images. Because of this property, the y-axis is called the axis of symmetry, or simply the axis of the parabola. The vertex of a parabola is the point where the parabola crosses its axis. In the case of, the vertex is the origin. The graph of, shown in the figure below, is a mirror image or congruent copy of the graph of, If the graph of is reflected across the x-axis, then the result is the graph of. 2
3 The two figures below show the effect of the value of a on the graph of an equation of the form. The graph of opens upward if and downward if. The larger is, the "narrower" the graph is. By graphing pairs of quadratic equations on the same axes, such as those in the six figures that follow, you may investigate the methods for graphing parabolas described in this lesson. A computer or a graphing calculator may be helpful. The two figures below illustrate the following method for graphing a parabola whose equation has the form ( ) To graph ( ), slide the graph of horizontally h units. If, slide it to the right; if slide it to the left. The graph has vertex (h, 0) and its axis is the line. y 1 2 (x 3)2 y 1 2 x ( 3) 2 3
4 Exercise 1: Fill in the table. Then graph, and ( ). Use the grid below. ( )
5 The two figures below illustrate the following method for graphing a parabola whose equation has the form To graph ( ), slide the graph of vertically k units. If, slide it upward; if, slide it downward. The graph has vertex (0, k) and its axis is the line (the y- axis). 1 1 y 3 2 x2 y ( 3) 2 x2 The two figures below illustrate the following method for graphing a parabola whose equation has the form ( ) To graph ( ), slide the graph of horizontally h units and vertically k units. The graph has vertex (h, k) and its axis is the line. y (x 4)2 y 3 1 (x + 2)2 2 5
6 Exercise 2: Fill in the table. Then graph, and. Use the grid below
7 Exercise 3: Fill in the table. Then graph, and ( + ). Use the grid below. ( + )
8 Exercise 4: What are the coordinates of the vertex of the graph in Exercise 1? What is the equation of the axis of symmetry of the graph in Exercise 1? Label the vertex and draw the axis of symmetry on the graph in Exercise 1. Exercise 5: What are the coordinates of the vertex of the graph in Exercise 2? What is the equation of the axis of symmetry of the graph in Exercise 2? Label the vertex and draw the axis of symmetry on the graph in Exercise 2. Exercise 6: What are the coordinates of the vertex of the graph in Exercise 3? What is the equation of the axis of symmetry of the graph in Exercise 3? Label the vertex and draw the axis of symmetry on the graph in Exercise 3. 8
9 Example 1: Graph ( + ). Label the vertex and axis. Solution Since, the parabola opens downward. Since and the vertex is ( ). The axis of symmetry is the line. Calculate a few convenient ordered pairs and plot the corresponding points. Also plot their images by reflection across the axis. Now draw the parabola by connecting the points with a smooth curve. x y
10 Intercepts: When graphing an equation in the coordinate plane, it is usually helpful to know the intercepts of the graph. The y-coordinate of a point where a graph crosses the y-axis is called the y-intercept. The x-coordinate of a point where a graph crosses the x-axis is called an x-intercept. A parabola may have no x-intercepts, one x-intercept, or two x-intercepts, as illustrated below. No x intercepts y intercept = 2 One x intercept y intercept = 1 Two x intercepts y intercept = 3 10
11 Determining the x-intercepts: 1. Set. ( ) 2. Transform the quadratic equation into standard form Find the discriminant where, + Discriminant < 0 If the discriminant is negative, the quadratic equation has no real roots, and the quadratic function has no x-intercepts. Discriminant = 0 If the discriminant is zero, the quadratic equation has one real root, and the quadratic function has one x-intercept. Discriminant > 0 If the discriminant is positive, the quadratic equation has two real roots, and the quadratic function has two x-intercepts. 11
12 Example 2: Graph + ( + ). Label the vertex and axis. Find all intercepts. Solution 1. Since, the parabola opens upward. Since and the vertex is ( ) The axis of symmetry is the line. 2. To find the y-intercept, set and solve for y. + ( + ) + y-intercept Therefore, the graph crosses the y-axis at ( ). Since ( ) is on the graph, so is its mirror image across the axis of symmetry, ( ). 3. To find any x-intercepts, set. Since + ( + ) + ( + ), the x-intercepts are, and 4. Plot the vertex ( ) and the intercepts. Then complete the curve using symmetry. The graph is shown on the next page. 12
13 Exercise 7: Find the vertex, axis of symmetry, y-intercept and all x-intercepts (if any) for the following quadratic functions: 1. + ( + ) 13
14 Exercise 7, continued: 2. ( ) 3. + ( ) 14
15 Example 3: Find an equation ( ) of the parabola having vertex (1, 2) and containing the point (3, 6). Solution Substitute (1, 2) for (h, k) in the equation ( ). ( ) ( ) + ( ) Since the parabola contains the point (3, 6), the coordinates of this point must satisfy the equation, + ( ) an equation of the parabola is + ( ) Exercise 8: 1. Find an equation ( ) of the parabola having vertex (5, 4) and containing the point (3, 8). 15
16 Exercise 8, continued: 2. Find an equation ( ) of the parabola having vertex ( 5, 4) and containing the point ( 4, 2). 3. Find an equation ( ) of the parabola having vertex ( 5, 4) and containing the point ( 1, 6). Class work: p 330 Oral Exercises: 1-17 Homework: p 331 Written Exercises: 1-30, P 332 Mixed Review:
Algebra II Quadratic Functions
1 Algebra II Quadratic Functions 2014-10-14 www.njctl.org 2 Ta b le o f C o n te n t Key Terms click on the topic to go to that section Explain Characteristics of Quadratic Functions Combining Transformations
More informationWarm-Up Exercises. Find the x-intercept and y-intercept 1. 3x 5y = 15 ANSWER 5; y = 2x + 7 ANSWER ; 7
Warm-Up Exercises Find the x-intercept and y-intercept 1. 3x 5y = 15 ANSWER 5; 3 2. y = 2x + 7 7 2 ANSWER ; 7 Chapter 1.1 Graph Quadratic Functions in Standard Form A quadratic function is a function that
More information9.1: GRAPHING QUADRATICS ALGEBRA 1
9.1: GRAPHING QUADRATICS ALGEBRA 1 OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form What does the graph of a quadratic look like? https://www.desmos.com/calculator
More informationUNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables
Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The
More informationSection 4.4: Parabolas
Objective: Graph parabolas using the vertex, x-intercepts, and y-intercept. Just as the graph of a linear equation y mx b can be drawn, the graph of a quadratic equation y ax bx c can be drawn. The graph
More informationSection 9.3 Graphing Quadratic Functions
Section 9.3 Graphing Quadratic Functions A Quadratic Function is an equation that can be written in the following Standard Form., where a 0. Every quadratic function has a U-shaped graph called a. If the
More informationUnit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form
Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Imagine the path of a basketball as it leaves a player s hand and swooshes through the net. Or, imagine the path of an Olympic diver
More informationUnit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1
Algebra I Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola Name Period Date Day #1 There are some important features about the graphs of quadratic functions we are going to explore over the
More informationUNIT 8: SOLVING AND GRAPHING QUADRATICS. 8-1 Factoring to Solve Quadratic Equations. Solve each equation:
UNIT 8: SOLVING AND GRAPHING QUADRATICS 8-1 Factoring to Solve Quadratic Equations Zero Product Property For all numbers a & b Solve each equation: If: ab 0, 1. (x + 3)(x 5) = 0 Then one of these is true:
More informationYimin Math Centre. Year 10 Term 2 Homework. 3.1 Graphs in the number plane The minimum and maximum value of a quadratic function...
Year 10 Term 2 Homework Student Name: Grade: Date: Score: Table of contents 3 Year 10 Term 2 Week 3 Homework 1 3.1 Graphs in the number plane................................. 1 3.1.1 The parabola....................................
More informationProperties of Quadratic functions
Name Today s Learning Goals: #1 How do we determine the axis of symmetry and vertex of a quadratic function? Properties of Quadratic functions Date 5-1 Properties of a Quadratic Function A quadratic equation
More information8.2 Graph and Write Equations of Parabolas
8.2 Graph and Write Equations of Parabolas Where is the focus and directrix compared to the vertex? How do you know what direction a parabola opens? How do you write the equation of a parabola given the
More informationAlgebra I. Slide 1 / 137. Slide 2 / 137. Slide 3 / 137. Quadratic & Non-Linear Functions. Table of Contents
Slide 1 / 137 Slide 2 / 137 Algebra I Quadratic & Non-Linear Functions 2015-11-04 www.njctl.org Table of Contents Slide 3 / 137 Click on the topic to go to that section Key Terms Explain Characteristics
More informationUnit: Quadratic Functions
Unit: Quadratic Functions Learning increases when you have a goal to work towards. Use this checklist as guide to track how well you are grasping the material. In the center column, rate your understand
More informationExploring Quadratic Graphs
Exploring Quadratic Graphs The general quadratic function is y=ax 2 +bx+c It has one of two basic graphs shapes, as shown below: It is a symmetrical "U"-shape or "hump"-shape, depending on the sign of
More informationQuadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0
Quadratic Equations Learning Objectives 1. Graph a quadratic function using transformations. Identify the vertex and axis of symmetry of a quadratic function 3. Graph a quadratic function using its vertex,
More informationSection 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
More informationSketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
More informationCHAPTER 6 Quadratic Functions
CHAPTER 6 Quadratic Functions Math 1201: Linear Functions is the linear term 3 is the leading coefficient 4 is the constant term Math 2201: Quadratic Functions Math 3201: Cubic, Quartic, Quintic Functions
More informationMATHS METHODS QUADRATICS REVIEW. A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation
MATHS METHODS QUADRATICS REVIEW LAWS OF EXPANSION A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation a) b) c) d) e) FACTORISING Exercise 4A Q6ace,7acegi
More informationStudent Exploration: Quadratics in Polynomial Form
Name: Date: Student Exploration: Quadratics in Polynomial Form Vocabulary: axis of symmetry, parabola, quadratic function, vertex of a parabola Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
More informationF.BF.B.3: Graphing Polynomial Functions
F.BF.B.3: Graphing Polynomial Functions 1 Given the graph of the line represented by the equation f(x) = 2x + b, if b is increased by 4 units, the graph of the new line would be shifted 4 units 1) right
More informationQuadratic Functions. *These are all examples of polynomial functions.
Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real
More informationGRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.
GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),
More information1.1 Functions. Cartesian Coordinate System
1.1 Functions This section deals with the topic of functions, one of the most important topics in all of mathematics. Let s discuss the idea of the Cartesian coordinate system first. Cartesian Coordinate
More informationA function: A mathematical relationship between two variables (x and y), where every input value (usually x) has one output value (usually y)
SESSION 9: FUNCTIONS KEY CONCEPTS: Definitions & Terminology Graphs of Functions - Straight line - Parabola - Hyperbola - Exponential Sketching graphs Finding Equations Combinations of graphs TERMINOLOGY
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.1 Quadratic Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic
More information2. From General Form: y = ax 2 + bx + c # of x-intercepts determined by the, D =
Alg2H 5-3 Using the Discriminant, x-intercepts, and the Quadratic Formula WK#6 Lesson / Homework --Complete without calculator Read p.181-p.186. Textbook required for reference as well as to check some
More informationEXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR
EXERCISE SET 10. STUDENT MATD 090 DUE DATE: INSTRUCTOR You have studied the method known as "completing the square" to solve quadratic equations. Another use for this method is in transforming the equation
More information6.4 Vertex Form of a Quadratic Function
6.4 Vertex Form of a Quadratic Function Recall from 6.1 and 6.2: Standard Form The standard form of a quadratic is: f(x) = ax 2 + bx + c or y = ax 2 + bx + c where a, b, and c are real numbers and a 0.
More informationWK # Given: f(x) = ax2 + bx + c
Alg2H Chapter 5 Review 1. Given: f(x) = ax2 + bx + c Date or y = ax2 + bx + c Related Formulas: y-intercept: ( 0, ) Equation of Axis of Symmetry: x = Vertex: (x,y) = (, ) Discriminant = x-intercepts: When
More information1.1 Graphing Quadratic Functions (p. 2) Definitions Standard form of quad. function Steps for graphing Minimums and maximums
1.1 Graphing Quadratic Functions (p. 2) Definitions Standard form of quad. function Steps for graphing Minimums and maximums Quadratic Function A function of the form y=ax 2 +bx+c where a 0 making a u-shaped
More informationMAC Learning Objectives. Module 4. Quadratic Functions and Equations. - Quadratic Functions - Solving Quadratic Equations
MAC 1105 Module 4 Quadratic Functions and Equations Learning Objectives Upon completing this module, you should be able to: 1. Understand basic concepts about quadratic functions and their graphs. 2. Complete
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 6: Analyzing Quadratic Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: factoring quadratic expressions finding the vertex of a quadratic function Introduction We have studied the key features of the
More information1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
More information2.5: GRAPHS OF EXPENSE AND REVENUE FUNCTIONS OBJECTIVES
Section 2.5: GRAPHS OF EXPENSE AND REVENUE FUNCTIONS OBJECTIVES Write, graph and interpret the expense function. Write, graph and interpret the revenue function. Identify the points of intersection of
More information8-4 Transforming Quadratic Functions
8-4 Transforming Quadratic Functions Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward
More informationObjective. 9-4 Transforming Quadratic Functions. Graph and transform quadratic functions.
Warm Up Lesson Presentation Lesson Quiz Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward. 1. y = x 2 + 3 2. y = 2x 2 x
More informationNotes Rules for Transformations of Functions If f x is the original functions, a > 0 and c > 0.
9.1.2 Parabola Investigation Do Now 1. Vertical means and horizontal is. 2. Another word for compress is. 3. Given the statement 0 < a < 1, a represents numbers like 4. Given the statement a > 1, a represents
More informationQuadratic Functions In Standard Form In Factored Form In Vertex Form Transforming Graphs. Math Background
Graphing In Standard Form In Factored Form In Vertex Form Transforming Graphs Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationChapter 2: Polynomial and Rational Functions Power Standard #7
Chapter 2: Polynomial and Rational s Power Standard #7 2.1 Quadratic s Lets glance at the finals. Learning Objective: In this lesson you learned how to sketch and analyze graphs of quadratic functions.
More information3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Finding the Zeros of a Quadratic Function Examples 1 and and more Find the zeros of f(x) = x x 6. Solution by Factoring f(x) = x x 6 = (x 3)(x + )
More information5.1 Introduction to the Graphs of Polynomials
Math 3201 5.1 Introduction to the Graphs of Polynomials In Math 1201/2201, we examined three types of polynomial functions: Constant Function - horizontal line such as y = 2 Linear Function - sloped line,
More informationSection 6 Quadratic Functions Part 2
Section 6 Quadratic Functions Part 2 The following Mathematics Florida Standards will be covered in this section: MAFS.912.A-CED.1.2 Create equations in two or more variables to represent relationships
More informationUnit 5: Quadratic Functions
Unit 5: Quadratic Functions LESSON #5: THE PARABOLA GEOMETRIC DEFINITION DIRECTRIX FOCUS LATUS RECTUM Geometric Definition of a Parabola Quadratic Functions Geometrically, a parabola is the set of all
More informationAlgebra II Quadratic Functions and Equations - Extrema Unit 05b
Big Idea: Quadratic Functions can be used to find the maximum or minimum that relates to real world application such as determining the maximum height of a ball thrown into the air or solving problems
More informationReplacing f(x) with k f(x) and. Adapted from Walch Education
Replacing f(x) with k f(x) and f(k x) Adapted from Walch Education Graphing and Points of Interest In the graph of a function, there are key points of interest that define the graph and represent the characteristics
More informationUNIT 3B CREATING AND GRAPHING EQUATIONS Lesson 4: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing multiple equations on a graphing calculator graphing quadratic equations graphing linear equations Introduction A system
More informationMS Algebra Ch Graph ax 2 + bx + c. Mr. Deyo
MS Algebra Ch. 10.2 Graph ax 2 + bx + c Mr. Deyo Learning Target By the end of the period, I will graph quadratic equations in the standard form of y = ax 2 + bx + c. I will demonstrate this by completing
More informationQuadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31
CHAPTER Quadratic Functions Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in architecture by the Mesopotamians over 4000 years ago. Later, the Romans
More informationGraphing Absolute Value Functions
Graphing Absolute Value Functions To graph an absolute value equation, make an x/y table and plot the points. Graph y = x (Parent graph) x y -2 2-1 1 0 0 1 1 2 2 Do we see a pattern? Desmos activity: 1.
More information2.2 Transformers: More Than Meets the y s
10 SECONDARY MATH II // MODULE 2 STRUCTURES OF EXPRESSIONS 2.2 Transformers: More Than Meets the y s A Solidify Understanding Task Writetheequationforeachproblembelow.Useasecond representationtocheckyourequation.
More informationName: Chapter 7 Review: Graphing Quadratic Functions
Name: Chapter Review: Graphing Quadratic Functions A. Intro to Graphs of Quadratic Equations: = ax + bx+ c A is a function that can be written in the form = ax + bx+ c where a, b, and c are real numbers
More information2. The diagram shows part of the graph of y = a (x h) 2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0).
Quadratics Vertex Form 1. Part of the graph of the function y = d (x m) + p is given in the diagram below. The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, ). (a) Write down the value of (i)
More informationSection 2.1 Graphs. The Coordinate Plane
Section 2.1 Graphs The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of numbers to form
More informationx 2 + 8x - 12 = 0 Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials
Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials Do Now - Solve using any strategy. If irrational, express in simplest radical form x 2 + 8x - 12 = 0 Review Topic Index 1.
More informationAlgebra 1 STAAR EOC Review #9 Reporting Category 5: Quadratic and Other Nonlinear Functions
Name Class Date RC9 A.09B Algebra 1 STAAR EOC Review #9 Reporting Category 5: Quadratic and Other Nonlinear Functions 1. Which shows the functions correctly listed in order from widest to narrowest graph?
More information3.1 Investigating Quadratic Functions in Vertex Form
Math 2200 Date: 3.1 Investigating Quadratic Functions in Vertex Form Degree of a Function - refers to the highest exponent on the variable in an expression or equation. In Math 1201, you learned about
More informationModule 3: Graphing Quadratic Functions
Haberman MTH 95 Section V Quadratic Equations and Functions Module 3 Graphing Quadratic Functions In this module, we'll review the graphing quadratic functions (you should have studied the graphs of quadratic
More informationInvestigating Transformations With DESMOS
MPM D0 Date: Investigating Transformations With DESMOS INVESTIGATION Part A: What if we add a constant to the x in y = x? 1. Use DESMOS to graph the following quadratic functions on the same grid. Graph
More informationSection 7.2 Characteristics of Quadratic Functions
Section 7. Characteristics of Quadratic Functions A QUADRATIC FUNCTION is a function of the form " # $ N# 1 & ;# & 0 Characteristics Include:! Three distinct terms each with its own coefficient:! An x
More informationAlgebra II Chapter 4: Quadratic Functions and Factoring Part 1
Algebra II Chapter 4: Quadratic Functions and Factoring Part 1 Chapter 4 Lesson 1 Graph Quadratic Functions in Standard Form Vocabulary 1 Example 1: Graph a Function of the Form y = ax 2 Steps: 1. Make
More informationMAC Rev.S Learning Objectives. Learning Objectives (Cont.) Module 4 Quadratic Functions and Equations
MAC 1140 Module 4 Quadratic Functions and Equations Learning Objectives Upon completing this module, you should be able to 1. understand basic concepts about quadratic functions and their graphs.. complete
More informationChapter 12: Quadratic and Cubic Graphs
Chapter 12: Quadratic and Cubic Graphs Section 12.1 Quadratic Graphs x 2 + 2 a 2 + 2a - 6 r r 2 x 2 5x + 8 2y 2 + 9y + 2 All the above equations contain a squared number. They are therefore called quadratic
More informationProperties of Graphs of Quadratic Functions
H e i g h t (f t ) Lesson 2 Goal: Properties of Graphs of Quadratic Functions Identify the characteristics of graphs of quadratic functions: Vertex Intercepts Domain and Range Axis of Symmetry and use
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationLesson 6. Opening Exercise 1. Without graphing, state the vertex for each of the following quadratic equations.
: Further Exploration of Vertex Form, yy = aa(xx hh) + kk Opening Exercise 1. Without graphing, state the vertex for each of the following quadratic equations. A. yy = (xx 5) + 3 B. yy = xx.5 C. yy = (xx
More informationGraphs of Equations. MATH 160, Precalculus. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Graphs of Equations
Graphs of Equations MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: sketch the graphs of equations, find the x- and y-intercepts
More informationWarm Up. Factor the following numbers and expressions. Multiply the following factors using either FOIL or Box Method
Warm Up Factor the following numbers and expressions 1. 36 2. 36x 3 + 48x 2 + 24x Multiply the following factors using either FOIL or Box Method 3. (3x 2)(x 1) 4. (x 2)(x + 3) Objectives Recognize standard
More informationQuadratic Functions (Section 2-1)
Quadratic Functions (Section 2-1) Section 2.1, Definition of Polynomial Function f(x) = a is the constant function f(x) = mx + b where m 0 is a linear function f(x) = ax 2 + bx + c with a 0 is a quadratic
More informationAssignment 3/17/15. Section 10.2(p 568) 2 12 (E) (E)
Section 10.2 Warm Up Assignment 3/17/15 Section 10.2(p 568) 2 12 (E) 24 40 (E) Objective We are going to find equations for parabolas identify the vertex, focus, and directrix of a parabola The parabola
More informationChapter 2. Polynomial and Rational Functions. 2.2 Quadratic Functions
Chapter 2 Polynomial and Rational Functions 2.2 Quadratic Functions 1 /27 Chapter 2 Homework 2.2 p298 1, 5, 17, 31, 37, 41, 43, 45, 47, 49, 53, 55 2 /27 Chapter 2 Objectives Recognize characteristics of
More informationQuadratic Functions. Full Set of Notes. No Solutions
Quadratic Functions Full Set of Notes No Solutions Graphing Quadratic Functions The graph of a quadratic function is called a parabola. Applications of Parabolas: http://www.doe.virginia.gov/div/winchester/jhhs/math/lessons/calc2004/appparab.html
More informationChapter 10. Homework
Chapter 0 Homework Lesson 0- pages 538 5 Exercises. 2. Hyperbola: center (0, 0), y-intercepts at ±, no x-intercepts, the lines of symmetry are the x- and y-axes; domain: all real numbers, range: y 5 3
More information4.1 Graph Quadratic Functions in
4. Graph Quadratic Functions in Standard Form Goal p Graph quadratic functions. Your Notes VOCABULARY Quadratic function Parabola Verte Ais of smmetr Minimum and maimum value PARENT FUNCTION FOR QUADRATIC
More informationRational Numbers on the Coordinate Plane. 6.NS.C.6c
Rational Numbers on the Coordinate Plane 6.NS.C.6c Copy all slides into your composition notebook. Lesson 14 Ordered Pairs Objective: I can use ordered pairs to locate points on the coordinate plane. Guiding
More information3.1 Quadratic Functions and Models
3.1 Quadratic Functions and Models Objectives: 1. Identify the vertex & axis of symmetry of a quadratic function. 2. Graph a quadratic function using its vertex, axis and intercepts. 3. Use the maximum
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 7: Building Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: multiplying linear expressions factoring quadratic equations finding the value of a in the vertex form of a quadratic equation
More informationQUADRATIC FUNCTIONS Investigating Quadratic Functions in Vertex Form
QUADRATIC FUNCTIONS Investigating Quadratic Functions in Verte Form The two forms of a quadratic function that have been eplored previousl are: Factored form: f ( ) a( r)( s) Standard form: f ( ) a b c
More informationNO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED
Algebra II (Wilsen) Midterm Review NO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED Remember: Though the problems in this packet are a good representation of many of the topics that will be on the exam, this
More informationBut a vertex has two coordinates, an x and a y coordinate. So how would you find the corresponding y-value?
We will work with the vertex, orientation, and x- and y-intercepts of these functions. Intermediate algebra Class notes More Graphs of Quadratic Functions (section 11.6) In the previous section, we investigated
More informationWarm Up Grab your calculator Find the vertex: y = 2x x + 53 (-5, 3)
Warm Up Grab your calculator Find the vertex: y = 2x 2 + 20x + 53 (-5, 3) Quiz will be next Tuesday, folks. Check HW/ New Section Another useful form of writing quadratic functions is the standard form.
More informationLesson 19: The Graph of a Linear Equation in Two Variables is a Line
Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Classwork Exercises Theorem: The graph of a linear equation y = mx + b is a non-vertical line with slope m and passing through (0, b),
More informationQUADRATIC FUNCTIONS. PROTOTYPE: f(x) = ax 2 + bx + c. (1) The leading coefficient a 0 is called the shape parameter.
QUADRATIC FUNCTIONS PROTOTYPE: f(x) = ax 2 + bx + c. (1) The leading coefficient a 0 is called the shape parameter. SHAPE-VERTEX FORMULA One can write any quadratic function (1) as f(x) = a(x h) 2 + k,
More informationTransformations with Quadratic Functions KEY
Algebra Unit: 05 Lesson: 0 TRY THIS! Use a calculator to generate a table of values for the function y = ( x 3) + 4 y = ( x 3) x + y 4 Next, simplify the function by squaring, distributing, and collecting
More informationSection 6.2: Properties of Graphs of Quadratic Functions. Vertex:
Section 6.2: Properties of Graphs of Quadratic Functions determine the vertex of a quadratic in standard form sketch the graph determine the y intercept, x intercept(s), the equation of the axis of symmetry,
More informationChapter 10. Exploring Conic Sections
Chapter 10 Exploring Conic Sections Conics A conic section is a curve formed by the intersection of a plane and a hollow cone. Each of these shapes are made by slicing the cone and observing the shape
More informationLesson 1: Analyzing Quadratic Functions
UNIT QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Common Core State Standards F IF.7 F IF.8 Essential Questions Graph functions expressed symbolically and show key features
More information9.1 Linear Inequalities in Two Variables Date: 2. Decide whether to use a solid line or dotted line:
9.1 Linear Inequalities in Two Variables Date: Key Ideas: Example Solve the inequality by graphing 3y 2x 6. steps 1. Rearrange the inequality so it s in mx ± b form. Don t forget to flip the inequality
More informationUnit 1 Quadratic Functions
Unit 1 Quadratic Functions This unit extends the study of quadratic functions to include in-depth analysis of general quadratic functions in both the standard form f ( x) = ax + bx + c and in the vertex
More informationAlgebra II Lesson 4.1 and 4.2 Review
Name: Class: Date: Algebra II Lesson 4.1 and 4.2 Review 1. Graph y = 1 4 x 2. a. c. b. d. Graph. 2. y = x 2 3 a. c. b. d. 1 Name: 3. y = 3x 2 + x + 1 a. c. b. d. 4. y = 2x 2 + x + 3 5. How would you translate
More informationMore Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a
More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a Example: Solve using the square root property. a) x 2 144 = 0 b) x 2 + 144 = 0 c) (x + 1) 2 = 12 Completing
More informationMathematical Reasoning. Lesson 37: Graphing Quadratic Equations. LESSON 37: Graphing Quadratic Equations
LESSON 37: Graphing Quadratic Equations Weekly Focus: quadratic equations Weekly Skill: graphing Lesson Summary: For the warm-up, students will solve a problem about mean, median, and mode. In Activity
More informationLesson 3.1 Vertices and Intercepts. Important Features of Parabolas
Lesson 3.1 Vertices and Intercepts Name: _ Learning Objective: Students will be able to identify the vertex and intercepts of a parabola from its equation. CCSS.MATH.CONTENT.HSF.IF.C.7.A Graph linear and
More informationIt is than the graph of y= x if a > 1.
Chapter 8 Quadratic Functions and Equations Name: Instructor: 8.1 Quadratic Functions and Their Graphs Graphs of Quadratic Functions Basic Transformations of Graphs More About Graphing Quadratic Functions
More informationSection 6: Quadratic Equations and Functions Part 2
Section 6: Quadratic Equations and Functions Part 2 Topic 1: Observations from a Graph of a Quadratic Function... 147 Topic 2: Nature of the Solutions of Quadratic Equations and Functions... 150 Topic
More informationPolynomial Functions Graphing Investigation Unit 3 Part B Day 1. Graph 1: y = (x 1) Graph 2: y = (x 1)(x + 2) Graph 3: y =(x 1)(x + 2)(x 3)
Part I: Polynomial Functions when a = 1 Directions: Polynomial Functions Graphing Investigation Unit 3 Part B Day 1 1. For each set of factors, graph the zeros first, then use your calculator to determine
More informationKEY Algebra: Unit 10 Graphing Quadratic Equations & other Relations
Name: KEY Algebra: Unit 10 Graphing Quadratic Equations & other Relations Date: Test Bank Part I: Answer all 15 questions in this part. Each correct answer will receive credits. No partial credit will
More informationUnit 2 Day 5. Characteristics of Quadratic Functions
Unit 2 Day 5 Characteristics of Quadratic Functions 1 Warm Up 1.) Jason and Jim jumped off a cliff into the ocean in Acapulco while vacationing. Jason s height as a function of time could be modeled by
More information4-1 (Part 2) Graphing Quadratics, Interpreting Parabolas
4-1 (Part 2) Graphing Quadratics, Interpreting Parabolas Objectives Students will be able to: Find the vertex and y-intercept of a parabola Graph a parabola Use quadratic models to analyze problem situations.
More information