8.2 Graph and Write Equations of Parabolas
|
|
|
- Dwain Strickland
- 6 years ago
- Views:
Transcription
1 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 focus/directrix? What is the general equation for a parabola?
2 Parabolas A parabola is defined in terms of a fixed point, called the focus, and a fixed line, called the directrix. A parabola is the set of all points P(x,y) in the plane whose distance to the focus equals its distance to the directrix. directrix focus axis of symmetry
3 Horizontal Directrix Standard Equation of a parabola with its vertex at the origin is x 2 = 4py y p > 0: opens upward p < 0: opens downward focus: (0, p) P(x, y) F(0, p) directrix: y = p axis of symmetry: y-axis O D(x, p) y = p x
4 Vertical Directrix Standard Equation of a parabola with its vertex at the origin is y 2 = 4px y p > 0: opens right p < 0: opens left focus: (p, 0) directrix: x = p axis of symmetry: x-axis D(x, p) x = p O P(x, y) F(p, 0) x
5 Example 1 Graph directrix.. Label the vertex, focus, and y 2 = 4px Identify p. y 2 = 4(1)x So, p = Since p > 0, the parabola opens to the right Vertex: (0,0) Focus: (1,0) Directrix: x = -1
6 Example 1 Graph directrix.. Label the vertex, focus, and Y 2 = 4x Use a table to sketch a graph 4 y x
7 Graph x = ⅛y 2. Identify the focus, directrix, and axis of symmetry. SOLUTION STEP 1 Rewrite the equation in standard form. x = 1 y 2 8 Write original equation. 8x = y 2 STEP 2 Multiply each side by 8. Identify the focus, directrix, and axis of symmetry. The equation has the form y 2 = 4px where p = 2. The focus is (p, 0), or ( 2, 0). The directrix is x = p, or x = 2. Because y is squared, the axis of symmetry is the x - axis.
8 STEP 3 8x = y 2 Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative x - values.
9 Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 1. Y 2 = 6x SOLUTION STEP 1 Rewrite the equation in standard form. Y 2 = 4 ( 3 2 )x STEP 2 Identify the focus, directrix, and axis of symmetry. The equation has the form y 2 = 4px where p = 3. The focus 2 is (p, 0), or ( 3 3, 0). The directrix is x = p, or x =. 2 2 Because y is squared, the axis of symmetry is the x - axis.
10 STEP 3 Y 2 = 6x Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative x - values
11 Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola y = x 2 4 SOLUTION STEP 1 Rewrite the equation in standard form. y = 1 x 2 Write original equation. 4 Multiply each side by 4. 4y = x 2 STEP 2 equation focus directrix axis of symmetry x 2 = 4y (0, 1) y = 1 Vertical x = 0
12 STEP 3 x 2 = 4y Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative y - values. y x
13 Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 4. x = 1 y 2 3 SOLUTION STEP 1 Rewrite the equation in standard form. 1 x = Y 2 Write original equation. 3 3x = y 2 Multiply each side by 3. STEP 2 equation Y 2 3 = 4 4 x focus directrix axis of symmetry 3,0 x = 3 Horizontal y = 0 4 4
14 STEP 3 3x = y 2 Draw the parabola by making a table of values and plotting points. Because p < 0, the parabola opens to the left. So, use only negative x - values. y x
15 Example 2 Write the standard equation of the parabola with its vertex at the origin and the directrix y = -6. Since the directrix is below the vertex, the parabola opens up Since y = -p and y = -6, p = 6 x 2 =4(6)y x 2 = 24y
16 Write an equation of the parabola shown. SOLUTION The graph shows that the vertex is (0, 0) and the directrix is y = p = 3 for p in the standard form 2 of the equation of a parabola. x 2 = 4py Standard form, vertical axis of symmetry x 2 = 4 ( 3 )y 2 Substitute 3 for p 2 x 2 = 6y Simplify.
17 Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus. 5. Directrix: y = 2 SOLUTION x 2 = 4py x 2 = 4 ( 2)y x 2 = 8y Standard form, vertical axis of symmetry Substitute 2 for p Simplify.
18 Write the standard form of the equation of the parabola with vertex at (0, 0) and the given directrix or focus. 8. Focus: (0, 3) SOLUTION x 2 = 4py x 2 = 4 (3)y x 2 = 12y Standard form, vertical axis of symmetry Substitute 3 for p Simplify.
19 Where is the focus and directrix compared to vertex? The focus is a point on the line of symmetry and the directrix is a line below the vertex. The focus and directrix are equidistance from the vertex. How do you know what direction a parabola opens? x 2, graph opens up or down, y 2, graph opens right or left How do you write the equation of a parabola given the focus/directrix? Find the distance from the focus/directrix to the vertex (p value) and substitute into the equation. What is the general equation for a parabola? x 2 = 4py (opens up [p>0] or down [p<0]), y 2 = 4px (opens right [p>0] or left [p<0])
20 Hw 8.2 p. 499, 3-17 odd, 26-33, NO WORK = NO CREDIT This is a long assignment with 24 problems! Start working on them today!
21 8.2 Graph and Write Equations of Parabolas day 2 What does it mean if a parabola has a translated vertex? What general equations can you use for a parabola when the vertex has been translated?
22 Standard Equation of a Translated Parabola Vertical axis: (x h) 2 = 4p(y k) vertex: (h, k) focus: (h, k + p) directrix: y = k p axis of symmetry: x = h
23 Standard Equation of a Translated Parabola Horizontal axis: (y k) 2 = 4p(x h) vertex: (h, k) focus: (h + p, k) directrix: x = h - p axis of symmetry: y = k
24 Example 3 Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4. Sketch the info. The parabola opens downward, so the equation is of the form (x h) 2 = 4p(y k) vertex: (-3,3) h = -3, k = 3 p = -1 (x + 3) 2 = 4( 1)(y 3)
25 SOLUTION STEP 1 STEP 2 Graph (x 2) 2 = 8 (y + 3). Compare the given equation to the standard form of an equation of a parabola. You can see that the graph is a parabola with vertex at (2, 3),focus (2, 1) and directrix y = 5 Draw the parabola by making a table of value and plot y point. Because p > 0, he parabola open to the right. So use only points x- value x y STEP 3 Draw a curve through the points.
26 Example 4 Write an equation of a parabola whose vertex is at ( 2,1) and whose focus is at ( 3, 1). Begin by sketching the parabola. Because the parabola opens to the left, it has the form (y k) 2 = 4p(x h) Find h and k: The vertex is at ( 2,1) so h = 2 and k = 1 Find p: The distance between the vertex ( 2,1) and the focus ( 3,1) by using the distance formula. p = 1 (y 1) 2 = 4(x + 2)
27 Write an equation of the parabola whose vertex is at ( 2, 3) and whose focus is at ( 4, 3). SOLUTION STEP 1 Determine the form of the equation. Begin by making a rough sketch of the parabola. Because the focus is to the left of the vertex, the parabola opens to the left, and its equation has the form (y k) 2 = 4p(x h) where p < 0. STEP 2 Identify h and k. The vertex is at ( 2, 3), so h = 2 and k = 3. STEP 3 Find p. The vertex ( 2, 3) and focus ( 4, 3) both lie on the line y = 3, so the distance between them is p = 4 ( 2) = 2, and thus p = +2. Because p < 0, it follows that p = 2, so 4p = 8. The standard form of the equation is (y 3) 2 = 8(x + 2).
28 Write the standard form of a parabola with vertex at (3, 1) and focus at (3, 2). SOLUTION STEP 1 Determine the form of the equation. Begin by making a rough sketch of the parabola. Because the focus is to the left of the vertex, the parabola opens to the left, and its equation has the form (x h) 2 = 4p(y k) where p > 0. STEP 2 Identify h and k. The vertex is at (3, 1), so h = 3 and k = 1. STEP 3 Find p. The vertex (3, 1) and focus (3, 2) both lie on the line x = 3, so the distance between them is p = 2 ( 1) = 3, and thus p = + 3. Because p > 0, it follows that p = 3, so 4p = 12. The standard form of the equation is (x 3) 2 = 12(y + 1)
29 What does it mean if a parabola has a translated vertex? It means that the vertex of the parabola has been moved from (0,0) to (h,k). What general equations can you use for a parabola when the vertex has been translated? (y-k) 2 =4p(x-h) (x-h) 2 =4p(y-k)
30 Hw 8.2 p. 499, 3-17 odd, 26-33, NO WORK = NO CREDIT This is a long assignment with 24 problems! Start working on them today!
Unit 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
Chapter 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
Warm-Up. Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) 2) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,2)
Warm-Up Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) ) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,) 8.4 Graph and Write Equations of Ellipses What are the major parts of
) 2 + (y 2. x 1. y c x2 = y
Graphing Parabola Parabolas A parabola is a set of points P whose distance from a fixed point, called the focus, is equal to the perpendicular distance from P to a line, called the directrix. Since this
This is called the vertex form of the quadratic equation. To graph the equation
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
UNIT 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
Acc. Pre Calculus Day 5 - Parabolas Notesheet PARABOLAS
Acc. Pre Calculus Day 5 - Parabolas Notesheet Name Date Block 1) Complete these truths about parabolas: * Parabolas are - shaped. PARABOLAS * Parabolas have a line of. * Parabolas are the graphs of functions.
Conic Sections. College Algebra
Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines
1.) Write the equation of a circle in standard form with radius 3 and center (-3,4). Then graph the circle.
Welcome to the world of conic sections! http://www.youtube.com/watch?v=bfonicn4bbg Some examples of conics in the real world: Parabolas Ellipse Hyperbola Your Assignment: Circle -Find at least four pictures
Assignment 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
y 1 ) 2 Mathematically, we write {(x, y)/! y = 1 } is the graph of a parabola with 4c x2 focus F(0, C) and directrix with equation y = c.
Ch. 10 Graphing Parabola Parabolas A parabola is a set of points P whose distance from a fixed point, called the focus, is equal to the perpendicular distance from P to a line, called the directrix. Since
Unit 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
Exploring 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
MATH 110 analytic geometry Conics. The Parabola
1 MATH 11 analytic geometry Conics The graph of a second-degree equation in the coordinates x and y is called a conic section or, more simply, a conic. This designation derives from the fact that the curve
We start by looking at a double cone. Think of this as two pointy ice cream cones that are connected at the small tips:
Math 1330 Chapter 8 Conic Sections In this chapter, we will study conic sections (or conics). It is helpful to know exactly what a conic section is. This topic is covered in Chapter 8 of the online text.
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
9.1: 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
Rewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 2 4 x 0
Pre-Calculus Section 1.1 Completing the Square Rewrite the equation in the left column into the format in the middle column. The answers are in the third column. 1. y 4y 4x 4 0 y k 4p x h y 4 x 0. 3x 3y
3.1 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 + )
Section 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
Sketching 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.
Quadratics and their Properties
Algebra 2 Quadratics and their Properties Name: Ms. Williams/Algebra 2 Pd: 1 Table of Contents Day 1: COMPLETING THE SQUARE AND SHIFTING PARABOLAS SWBAT: Write a quadratic from standard form to vertex
MATHS 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
Chapter 8.1 Conic Sections/Parabolas. Honors Pre-Calculus Rogers High School
Chapter 8.1 Conic Sections/Parabolas Honors Pre-Calculus Rogers High School Introduction to Conic Sections Conic sections are defined geometrically as the result of the intersection of a plane with a right
Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal
Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to
UNIT 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:
Name. Center axis. Introduction to Conic Sections
Name Introduction to Conic Sections Center axis This introduction to conic sections is going to focus on what they some of the skills needed to work with their equations and graphs. year, we will only
Unit #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
We start by looking at a double cone. Think of this as two pointy ice cream cones that are connected at the small tips:
Math 1330 Conic Sections In this chapter, we will study conic sections (or conics). It is helpful to know exactly what a conic section is. This topic is covered in Chapter 8 of the online text. We start
Section 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
What you will learn today
What you will learn today Conic Sections (in 2D coordinates) Cylinders (3D) Quadric Surfaces (3D) Vectors and the Geometry of Space 1/24 Parabolas ellipses Hyperbolas Shifted Conics Conic sections result
Quadratic Forms Formula Vertex Axis of Symmetry. 2. Write the equation in intercept form. 3. Identify the Vertex. 4. Identify the Axis of Symmetry.
CC Algebra II Test # Quadratic Functions - Review **Formulas Name Quadratic Forms Formula Vertex Axis of Symmetry Vertex Form f (x) = a(x h) + k Standard Form f (x) = ax + b x + c x = b a Intercept Form
ALGEBRA II UNIT X: Conic Sections Unit Notes Packet
Name: Period: ALGEBRA II UNIT X: Conic Sections Unit Notes Packet Algebra II Unit 10 Plan: This plan is subject to change at the teacher s discretion. Section Topic Formative Work Due Date 10.3 Circles
6.4 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.
Final Exam Review Algebra Semester 1
Final Exam Review Algebra 015-016 Semester 1 Name: Module 1 Find the inverse of each function. 1. f x 10 4x. g x 15x 10 Use compositions to check if the two functions are inverses. 3. s x 7 x and t(x)
2. 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)
WHAT 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
Section 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
Chapter 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.
Unit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =
Properties 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
F.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
7. r = r = r = r = r = 2 5
Exercise a: I. Write the equation in standard form of each circle with its center at the origin and the given radius.. r = 4. r = 6 3. r = 7 r = 5 5. r = 6. r = 6 7. r = 0.3 8. r =.5 9. r = 4 0. r = 3.
The point (x, y) lies on the circle of radius r and center (h, k) iff. x h y k r
NOTES +: ANALYTIC GEOMETRY NAME LESSON. GRAPHS OF EQUATIONS IN TWO VARIABLES (CIRCLES). Standard form of a Circle The point (x, y) lies on the circle of radius r and center (h, k) iff x h y k r Center:
Quadratic 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
Multivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
Slide 2 / 222. Algebra II. Quadratic Functions
Slide 1 / 222 Slide 2 / 222 Algebra II Quadratic Functions 2014-10-14 www.njctl.org Slide 3 / 222 Table of Contents Key Terms Explain Characteristics of Quadratic Functions Combining Transformations (review)
UNIT 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
Algebra II Chapter 10 Conics Notes Packet. Student Name Teacher Name
Algebra II Chapter 10 Conics Notes Packet Student Name Teacher Name 1 Conic Sections 2 Identifying Conics Ave both variables squared?' No PARABOLA y = a(x- h)z + k x = a(y- k)z + h YEs Put l'h squared!'erms
Quadratic 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
Section 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
Polynomial 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
Chapter 9 Topics in Analytic Geometry
Chapter 9 Topics in Analytic Geometry What You ll Learn: 9.1 Introduction to Conics: Parabolas 9.2 Ellipses 9.3 Hyperbolas 9.5 Parametric Equations 9.6 Polar Coordinates 9.7 Graphs of Polar Equations 9.1
UNIT 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
Substituting a 2 b 2 for c 2 and using a little algebra, we can then derive the standard equation for an ellipse centred at the origin,
Conics onic sections are the curves which result from the intersection of a plane with a cone. These curves were studied and revered by the ancient Greeks, and were written about extensively by both Euclid
Today is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class
Today is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class Back board says your name if you are on my roster. I need parent financial
1.1 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
Quadratic 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,
Transformations 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
CHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS
CHAPTER 8 QUADRATIC RELATIONS AND CONIC SECTIONS Big IDEAS: 1) Writing equations of conic sections ) Graphing equations of conic sections 3) Solving quadratic systems Section: Essential Question 8-1 Apply
Example 1: Give the coordinates of the points on the graph.
Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,
Module 3: Stand Up Conics
MATH55 Module 3: Stand Up Conics Main Math concepts: Conic Sections (i.e. Parabolas, Ellipses, Hyperbolas), nd degree equations Auxilliary ideas: Analytic vs. Co-ordinate-free Geometry, Parameters, Calculus.
Unit 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
Each point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
Unit 2: Functions and Graphs
AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to exactly one element in the range. The domain is the set of all possible
EXERCISE 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
Graphs and transformations, Mixed Exercise 4
Graphs and transformations, Mixed Exercise 4 a y = x (x ) 0 = x (x ) So x = 0 or x = The curve crosses the x-axis at (, 0) and touches it at (0, 0). y = x x = x( x) As a = is negative, the graph has a
GRAPHING 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),
8.5 Graph and Write Equations of Hyperbolas p.518 What are the parts of a hyperbola? What are the standard form equations of a hyperbola?
8.5 Graph and Write Equations of Hyperbolas p.518 What are the parts of a hyperbola? What are the standard form equations of a hyperbola? How do you know which way it opens? Given a & b, how do you find
Student 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.)
13.1 2/20/2018. Conic Sections. Conic Sections: Parabolas and Circles
13 Conic Sections 13.1 Conic Sections: Parabolas and Circles 13.2 Conic Sections: Ellipses 13.3 Conic Sections: Hyperbolas 13.4 Nonlinear Systems of Equations 13.1 Conic Sections: Parabolas and Circles
Assignment Assignment for Lesson 14.1
Assignment Assignment for Lesson.1 Name Date The Origin of Parabolas Parabolas Centered at the Origin 1. Consider the parabola represented by the equation y 2 12x 0. a. Write the equation of the parabola
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Pre-Calculus Mid Term Review. January 2014 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the graph of the function f, plotted with a solid
3. parallel: (b) and (c); perpendicular (a) and (b), (a) and (c)
SECTION 1.1 1. Plot the points (0, 4), ( 2, 3), (1.5, 1), and ( 3, 0.5) in the Cartesian plane. 2. Simplify the expression 13 7 2. 3. Use the 3 lines whose equations are given. Which are parallel? Which
Quadratic 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
Module 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
Conic Sections: Parabolas
Conic Sections: Parabolas Why are the graphs of parabolas, ellipses, and hyperbolas called 'conic sections'? Because if you pass a plane through a double cone, the intersection of the plane and the cone
Pre-Calculus Guided Notes: Chapter 10 Conics. A circle is
Name: Pre-Calculus Guided Notes: Chapter 10 Conics Section Circles A circle is _ Example 1 Write an equation for the circle with center (3, ) and radius 5. To do this, we ll need the x1 y y1 distance formula:
KEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila
KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila January 26, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic
WK # 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
Section 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
Geometry: Conic Sections
Conic Sections Introduction When a right circular cone is intersected by a plane, as in figure 1 below, a family of four types of curves results. Because of their relationship to the cone, they are called
Mid-Chapter Quiz: Lessons 7-1 through 7-3
Write an equation for and graph a parabola with the given focus F and vertex V 1. F(1, 5), V(1, 3) Because the focus and vertex share the same x coordinate, the graph is vertical. The focus is (h, k +
Unit 6 Quadratic Functions
Unit 6 Quadratic Functions 12.1 & 12.2 Introduction to Quadratic Functions What is A Quadratic Function? How do I tell if a Function is Quadratic? From a Graph The shape of a quadratic function is called
, minor axis of length 12. , asymptotes y 2x. 16y
Math 4 Midterm 1 Review CONICS [1] Find the equations of the following conics. If the equation corresponds to a circle find its center & radius. If the equation corresponds to a parabola find its focus
Warm-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
MAC Learning Objectives. Transformation of Graphs. Module 5 Transformation of Graphs. - A Library of Functions - Transformation of Graphs
MAC 1105 Module 5 Transformation of Graphs Learning Objectives Upon completing this module, you should be able to: 1. Recognize the characteristics common to families of functions. 2. Evaluate and graph
MAC Module 5 Transformation of Graphs. Rev.S08
MAC 1105 Module 5 Transformation of Graphs Learning Objectives Upon completing this module, you should be able to: 1. Recognize the characteristics common to families of functions. 2. Evaluate and graph
Put your initials on the top of every page, in case the pages become separated.
Math 1201, Fall 2016 Name (print): Dr. Jo Nelson s Calculus III Practice for 1/2 of Final, Midterm 1 Material Time Limit: 90 minutes DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO. This exam contains
Quadratic 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
2.) Write the standard form of the equation of a circle whose endpoints of diameter are (4, 7) and (2,3).
Ch 10: Conic Sections Name: Objectives: Students will be able to: -graph parabolas, hyperbolas and ellipses and answer characteristic questions about these graphs. -write equations of conic sections Dec
Algebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor
Algebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor In this section, we will look at the hyperbola. A hyperbola is a set of points P in a plane such that the absolute value of the difference between
Section Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
But 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
3.5D Graphing Rational Functions
3.5D Graphing Rational Functions A. Strategy 1. Find all asymptotes (vertical, horizontal, oblique, curvilinear) and holes for the function. 2. Find the and intercepts. 3. Plot the and intercepts, draw
Complex Numbers, Polar Equations, and Parametric Equations. Copyright 2017, 2013, 2009 Pearson Education, Inc.
8 Complex Numbers, Polar Equations, and Parametric Equations Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 8.5 Polar Equations and Graphs Polar Coordinate System Graphs of Polar Equations Conversion
Name: Date: Period: Parabolas Day One Activity Sheet and Homework. The diagram below depicts a parabola with its directrix, focus, and vertex.
Name: ate: Period: Parabolas ay One Activity Sheet and Homework The diagram below depicts a parabola with its directrix, focus, and vertex. 1. or each of the parabolas below, two of the three parts is
1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
8.4 Directing Our Focus A Develop Understanding Task
16 8.4 Directing Our Focus A Develop Understanding Task On a board in your classroom, your teacher has set up a point and a line like this: Focus (point A) www.flickr.com/photos/pixelthing directrix (line
9.1 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