Lesson 8: Graphs and Graphing Linear Equations
|
|
|
- Darcy Nichols
- 5 years ago
- Views:
Transcription
1 In this chapter, we will begin looking at the relationships between two variables. Typically one variable is considered to be the input, and the other is called the output. The input is the value that is considered first, and the output is the value that corresponds to or is matched with the input. The input/output designation may represent a cause/effect relationship, but that is not always the case. A critical skill required for the study of algebra is the ability to construct and interpret graphs. In this lesson we will learn how the Cartesian plane is used for constructing graphs and plotting data. We will interpret the points and behavior of a graph with respect to the input and output variables. We will also learn the rules/ guidelines for constructing good graphs so that the graphs we create can be easily read and understood. Section 8.1: The Cartesian Plane Section 8.2: Constructing Good Graphs from Data Section 8.3: Linear Equations Two Variables Section 8.4: Graphing Linear Equations by Plotting Points Section 8.5: Intercepts Section 8.6: Horizontal and Vertical Lines Scottsdale Community College Page 147 Introductory Algebra
2 Lesson 8 Checklist Component Required? Y or N Comments Due Score Online Homework Online Quiz Online Test Practice Problems Lesson Assessment Scottsdale Community College Page 148 Introductory Algebra
3 Name: Date: 8 Section 8.1: The Cartesian Plane In this chapter, we will begin looking at the relationships between two variables. Typically one variable is considered to be the INPUT, and the other is called the OUTPUT. The input is the value that is considered first, and the output is the value that corresponds to or is matched with the input. The input/output designation may represent a cause/effect relationship, but that is not always the case. Ordered Pairs Example 1: Ordered Pairs (input value, corresponding output value) Input Output Ordered Pairs (input, output) 4!3 5 8 ( 0,!4) (!2, 6) Example 2: The Rectangular Coordinate System (Cartesian Coordinate System) Scottsdale Community College Page 149 Introductory Algebra
4 Plot and label the points. A. (-4, 2) B. (3, 8) C. (0, -5) D. (-6, -4) E. (5, 0) F. (2, -8) G. (0, 0) Quadrants Quadrant Coordinates I (+, +) II (, +) III (, ) IV (+, ) 1. You Try Plot and label the points. A. ( 6,!3) B. ( 1, 9) C. (!4, 0) D. (!2,!8) E. ( 0, 5) Scottsdale Community College Page 150 Introductory Algebra
5 Section 8.2: Constructing a Graph from Data Criteria for a Good Graph 1. The horizontal axis should be properly labeled with the name and units of the input variable. 2. The vertical axis should be properly labeled with the name and units of the output variable. 3. Use an appropriate scale. a. Start at or just below the lowest value. b. End at or just above the highest value. c. Scale the graph so the adjacent tick marks are equal distance apart. d. Use numbers that make sense for the given data set. e. The axes meet at (0,0). Use a // between the origin and the first tick mark if the scale does not begin at All points should be plotted correctly, and the graph should make use of the available space. Example 1: The table below shows the total distance (including reaction time and deceleration time) it takes a car traveling at various speeds to come to a complete stop. Speed (miles per hour) Stopping Distance (ft) Scottsdale Community College Page 151 Introductory Algebra
6 You Try 2. Consider the following data set. Elapsed time (seconds) Height of Golf Ball (feet) a. What is the input variable? b. What was the height of the ball after 3 seconds? c. After how many seconds was the ball 50 feet in the air? d. In a complete sentence, interpret the meaning of the ordered pair (1, 43). e. Construct a good graph of this data. Scottsdale Community College Page 152 Introductory Algebra
7 Section 8.3: Linear Equations - Two Variables Linear Equations Example 1: Verify that the ordered pairs below satisfy the equation y = 2x + 3. (!2,!1) ( 0,3) ( 1, 5) Scottsdale Community College Page 153 Introductory Algebra
8 Example 2: Verify that the ordered pairs below satisfy the equation 3x + 2y = 6. (!2, 6) ( 0,3) ( 2, 0) You Try 3. Verify that the ordered pairs below satisfy the equation y = 3x 2? (!2,!8) ( 3, 7) ( 0,!2) Scottsdale Community College Page 154 Introductory Algebra
9 Section 8.4: Graphing Linear Equations - Two Variables Graphing Linear Equations by Plotting Points Step 1: Choose two or more values for the input variable, and list them in a table of values. Step 2: Substitute each input value into the equation and compute the corresponding output values. List these values in the table. Step 3: Write each input-output pair in the table as an ordered pair. Step 4: Plot the ordered pairs, connect them, and extend the line beyond the points to show that the pattern continues Example 1: Graph the equation y = 3x 2 x y Ordered Pair Example 2: Graph the equation y = 2 3 x!1 x y Ordered Pair Scottsdale Community College Page 155 Introductory Algebra
10 Example 3: Graph the equation 4x + 2y = 10 x y Ordered Pair 4. Graph the equation y = 1 2 x + 3 You Try x y Ordered Pair Scottsdale Community College Page 156 Introductory Algebra
11 Section 8.5: Intercepts Vertical and Horizontal Intercepts The vertical intercept (y-intercept) is the point at which the graph crosses the vertical axis. The input value of the vertical intercept is always The coordinates of the vertical intercept will be To determine the vertical intercept: The horizontal intercept (x-intercept) is the point at which the graph crosses the horizontal axis. The output value of the horizontal intercept is always The coordinates of the horizontal intercept will be To determine the horizontal intercept: Example 1: Determine the vertical and horizontal intercepts for y = 3x 2. Example 2: Determine the vertical and horizontal intercepts for 4x 2y = 10. Scottsdale Community College Page 157 Introductory Algebra
12 You Try 5. Complete the table below. Write the intercepts as ordered pairs. Equation Vertical Intercept Horizontal Intercept y = 24! 6x 5x! 3y = 30 Scottsdale Community College Page 158 Introductory Algebra
13 Section 8.6: Horizontal and Vertical Lines Horizontal Lines y = b, where b is a real constant Example 1: Graph the equation y = 2. x y Ordered Pair Vertical Lines x = k, where k is a real constant Example 2: Graph the equation x =!3. x y Ordered Pair You Try 6. Graph the equation y =!2. Graph the equation x = 4. Scottsdale Community College Page 159 Introductory Algebra
14 Scottsdale Community College Page 160 Introductory Algebra
Lesson 8: Graphs and Graphing Linear Equations
A critical skill required for the study of algebra is the ability to construct and interpret graphs. In this lesson we will learn how the Cartesian plane is used for constructing graphs and plotting data.
Lesson 8 Introduction to Quadratic Functions
Lesson 8 Introduction to Quadratic Functions We are leaving exponential and logarithmic functions behind and entering an entirely different world. As you work through this lesson, you will learn to identify
Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations origin (x, y) Ordered pair (x-coordinate, y-coordinate) (abscissa, ordinate) x axis Rectangular or
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,
The Rectangular Coordinate System and Equations of Lines. College Algebra
The Rectangular Coordinate System and Equations of Lines College Algebra Cartesian Coordinate System A grid system based on a two-dimensional plane with perpendicular axes: horizontal axis is the x-axis
Lesson 11 Rational Functions
Lesson 11 Rational Functions In this lesson, you will embark on a study of rational functions. These may be unlike any function you have ever seen. Rational functions look different because they are in
UNIT 4 DESCRIPTIVE STATISTICS Lesson 2: Working with Two Categorical and Quantitative Variables Instruction
Prerequisite Skills This lesson requires the use of the following skills: plotting points on the coordinate plane, given data in a table plotting the graph of a linear function, given an equation plotting
Properties 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
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
UNIT 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
Lesson 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
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
Lesson 12 Course Review
In this lesson, we will review the topics and applications from Lessons 1-11. We will begin with a review of the different types of functions, and then apply each of them to a set of application problems.
Graphs 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
3-6 Lines in the Coordinate Plane
3-6 Lines in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 1. m = 2, x = 3, and
2.1 Linear Equations in Two Variables
2.1 Linear Equations in Two Variables Concept 1: The Rectangular Coordinate System 2. Let a and b represent nonzero real numbers. Then 1. An ordered pair of the form (0, b) represents a point on which
Section 6.1: Quadratic Functions and their Characteristics Vertical Intercept Vertex Axis of Symmetry Domain and Range Horizontal Intercepts
Lesson 6 Quadratic Functions and Equations Lesson 6 Quadratic Functions and Equations We are leaving exponential functions behind and entering an entirely different world. As you work through this lesson,
Lesson 6a Exponents and Rational Functions
Lesson 6a Eponents and Rational Functions In this lesson, we put quadratics aside for the most part (not entirely) in this lesson and move to a study of eponents and rational functions. The rules of eponents
Graphing Linear Equations
Graphing Linear Equations Question 1: What is a rectangular coordinate system? Answer 1: The rectangular coordinate system is used to graph points and equations. To create the rectangular coordinate system,
8-4 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
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 + )
Lesson 8 Practice Problems
Name: Date: Lesson 8 Section 8.1: Characteristics of Quadratic Functions 1. For each of the following quadratic functions: Identify the coefficients a, b, c Determine if the parabola opens up or down and
Lesson 6 - Practice Problems
Lesson 6 - Practice Problems Section 6.1: Characteristics of Quadratic Functions 1. For each of the following quadratic functions: Identify the coefficients a, b and c. Determine if the parabola opens
Did You Find a Parking Space?
Lesson.4 Skills Practice Name Date Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane Vocabulary Complete the sentence. 1. The point-slope form of the equation of the
Section 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,
QUADRATICS Graphing Quadratic Functions Common Core Standard
H Quadratics, Lesson 6, Graphing Quadratic Functions (r. 2018) QUADRATICS Graphing Quadratic Functions Common Core Standard Next Generation Standard F-IF.B.4 For a function that models a relationship between
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.
4.1 Ordered Pairs and Graphs. Copyright Cengage Learning. All rights reserved.
4.1 Ordered Pairs and Graphs Copyright Cengage Learning. All rights reserved. 1 What You Will Learn Plot points on a rectangular coordinate system Determine whether ordered pairs are solutions of equations
Linear, Quadratic, and Exponential Models Attendance Problems 1. Find the slope and y-intercept of the line that passes through (4, 20) and (20, 24).
Page 1 of 13 Linear, Quadratic, and Exponential Models Attendance Problems 1. Find the slope and y-intercept of the line that passes through (4, 20) and (20, 24). The population of a town is decreasing
To sketch the graph we need to evaluate the parameter t within the given interval to create our x and y values.
Module 10 lesson 6 Parametric Equations. When modeling the path of an object, it is useful to use equations called Parametric equations. Instead of using one equation with two variables, we will use two
+ b. From this we can derive the following equations:
A. GEOMETRY REVIEW Pythagorean Theorem (A. p. 58) Hypotenuse c Leg a 9º Leg b The Pythagorean Theorem is a statement about right triangles. A right triangle is one that contains a right angle, that is,
It 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
L13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11, page 1 L13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11
L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 1 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11 1.1 Rectangular Coordinate System: Suppose we know the sum of
Lesson 8 Practice Problems
Name: Date: Lesson 8 Skills Practice 1. Plot and label the points. A. (8, 2) B. (0, 0) C. (0, 5) D. (10, 10) E. ( 4, 4) F. ( 9, 1) G. ( 5, 0) H. (2, 8) 2. Give the coordinates of each of the points shown
Chapter 11 GRAPHS OF LINEAR EQUATIONS
Chapter 11 GRAPHS OF LINEAR EQUATIONS 11.1 Graphs and Applications of Linear Equations Learning Objectives A Plot points associated with ordered pairs of numbers; determine the quadrant in which a point
Lesson 11 Practice Problems
Name: Date: Lesson 11 Section 11.1: Characteristics of Rational Functions 1. Complete the table below. Function Domain a) = 4 6 2 b) = 8 + 2 3 9 c) = d) = 12 6 e) = 2 1 4 2 f) = 3 4 2 + 3 g) = 8 3 Page
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
Algebra 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
Section 4.4 Quadratic Functions in Standard Form
Section 4.4 Quadratic Functions in Standard Form A quadratic function written in the form y ax bx c or f x ax bx c is written in standard form. It s not right to write a quadratic function in either vertex
6.1 The Rectangular Coordinate System
6.1 The Rectangular Coordinate System In this chapter we want to take a look at the connection between algebra and geometry. This connection is made in the graphing of equations. We will start by looking
Algebra 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
Section 3.1. Reading graphs and the Rectangular Coordinate System
Section 3.1 Reading graphs and the Rectangular Coordinate System Learning objectives Read bar & line graphs Introduce the rectangular coordinate system Graph paired data points on the rectangular coordinate
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)
Chapter 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
Lesson 2b Functions and Function Operations
As we continue to work with more complex functions it is important that we are comfortable with Function Notation, opertions on Functions and opertions involving more than one function. In this lesson,
Quiz 1 Review: Quadratics through 4.2.2
Name: Class: Date: ID: A Quiz 1 Review: Quadratics 4.1.1 through 4.2.2 Graph each function. How is each graph a translation of f(x) = x 2? 1. y = x 2 + 2 2. y = (x 3) 2 3. y = (x + 3) 2 + 4 4. Which is
Intro. To Graphing Linear Equations
Intro. To Graphing Linear Equations The Coordinate Plane A. The coordinate plane has 4 quadrants. B. Each point in the coordinate plain has an x-coordinate (the abscissa) and a y-coordinate (the ordinate).
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
Objective. 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
1.6 Modeling with Equations
1.6 Modeling with Equations Steps to Modeling Problems with Equations 1. Identify the variable you want to solve for. 2. Express all unknown quantities in terms of this variable. 3. Set up the model by
Standard Form v. Vertex Form
Standard Form v. Vertex Form The Standard Form of a quadratic equation is:. The Vertex Form of a quadratic equation is where represents the vertex of an equation and is the same a value used in the Standard
3.1 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
6.7. Graph Linear Inequalities in Two Variables. Warm Up Lesson Presentation Lesson Quiz
6.7 Graph Linear Inequalities in Two Variables Warm Up Lesson Presentation Lesson Quiz 6.7 Warm-Up Tell whether the ordered pair is a solution of the equation. 1. x + 2y = 4; (2, 1) no 2. 4x + 3y = 22;
Notes Lesson 3 4. Positive. Coordinate. lines in the plane can be written in standard form. Horizontal
A, B, C are Notes Lesson 3 4 Standard Form of an Equation: Integers Ax + By = C Sometimes it is preferred that A is Positive All lines in the plane can be written in standard form. Oblique Coordinate Horizontal
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
Section 9.3: Functions and their Graphs
Section 9.: Functions and their Graphs Graphs provide a wa of displaing, interpreting, and analzing data in a visual format. In man problems, we will consider two variables. Therefore, we will need to
Applied Parabolas: Catapult (one test grade)
Name: I. Overview: PreCalculus Applied Parabola Project Applied Parabolas: Catapult (one test grade) You will use catapults to launch candy into the air. Using a stopwatch, you will time how long the projectile
Mathematical 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
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
Section 3.1: Introduction to Linear Equations in 2 Variables Section 3.2: Graphing by Plotting Points and Finding Intercepts
Remember to read the tetbook before attempting to do our homework. Section 3.1: Introduction to Linear Equations in 2 Variables Section 3.2: Graphing b Plotting Points and Finding Intercepts Rectangular
Quadratics Functions: Review
Quadratics Functions: Review Name Per Review outline Quadratic function general form: Quadratic function tables and graphs (parabolas) Important places on the parabola graph [see chart below] vertex (minimum
Name Class Date. Using Graphs to Relate Two Quantities
4-1 Reteaching Using Graphs to Relate Two Quantities An important life skill is to be able to a read graph. When looking at a graph, you should check the title, the labels on the axes, and the general
Review for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
Sample: Do Not Reproduce QUAD4 STUDENT PAGES. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications
Name Period Date QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications QUAD 4.1 Vertex Form of a Quadratic Function 1 Explore how changing the values of h and
E Linear Equations, Lesson 2, Graphing Linear Functions (r. 2018) LINEAR EQUATIONS Graphing Linear Functions Common Core Standards
E Linear Equations, Lesson 2, Graphing Linear Functions (r. 2018) LINEAR EQUATIONS Graphing Linear Functions Common Core Standards Next Generation Standards A-CED.A.2 Create equations in two or more variables
Chpt 1. Functions and Graphs. 1.1 Graphs and Graphing Utilities 1 /19
Chpt 1 Functions and Graphs 1.1 Graphs and Graphing Utilities 1 /19 Chpt 1 Homework 1.1 14, 18, 22, 24, 28, 42, 46, 52, 54, 56, 78, 79, 80, 82 2 /19 Objectives Functions and Graphs Plot points in the rectangular
Section 7.6 Graphs of the Sine and Cosine Functions
Section 7.6 Graphs of the Sine and Cosine Functions We are going to learn how to graph the sine and cosine functions on the xy-plane. Just like with any other function, it is easy to do by plotting points.
Chapter 3 Practice Test
1. Complete parts a c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex.
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.
( )! 1! 3 = x + 1. ( ) =! x + 2
7.5 Graphing Parabolas 1. First complete the square: y = x 2 + 2x! 3 = x 2 + 2x + 1 ( )! 1! 3 = x + 1 ( ) 2! 4 The x-intercepts are 3,1 and the vertex is ( 1, 4). Graphing the parabola: 3. First complete
7/7/2016 Unit 4: Linear Relations Grade 9 Mathematics
Rene Descartes, a mathematician who lived during the 17 th century, developed a system for graphing ordered pairs on a grid. This system is called the Cartesian Coordinate System. 1 In this system, ordered
BLM Answers. BLM 4-1 Prerequisite Skills. BLM 4-3 Section 4.1 Modelling With Quadratic Relations. 10. a)
BLM Answers (page 1) BLM 4-1 Prerequisite Skills 1. a) 11.1 2.7 9.0 d) 20.2 2. a) 1.7 10.7 6.5 d) 25.1 3. a) 9.5 20.7 96 d) 31.85 4. a) 3x 6x 2 + 6x + 5 10x 2 2x + 6 d) 12x 2 + 10x 6 5. a) 5 0 12 d) 2
, 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
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
By naming a function f, you can write the function using function notation. Function notation. ACTIVITY: Matching Functions with Their Graphs
5. Function Notation represent a function? How can ou use function notation to B naming a function f, ou can write the function using function notation. f () = Function notation This is read as f of equals
Meeting 1 Introduction to Functions. Part 1 Graphing Points on a Plane (REVIEW) Part 2 What is a function?
Meeting 1 Introduction to Functions Part 1 Graphing Points on a Plane (REVIEW) A plane is a flat, two-dimensional surface. We describe particular locations, or points, on a plane relative to two number
Quadratic 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
FOA/Algebra 1. Unit 2B Review - Linear Functions
FOA/Algebra Unit B Review Name: Date: Block: Unit B Review - Linear Functions What you need to know & be able to do. Determine if a relation is a Things to remember Every input only has one output (each
Parallel lines are lines that never intersect and are always the same distance apart. Parallel Lines
Lesson 4.6 Objectives Determine if two lines are parallel or perpendicular. Write equations of parallel and perpendicular lines. Slopes of Parallel and Perpendicular Lines Parallel and perpendicular lines
February 13, notebook
Module 12 Lesson 1: Graphing on the coordinate plane Lesson 2: Independent and dependent variables in tables and graphs Lesson 3: Writing equations from tables Lesson 4: Representing Algebraic relationships
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)
NO 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
Chapter 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
Name Course Days/Start Time
Name Course Days/Start Time Mini-Project : The Library of Functions In your previous math class, you learned to graph equations containing two variables by finding and plotting points. In this class, we
Assignments for Algebra 1 Unit 9 Quadratics, Part 1
Name: Assignments for Algebra 1 Unit 9 Quadratics, Part 1 Day 1, Quadratic Transformations: p.1-2 Day 2, Vertex Form of Quadratics: p. 3 Day 3, Solving Quadratics: p. 4-5 Day 4, No Homework (be sure you
The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.
We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section
10-2 Circles. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra2 2
10-2 Circles Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 2 Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and ( 1, 6) 1 6 2. (3, 4) and ( 4, 3) 1 Warm Up Find
Unit: 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
Lesson 10 Rational Functions and Equations
Lesson 10 Rational Functions and Equations Lesson 10 Rational Functions and Equations In this lesson, you will embark on a study of rational functions. Rational functions look different because they are
Algebra Unit 2: Linear Functions Notes. Slope Notes. 4 Types of Slope. Slope from a Formula
Undefined Slope Notes Types of Slope Zero Slope Slope can be described in several ways: Steepness of a line Rate of change rate of increase or decrease Rise Run Change (difference) in y over change (difference)
Lesson 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),
Translation of graphs (2) The exponential function and trigonometric function
Lesson 35 Translation of graphs (2) The exponential function and trigonometric function Learning Outcomes and Assessment Standards Learning Outcome 2: Functions and Algebra Assessment Standard Generate
7-5 Parametric Equations
3. Sketch the curve given by each pair of parametric equations over the given interval. Make a table of values for 6 t 6. t x y 6 19 28 5 16.5 17 4 14 8 3 11.5 1 2 9 4 1 6.5 7 0 4 8 1 1.5 7 2 1 4 3 3.5
Rational Numbers: Graphing: The Coordinate Plane
Rational Numbers: Graphing: The Coordinate Plane A special kind of plane used in mathematics is the coordinate plane, sometimes called the Cartesian plane after its inventor, René Descartes. It is one
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
MATH ALGEBRA AND FUNCTIONS 5 Performance Objective Task Analysis Benchmarks/Assessment Students:
Students: 1. Use information taken from a graph or Which table, a or b, matches the linear equation to answer questions about a graph? problem situation. y 1. Students use variables in simple expressions,
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
REMARKS. 8.2 Graphs of Quadratic Functions. A Graph of y = ax 2 + bx + c, where a > 0
8. Graphs of Quadratic Functions In an earlier section, we have learned that the graph of the linear function = m + b, where the highest power of is 1, is a straight line. What would the shape of the graph
A I only B II only C II and IV D I and III B. 5 C. -8
1. (7A) Points (3, 2) and (7, 2) are on the graphs of both quadratic functions f and g. The graph of f opens downward, and the graph of g opens upward. Which of these statements are true? I. The graphs
Math 2201 Unit 4: Quadratic Functions. 16 Hours
Math 2201 Unit 4: Quadratic Functions 16 Hours 6.1: Exploring Quadratic Relations Quadratic Relation: A relation that can be written in the standard form y = ax 2 + bx + c Ex: y = 4x 2 + 2x + 1 ax 2 is