Using the data above Height range: 6 1to 74 inches Weight range: 95 to 205

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Using the data above Height range: 6 1to 74 inches Weight range: 95 to 205"

Transcription

1 Plotting When plotting data, ou will normall be using two numbers, one for the coordinate, another for the coordinate. In some cases, like the first assignment, ou ma have onl one value. There, the second coordinate has no meaning. The data below is provided as an eample. Suppose ou were to plot height vs weight. Number Height Weight Age Number Height Weight Age First ou would need to set up a scale on the paper. Each space on the paper must represent the same amount. So the points will not normall be right on the grid lines. The plot must fit on our paper, so first look at the range of values ou will represent. Using the data above Height range: 6 1to 7 inches Weight range: 9 to 0 Your scale then needs to etend AT LEAST from 61 to 7 inches on one ais and AT LEAST from 9 to 0lb on the other. It does not need to go through zero pounds or zero inches. You know there will never be a need to plot data for a zero height person. It is our choice of what number of inches correspond to the major grid lines. You should make our plot take up enough of the paper that it is eas to see what ou are plotting. It is good to make the grid marks correspond to an even number of units, just to save effort in counting and to avoid making errors. There is no reason to use the same number of squares on the grid for the two aes. In the eample, the range in number of pounds is much larger than the range in inches. There is ever reason to use different numbers of pounds for each scale AND to use the larger dimension for the pounds where ou are likel to need more space. Don t label the points. Linear vs Logarithmic Plot In a normal linear plot, each space represents the same amount. In a logarithmic plot, equal spaces represent the same power of 10. So tpical major intervals on a logarithmic scale are as shown

2 below To use a logarithmic plot b hand, put numbers into scientific notation first. Then look for the sane power of 10 as in the scientific notation. The number goes ABOVE the power of 10. The plot below shows where some points would be plotted. You would NOT generall epect to see the power of 10 on the intermediate tic marks. You would not be epected to label the points. This is just so that ou can see how it should be done , , , , Labels and Aes Each ais should have A title indicating what is being plotted and the units used (e.g. Weight in pounds) Numbers at regular intervals (e.g. 9, 100, 110 ). Tic marks, marks at these same intervals. The numbers and tic marks should continue along the entire ais. Making a Curve from Data As scientists find out information, the do not necessaril know what it means beond that there are a bunch of disconnected facts. In order to make sense of the information, the (and we) tr to find patterns. The idea is perhaps the pattern will show the relationship among all the seemingl disconnected information. One wa we seek patterns is to make graphs of the information we have gathered. If there is a pattern (and if we have graphed the right quantities with respect to one another), the points on the graph will not just fall all over the place. The might fall in a glob (i.e. in a small region of the graph paper), the might fall on a straight line, or the might fall on a curved line. In order to determine the relationship between quantities, we need an equation which describes the

3 best fit line or curve. There are rigorous was to prove that the line or curve is the best. For the purposes of this class, we will usuall use our best judgment, rather than a rigorous computation t o find the best line. Suppose that ou have some data and ou plot it. The result looks like the following figure. The dots are the measured points. The bars represent the uncertainties of the points. (Note that the scales need not start at 0.) - What is the best line or curve to fit this set of data? Is it a straight line like the following?

4 Or is it like? Or like

5 Given this data, the last solution, the parabola, is not justified, since there are no points between 0 and approimatel -0.6 on the descending side of the curve. On the other hand, the data does not rule it out. In general, ou do not use a curve like the one above based on the data shown, unless there is some other reason to know the shape of the curve (e.g. the theor sas so, and the data does not rule out other possibilities). Returning to the straight line fits. Both lines have been plotted on the same graph below. Either of these straight lines could be the best estimate of an honest, independent observer eeballing the data. We could use the difference between these lines to estimate the uncertaint in the fits. Line number 1 Line number But how can we talk about the straight line to discuss whether the are the same or different?

6 Each straight line can be written as the equation: = m + b, which is the usual form, or equivalentl: = /m -b/m In the first case, = a + b, the coordinate of each point is related to the coordinate of the same point, b the constants m and b. The constant, m, is called the slope of the line. It tells what angle the line makes with the aes. The constant, b, is called the intercept. It tells what value of results when equals zero. So lets tr to find the equations of the lines in the figure above. Lets consider line number 1 first. The value of for equal zero (on the line, not of an of the points) is. So in our potential equation for the line, we have =0, =, the equation should give =m+b. Substituting for and produces, = m*0 +b =b (es this is a trick, but it s ok ) To find m, choose an other, point on the line. For eample, line number 1 goes through the point (1,), that is =1, =. Substitute into the equation to produce: = m *1 + There is onl one unknown thing left, m. So ou surel can solve for it. Rearranging produces: - =m = m Yes! The slope can be negative. Negative slope means that the bigger that gets, the smaller that gets. Is this true of the data shown here? So ou now have both m and b for line 1. The equation for line 1 is =-+. Find the equation for line. The differences in the values of m for the two lines and for b for the two lines is a measure of the uncertaint in the equations which describe the relation of the data. for line : m=-0.7, b=.

2.1 Equations of Lines

2.1 Equations of Lines Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

More information

5. Linear regression and correlation

5. Linear regression and correlation Statistics for Engineers 5-1 5. Linear regression and correlation If we measure a response variable at various values of a controlled variable, linear regression is the process of fitting a straight line

More information

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0 LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

More information

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304 Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

More information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

Section C Non Linear Graphs

Section C Non Linear Graphs 1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1.

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1. Joint probabilit is the probabilit that the RVs & Y take values &. like the PDF of the two events, and. We will denote a joint probabilit function as P,Y (,) = P(= Y=) Marginal probabilit of is the probabilit

More information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 Inequalities with Absolute Value and Quadratic Functions 08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

More information

Between Curves. Definition. The average value of a continuous function on an interval [a, b] is given by: average value on [a, b] = 1 b a

Between Curves. Definition. The average value of a continuous function on an interval [a, b] is given by: average value on [a, b] = 1 b a Section 5.: Average Value and Area Between Curves Definition. The average value of a continuous function on an interval [a, b] is given b: average value on [a, b] = b a b a f() d. Eample. Find the average

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

Transformations of Function Graphs

Transformations of Function Graphs - - - 0 - - - - - - - Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),

More information

Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. Slope-Intercept Form

Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. Slope-Intercept Form 8 (-) Chapter Linear Equations in Two Variables and Their Graphs In this section Slope-Intercept Form Standard Form Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications

More information

Write seven terms of the Fourier series given the following coefficients. 1. a 0 4, a 1 3, a 2 2, a 3 1; b 1 4, b 2 3, b 3 2

Write seven terms of the Fourier series given the following coefficients. 1. a 0 4, a 1 3, a 2 2, a 3 1; b 1 4, b 2 3, b 3 2 36 Chapter 37 Infinite Series Eercise 5 Fourier Series Write seven terms of the Fourier series given the following coefficients.. a 4, a 3, a, a 3 ; b 4, b 3, b 3. a.6, a 5., a 3., a 3.4; b 7.5, b 5.3,

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

P1. Plot the following points on the real. P2. Determine which of the following are solutions

P1. Plot the following points on the real. P2. Determine which of the following are solutions Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in

More information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs

STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs 6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous

More information

MATH Area Between Curves

MATH Area Between Curves MATH - Area Between Curves Philippe Laval September, 8 Abstract This handout discusses techniques used to nd the area of regions which lie between two curves. Area Between Curves. Theor Given two functions

More information

HI-RES STILL TO BE SUPPLIED

HI-RES STILL TO BE SUPPLIED 1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

More information

4.1 Piecewise-Defined Functions

4.1 Piecewise-Defined Functions Section 4.1 Piecewise-Defined Functions 335 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

More information

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2).

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2). Chapter Section : Equations of Lines Answers to Problems For problems -, put our answers into slope intercept form..* Find the equation of the line with slope, and passing through the point (,0).. Find

More information

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM . Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

More information

Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY

Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

2.3 Writing Equations of Lines

2.3 Writing Equations of Lines . Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

More information

Key Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function)

Key Terms: Quadratic function. Parabola. Vertex (of a parabola) Minimum value. Maximum value. Axis of symmetry. Vertex form (of a quadratic function) Outcome R3 Quadratic Functions McGraw-Hill 3.1, 3.2 Key Terms: Quadratic function Parabola Vertex (of a parabola) Minimum value Maximum value Axis of symmetry Vertex form (of a quadratic function) Standard

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

6.3 Parametric Equations and Motion

6.3 Parametric Equations and Motion SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest

More information

Quadratic Functions. MathsStart. Topic 3

Quadratic Functions. MathsStart. Topic 3 MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

More information

Math 152, Intermediate Algebra Practice Problems #1

Math 152, Intermediate Algebra Practice Problems #1 Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work

More information

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

8. Bilateral symmetry

8. Bilateral symmetry . Bilateral smmetr Our purpose here is to investigate the notion of bilateral smmetr both geometricall and algebraicall. Actuall there's another absolutel huge idea that makes an appearance here and that's

More information

Linearizing Equations Handout Wilfrid Laurier University

Linearizing Equations Handout Wilfrid Laurier University Linearizing Equations Handout Wilfrid Laurier University c Terry Sturtevant 1 2 1 Physics Lab Supervisor 2 This document may be freely copied as long as this page is included. ii Chapter 1 Linearizing

More information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

More information

Appendix C: Graphs. Vern Lindberg

Appendix C: Graphs. Vern Lindberg Vern Lindberg 1 Making Graphs A picture is worth a thousand words. Graphical presentation of data is a vital tool in the sciences and engineering. Good graphs convey a great deal of information and can

More information

Packet: Lines (Part 1) Standards covered:

Packet: Lines (Part 1) Standards covered: Packet: Lines (Part 1) Standards covered: *(2)MA.912.A.3.8 Graph a line given any of the following information: a table of values, the x and y- intercepts, two points, the slope and a point, the equation

More information

2 Analysis of Graphs of

2 Analysis of Graphs of ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

More information

Coordinate Geometry. Positive gradients: Negative gradients:

Coordinate Geometry. Positive gradients: Negative gradients: 8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between

More information

ALGEBRA. Generate points and plot graphs of functions

ALGEBRA. Generate points and plot graphs of functions ALGEBRA Pupils should be taught to: Generate points and plot graphs of functions As outcomes, Year 7 pupils should, for eample: Use, read and write, spelling correctl: coordinates, coordinate pair/point,

More information

Families of Quadratics

Families of Quadratics Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish

More information

Chapter 3A - Rectangular Coordinate System

Chapter 3A - Rectangular Coordinate System - Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been

More information

5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED

5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given

More information

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 ) SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0 P(X, Y) X

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0 P(X, Y) X Rotation of Aes For a discussion of conic sections, see Appendi. In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general second-degree

More information

2-5. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math

2-5. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math Chapter 2 Lesson 2-5 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

x 2. 4x x 4x 1 x² = 0 x = 0 There is only one x-intercept. (There can never be more than one y-intercept; do you know why?)

x 2. 4x x 4x 1 x² = 0 x = 0 There is only one x-intercept. (There can never be more than one y-intercept; do you know why?) Math Learning Centre Curve Sketching A good graphing calculator can show ou the shape of a graph, but it doesn t alwas give ou all the useful information about a function, such as its critical points and

More information

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8. Lines and Planes. By the end of this chapter, you will Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

More information

Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

More information

7.3 Graphing Rational Functions

7.3 Graphing Rational Functions Section 7.3 Graphing Rational Functions 639 7.3 Graphing Rational Functions We ve seen that the denominator of a rational function is never allowed to equal zero; division b zero is not defined. So, with

More information

Find the Relationship: An Exercise in Graphing Analysis

Find the Relationship: An Exercise in Graphing Analysis Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course

More information

GRAPHS OF RATIONAL FUNCTIONS

GRAPHS OF RATIONAL FUNCTIONS 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique

More information

Straight Line Graphs. Teachers Teaching with Technology. Scotland T 3. y = mx + c. Teachers Teaching with Technology (Scotland)

Straight Line Graphs. Teachers Teaching with Technology. Scotland T 3. y = mx + c. Teachers Teaching with Technology (Scotland) Teachers Teaching with Technolog (Scotland) Teachers Teaching with Technolog T 3 Scotland Straight Line Graphs = m + c Teachers Teaching with Technolog (Scotland) STRAIGHT LINE GRAPHS Aim The aim of this

More information

Introduction. Introduction

Introduction. Introduction Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation

More information

Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations

Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.

More information

Supplementary Lesson: Log-log and Semilog Graph Paper

Supplementary Lesson: Log-log and Semilog Graph Paper Supplementar Lesson: Log-log and Semilog Graph Paper Chapter 7 looks at some elementar functions of algebra, including linear, quadratic, power, eponential, and logarithmic. The following supplementar

More information

REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

More information

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model . Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

More information

Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

More information

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship 6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes $25 an hour as an electrician. If he works

More information

San Jose State University Engineering 10 1

San Jose State University Engineering 10 1 KY San Jose State University Engineering 10 1 Select Insert from the main menu Plotting in Excel Select All Chart Types San Jose State University Engineering 10 2 Definition: A chart that consists of multiple

More information

17.1 Connecting Intercepts and Zeros

17.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,

More information

GRAPH OF A RATIONAL FUNCTION

GRAPH OF A RATIONAL FUNCTION GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find

More information

1.6. Determine a Quadratic Equation Given Its Roots. Investigate

1.6. Determine a Quadratic Equation Given Its Roots. Investigate 1.6 Determine a Quadratic Equation Given Its Roots Bridges like the one shown often have supports in the shape of parabolas. If the anchors at either side of the bridge are 4 m apart and the maximum height

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

Solving Absolute Value Equations and Inequalities Graphically

Solving Absolute Value Equations and Inequalities Graphically 4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value

More information

13 Graphs, Equations and Inequalities

13 Graphs, Equations and Inequalities 13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,

More information

MULTIPLE REPRESENTATIONS through 4.1.7

MULTIPLE REPRESENTATIONS through 4.1.7 MULTIPLE REPRESENTATIONS 4.1.1 through 4.1.7 The first part of Chapter 4 ties together several was to represent the same relationship. The basis for an relationship is a consistent pattern that connects

More information

Math 1400/1650 (Cherry): Quadratic Functions

Math 1400/1650 (Cherry): Quadratic Functions Math 100/1650 (Cherr): Quadratic Functions A quadratic function is a function Q() of the form Q() = a + b + c with a 0. For eample, Q() = 3 7 + 5 is a quadratic function, and here a = 3, b = 7 and c =

More information

sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj

sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models

More information

2.5 Library of Functions; Piecewise-defined Functions

2.5 Library of Functions; Piecewise-defined Functions SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,

More information

8 Graphs of Quadratic Expressions: The Parabola

8 Graphs of Quadratic Expressions: The Parabola 8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be

More information

Packet 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2

Packet 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2 Packet 1 for Unit Intercept Form of a Quadratic Function M Alg 1 Assignment A: Graphs of Quadratic Functions in Intercept Form (Section 4.) In this lesson, you will: Determine whether a function is linear

More information

Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line

Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

THE POINT-SLOPE FORM

THE POINT-SLOPE FORM . The Point-Slope Form (-) 67. THE POINT-SLOPE FORM In this section In Section. we wrote the equation of a line given its slope and -intercept. In this section ou will learn to write the equation of a

More information

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) = Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

More information

Rationale/Lesson Abstract: Students will graph exponential functions, identify key features and learn how the variables a, h and k in f x a b

Rationale/Lesson Abstract: Students will graph exponential functions, identify key features and learn how the variables a, h and k in f x a b Grade Level/Course: Algebra Lesson/Unit Plan Name: Graphing Eponential Functions Rationale/Lesson Abstract: Students will graph eponential functions, identif ke features h and learn how the variables a,

More information

4.9 Graph and Solve Quadratic

4.9 Graph and Solve Quadratic 4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC

More information

SAMPLE. Polynomial functions

SAMPLE. Polynomial functions Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

More information

EXPLORE EXPLAIN 1. Representing an Interval on a Number Line INTEGRATE TECHNOLOGY. INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling

EXPLORE EXPLAIN 1. Representing an Interval on a Number Line INTEGRATE TECHNOLOGY. INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Locker LESSON 1.1 Domain, Range, and End Behavior Teas Math Standards The student is epected to: A.7.1 Write the domain and range of a function in interval notation, inequalities, and set notation. Mathematical

More information

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving linear sstems of equations b graphing Common Core Standards Algebra: Solve sstems of equations.

More information

The Slope-Intercept Form

The Slope-Intercept Form 7.1 The Slope-Intercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph

More information

The Distance Formula and the Circle

The Distance Formula and the Circle 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

More information