# -2- Reason: This is harder. I'll give an argument in an Addendum to this handout.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 LINES Slope The slope of a nonvertical line in a coordinate plane is defined as follows: Let P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) be any two points on the line. Then slope of the line = y 2 y 1 change in y x 2 x = 1 change in x = rise run. A vertical line has no slope. Fact: A nonvertical line has only one slope. That is, it makes no difference which pair of points you choose on the line to compute the slope because you'll always get the same result. Reason: The triangles at right are similar so rise run = rise' run '. When finding (or estimating) slopes, if you take the run to be positive, then the "rise" can be positive, negative, or zero. Also, if you take the run to be 1, then the rise is actually the slope of the line. When estimating slopes, you have to pay attention to the scales on the two axes. If the scales are the same, slopes are easy to estimate. If the scales are different, you'll have to use the scales to estimate the run and the corresponding rise. For example, in the first picture at right, the scales are the same. For a run of 1, the "rise" is about 2 so the slope is about 2. In the second picture, the scales are not the same. With the line shown, for a run of 5, the rise is about 20 (by eyeball) so the slope is about 4. Since most coordinate systems have equal scales, you should be able to approximate slopes of lines by eyeball. The basic slopes for lines thru the origin are shown at right. The slope for a line not thru the origin is easy to estimate by visualizing a parallel line which does contain the origin. Fact: Nonvertical parallel lines have the same slope. Conversely, nonvertical lines with the same slope are parallel. Reason: l 1 and l 2 nonvertical lines with slopes s 1 and s 2. If l 1 l 2, then a 1 = a 2 so the triangles are congruent and therefore s 1 = s 2. If, conversely, s 1 = s 2, then the triangles are congruent so a 1 = a 2 and therefore l 1 l 2. [This argument assumed the slopes were positive but the same conclusion holds when they're zero or negative.]

2 -2- Fact: Two nonvertical lines are perpendicular if, and only if, their slopes are negative reciprocals of one another. In the figure at right, l 1 l 2 iff s 1 s 2 = 1. Reason: This is harder. I'll give an argument in an Addendum to this handout. After all this fuss about slopes, you probably wonder why slopes are so important. We'll return to that question in a bit -- after looking at equations for lines. Equations for Lines We'll take the following as axiomatic: Basic Fact: Any equation that can be put in the form Ax + By = C (A,B,C are constants, and not both A and B are zero) has a graph which is a line. Conversely, every line is the graph of an equation that can be put in this form. [This is often called the general form for the equation of a line.] Fact: Given the point (a, b). The equation x = a is the equation of the vertical line thru the point (a, b), and the equation y = b is the equation for the horizontal line thru the point (a, b). Reason: Written in the general form, these equations are 1x + 0y = a and 0x + 1y = b. Both equations therefore have graphs that are lines. If you think about the ordered pairs that make the equation 1x + 0y = a true, you should be able to see why the graph is the vertical line thru (a, b). Similarly for the equation y = b. Some general things about graphs: (1) A y-intercept is a point where the graph intersects the y-axis. To find the y-intercepts from an equation, put 0 in for x and solve for y. An x-intercept is a point where the graph hits the x-axis. To find x-intercepts, substitute 0 for y and solve for x. [To remember which variable gets replaced by zero, remember what intercept means and which variable must be zero as in the figure.] (2) The graph of an equation is the set of all ordered pairs that satisfy the equation. For example, given the equation y = rx + s. The point with x-coordinate w on the graph of this equation has y-coordinate rw + s. That is, the point (w, rw + s) is on this line. Similarly, the point with y-coordinate u which is on the graph has x-coordinate (u s)/r. That is, the point ((u s)/r, u) is on the line.

3 Fact: An equation that can be put in the form y = rx + s (r, s are constants) has a graph that is a line. The slope of the line is the number r and the y-intercept is (0, s). Reason: The equation can be written in the form rx + ( 1)y = s so its graph is a line by the Basic Fact on page 2. Next take two points on the line, say (w, rw + s) and (u, ru + s). If you use these two points to compute the slope [Don't forget parentheses!], the result will be the number r. [Do it.] And to find the y-intercept, substituting 0 for x in the equation gives s for y. -3- The form y = rx + s is called the slope-intercept form for the equation of a line. When you are asked to find the equation of a line, your result should be given in the slope-intercept form unless otherwise specified. [Note: There is nothing special about the constants r and s -- nor about m and b. (Some of you might think that "m" is synonymous with "slope". It is not.) The important thing is that, when an equation is in the slope-intercept form, the coefficient of x is the slope and the constant term is the y-value of the y-intercept.] Things you must be able to do: 1. Find the equation of the line thru a given point with a given slope. Example: Find the equation of the line thru the point (a, b) with slope k. Here's how: (1) Draw a rough picture. [Important!] (2) Take an "arbitrary" (i.e., general) point (x, y) on the line. [This is supposed to represent any point on the line.] (3) Use the arbitrary point and the given point to write the slope of the line: (4) Set this equal to the given slope. (A line has only one slope so the slopey b you calculated must be the given slope k.) x a y b x a = k (5) Simplify the equation, writing the result in the slope-intercept form: y b = k(x a) iff y = kx ka + b iff y = kx + (b ka). (6) CHECK. [This is very important. And it does not mean to look in the answer book to see if you agree with the book!] To check in the example, the slope is the coefficient of x which is k, and that is what was given. Also, when x = a, y = ka + (b ka) = b so the point (a, b) is on the line as was given. 2. Find the equation of the line thru two given points. Example. Find the equation of the line thru the points (r, m) and (s, n). [If r = s, the equation is x = r. Do you see why?] (1) Draw a rough picture. (2) Use the two given points to compute the slope of the line: slope = n m s r (3) Now use this slope and either of the two given points to find the equation as in problem 1. n m s r = y r x m iff y = n m (x m) + r iff etc. s r (4) Check. [Do the two given points satisfy the final equation?]

4 3. Variations on the basic theme. -4- Example 1: Find the equation of the line thru the point (a,b) which is perpendicular to the line with equation Ax + By = C (B 0). (1) Draw a picture. (2) Find the slope of the given line by putting the equation in the slope-intercept form: y = A B x+ C. B (3) The slope of the desired line is therefore B/A. Now use this slope and the given point to find the equation of the desired line. Example 2: Car sales at Acme Used Car Co. have increased linearly since the base year of 1980 when 200 cars were sold. In 1988, a total of 320 cars were sold. (a) Write an equation that gives the number of cars sold each year since (b) Draw a graph expressing this relationship. (c) How many cars will be sold in 1992, assuming a continued linear growth? (d) In what year will 470 cars be sold, assuming a continued linear growth? Solution: The key here is that the problem says there's been a linear increase which means that we should find a linear equation that relates the two quantities in question. As always, define the variables. Let y be the number of cars sold x years after the base year of We know that when x = 0, y = 200 and when x = 8, y = 320. We want the equation of the line thru (0,200) and (8,320). Slope is y 200 x = = 15 =15 y 200= 15x y = 15x is the equation. We have the equation so now we ll draw the graph. [Comments: Label axes on your graphs when doing an applied problem. Use all of the coordinate systems that makes sense for the problem when choosing your scale. Do not include unnecessary numbers - like zero to 200 on the y-axis. To indicate you're omitting numbers, use as I did. (c) The question is: When x = = 12, what's y? Answer: y = = = 380. About 380 cars will be sold in 1992, assuming continued linear growth. (d) The question is: When y = 470, what's x? Answer: Solve 470 = 15x to get x = 18. The year is = cars will be sold in 1998, again assuming continued linear growth.

5 -5- Example 3. I'm driving on the interstate at a constant speed of 55 miles per hour. At noon, I see that I've just passed mile marker number 82.5 (and the next marker is 82.6) Write an equation that tells my position on the highway using noon as the base time. Solution: Let s be my position t hours from noon. Since speed is constant, then distance = rate time so my position will be given by the equation s = 55t A graph of the equation is given at right. The Significance of Slope A linear equation gives a relationship between two quantities, the independent quantity (x) and the dependent quantity (y). The slope of the line tells the rate of change of the dependent quantity with respect to the independent quantity. It does this by telling the change in y for a unit change in x. Some examples: Look at Example 3 above. If you use the points (0, 82.5) and (2, 192.5) to compute the slope, you're really finding the average speed from time t = 0 to time t = 2 because you're computing ( )/(2 0) = distance / time = average speed. Similarly, if you use the points say (1, 137.5) and (4, 302.5), you'll get the average speed between 1 p.m. and 4 p.m. No matter which pair of points you use, you'll get 55 which is the average speed for any interval of time on this trip. (The reason is that the speed was constant.) Please note that, since the position was measured in miles and the time in hours, then the rate of change is given in miles per hour. Look at Example 2 on the previous page. Using any pair of points on the graph to compute the slope, the value will always be 15. What this means is that, for any year after 1980, Acme's sales have increased at the rate of 15 cars per year. Note again that the dependent variable is number of cars sold and the independent variable is years so the slope, the rate of change, is measured in number of cars sold per year. Here are a few more examples. Apples are 25 cents each. y is the cost of x apples. The equation relating x and y is y = 25x. The slope is 25. It means that the rate for apples is 25 cents per apple. A telemarketer finds that her sales increase linearly as a function of the number of calls she makes. The graph is given at right. Using the two given points to compute the slope, we get slope = This means that she averages \$22.50 in sales per call.

6 -6- Yesterday, the time and temperature were given by the graph at right. Suppose you're interested in the average rate if change of the temperature between 4 a.m. and 10 a.m. Using the approximate points (4, 10) and (10, 2), we get Avgtemp change = 2 ( 10) 10 4 =2 This means that, between 4 a.m. and 10 a.m., the temperature rose an average of 2 degrees per hour. If we're concerned about how the temperature changed between 4 p.m. and midnight, we'd use the (estimated) points (16, 19) and (24, 7) to compute Avg tempchange = = 1.5 Thus between 4 p.m. and midnight, the temperature dropped (note the negative) at the average rate of 1.5 degrees per hour. In calculus, you will learn to find instantaneous rate of change of y with respect to x. This means, for example, that you will find the rate at which the temperature is changing at exactly 1 p.m. Or for a car that is not traveling at a constant speed, you will learn to compute the car's speed at a given instant in time. This is much more realistic since it is the instantaneous speed that your speedometer gives, not an average speed. To do these things, you will be computing the slope of the tangent line to the curve at a particular point on the curve. And that is why slope is so important! ***************************************************************************** Addendum The following is an argument for this fact: Given lines l 1 and l 2 with slopes s 1 and s 2. Then l 1 l 2 iff * s 1 s 2 = 1. I'll show only that if l 1 l 2, then s 1 s 2 = 1. Also, I'll only argue for the case when the lines intersect at the origin. Since parallel lines have the same slope, it's not hard to extend this argument to the case where the lines intersect elsewhere. Suppose that l 1 and l 2 are perpendicular and intersect at the origin. Assume also that l 1 has positive slope s 1. Then the slope, s 2, of the other line "obviously" ** must be negative. In the figure at right, AB is the vertical line thru the point (1, 0). Then AF = s 1 and FB = s 2. *** Since m AOF = 90 m A = m B, then AOF ~ OBF so AF OF = BF OF or s 1 1 = 1 s 2. Hence s 1 s 2 = 1. Footnotes: * The symbol "iff" is a shorthand way of writing "if, and only if,". Thus l 1 l 2 iff s 1 s 2 = 1 means l 1 l 2 if, and only if, s 1 s 2 = 1. This in turn means "If l 1 l 2, then s 1 s 2 = 1 and if s 1 s 2 = 1, then l 1 l 2." ** Always be suspicious when someone says "obviously." *** The number FB is the length of the side of a triangle so must be positive. Since s 2 is a negative number, its opposite, which is denoted by s 2, must be a positive number. [You should read " x" as "the opposite of x" rather than "negative x" since x could be a positive number. It really seems strange to say "negative x is a positive number."]

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### Graphing Linear Equations

Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

### Linear Equations Review

Linear Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The y-intercept of the line y = 4x 7 is a. 7 c. 4 b. 4 d. 7 2. What is the y-intercept

### GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: Slope-Intercept Form: y = mx+ b In an equation

### Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry 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

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

### CHAPTER 1 Linear Equations

CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

### Study Guide and Review - Chapter 4

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. The

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines

MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Rectangular Coordinate System 2 Graphing Lines 3 The Equation of

### GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

### Slope-Intercept Equation. Example

1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

### In this section, we ll review plotting points, slope of a line and different forms of an equation of a line.

Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

### 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

### Helpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize

### Section 3.4 The Slope Intercept Form: y = mx + b

Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept Reminding! m = y x = y 2 y 1 x 2 x 1 Slope of a horizontal line is 0 Slope of a vertical line is Undefined Graph a linear

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

### Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### 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

### Pre-Calculus III Linear Functions and Quadratic Functions

Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3

### Techniques of Differentiation Selected Problems. Matthew Staley

Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Write the Equation of the Line Review

Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

### What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

### 1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

### Lecture 9: Lines. m = y 2 y 1 x 2 x 1

Lecture 9: Lines If we have two distinct points in the Cartesian plane, there is a unique line which passes through the two points. We can construct it by joining the points with a straight edge and extending

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

### Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

### Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

### graphs, Equations, and inequalities

graphs, Equations, and inequalities You might think that New York or Los Angeles or Chicago has the busiest airport in the U.S., but actually it s Hartsfield-Jackson Airport in Atlanta, Georgia. In 010,

### 5 \$75 6 \$90 7 \$105. Name Hour. Review Slope & Equations of Lines. STANDARD FORM: Ax + By = C. 1. What is the slope of a vertical line?

Review Slope & Equations of Lines Name Hour STANDARD FORM: Ax + By = C 1. What is the slope of a vertical line? 2. What is the slope of a horizontal line? 3. Is y = 4 the equation of a horizontal or vertical

### 1.01 b) Operate with polynomials.

1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of exponents. There are a few rules that simplify our dealings with exponents. Given the same base, there are ways

### 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

### Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.

Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is

### 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.

### Sect The Slope-Intercept Form

Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

### Slope-Intercept Form of a Linear Equation Examples

Slope-Intercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the y-intercept of AB. Suppose you want to find an equation

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.

### Section 2.2 Equations of Lines

Section 2.2 Equations of Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes

### Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

### Elements of a graph. Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

### Math 113 Review for Exam I

Math 113 Review for Exam I Section 1.1 Cartesian Coordinate System, Slope, & Equation of a Line (1.) Rectangular or Cartesian Coordinate System You should be able to label the quadrants in the rectangular

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

### Section 3.2. Graphing linear equations

Section 3.2 Graphing linear equations Learning objectives Graph a linear equation by finding and plotting ordered pair solutions Graph a linear equation and use the equation to make predictions Vocabulary:

### Precalculus Workshop - Functions

Introduction to Functions A function f : D C is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set C. D is called the domain of f. C is called the codomain of f.

### 5.1: Rate of Change and Slope

5.1: Rate of Change and Slope Rate of Change shows relationship between changing quantities. On a graph, when we compare rise and run, we are talking about steepness of a line (slope). You can use and

### TEKS 2A.7.A Quadratic and square root functions: connect between the y = ax 2 + bx + c and the y = a (x - h) 2 + k symbolic representations.

Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary axis of symmetry standard form minimum value maximum

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.

Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line

### Name: Class: Date: Does the equation represent a direct variation? If so, find the constant of variation. c. yes; k = 5 3. c.

Name: Class: Date: Chapter 5 Test Multiple Choice Identify the choice that best completes the statement or answers the question. What is the slope of the line that passes through the pair of points? 1.

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Systems of Linear Equations in Two Variables

Chapter 6 Systems of Linear Equations in Two Variables Up to this point, when solving equations, we have always solved one equation for an answer. However, in the previous chapter, we saw that the equation

### Section 1.10 Lines. The Slope of a Line

Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through

### 2x - y 4 y -3x - 6 y < 2x 5x - 3y > 7

DETAILED SOLUTIONS AND CONCEPTS GRAPHICAL REPRESENTATION OF LINEAR INEQUALITIES IN TWO VARIABLES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu.

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Creating Equations. Set 3: Writing Linear Equations Instruction. Student Activities Overview and Answer Key

Creating Equations Instruction Goal: To provide opportunities for students to develop concepts and skills related to writing linear equations in slope-intercept and standard form given two points and a

### A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### 4.1 & Linear Equations in Slope-Intercept Form

4.1 & 4.2 - Linear Equations in Slope-Intercept Form Slope-Intercept Form: y = mx + b Ex 1: Write the equation of a line with a slope of -2 and a y-intercept of 5. Ex 2:Write an equation of the line shown

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

### Formula Sheet. 1 The Three Defining Properties of Real Numbers

Formula Sheet 1 The Three Defining Properties of Real Numbers For all real numbers a, b and c, the following properties hold true. 1. The commutative property: a + b = b + a ab = ba. The associative property:

### Unit III Practice Questions

Unit III Practice Questions 1. Write the ordered pair corresponding to each point plotted below. A F B E D C 2. Determine if the ordered pair ( 1, 2) is a solution of 2x + y = 4. Explain how you know.

### Linear Equations and Graphs

2.1-2.4 Linear Equations and Graphs Coordinate Plane Quadrants - The x-axis and y-axis form 4 "areas" known as quadrants. 1. I - The first quadrant has positive x and positive y points. 2. II - The second

### x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =

Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the

### Coordinate Plane, Slope, and Lines Long-Term Memory Review Review 1

Review. What does slope of a line mean?. How do you find the slope of a line? 4. Plot and label the points A (3, ) and B (, ). a. From point B to point A, by how much does the y-value change? b. From point

### Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope

Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope I. Linear Functions 1. A linear equation is an equation whose graph is a straight line. 2. A linear equation in standard form: Ax +By=C ex: 4x

1.5 ANALYZING GRAPHS OF FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which

Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x

### IV. ALGEBRAIC CONCEPTS

IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other

### Equations of Lines Derivations

Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

### Week 1: Functions and Equations

Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

### MATH 111: EXAM 02 SOLUTIONS

MATH 111: EXAM 02 SOLUTIONS BLAKE FARMAN UNIVERSITY OF SOUTH CAROLINA Answer the questions in the spaces provided on the question sheets and turn them in at the end of the class period Unless otherwise

### 5.1 Writing Linear Equations in Slope-Intercept Form. 1. Use slope-intercept form to write an equation of a line.

5.1 Writing Linear Equations in Slope-Intercept Form Objectives 1. Use slope-intercept form to write an equation of a line. 2. Model a real-life situation with a linear function. Key Terms Slope-Intercept

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Student Lesson: Absolute Value Functions

TEKS: a(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete,

### Patterns, Equations, and Graphs. Section 1-9

Patterns, Equations, and Graphs Section 1-9 Goals Goal To use tables, equations, and graphs to describe relationships. Vocabulary Solution of an equation Inductive reasoning Review: Graphing in the Coordinate

10.9 Systems Of Inequalities Copyright Cengage Learning. All rights reserved. Objectives Graphing an Inequality Systems of Inequalities Systems of Linear Inequalities Application: Feasible Regions 2 Graphing

### COMPARING LINEAR AND NONLINEAR FUNCTIONS

1 COMPARING LINEAR AND NONLINEAR FUNCTIONS LEARNING MAP INFORMATION STANDARDS 8.F.2 Compare two s, each in a way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example,

### 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

### 4.4 Transforming Circles

Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different

### Math 10 - Unit 7 Final Review - Coordinate Geometry

Class: Date: Math 10 - Unit Final Review - Coordinate Geometry Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine the slope of this line segment.

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires