SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE SLOW VEHICLES KEEP RIGHT


 Ann Manning
 2 years ago
 Views:
Transcription
1 . Slope of a Line () SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail and stud the concept of slope of a line. Slope Using Coordinates to Find Slope Parallel Lines Perpendicular Lines Applications of Slope Slope If a highwa has a 6% grade, then in 00 feet (measured horizontall) the road rises 6 feet (measured verticall). See Fig..0. The ratio of 6 to 00 is 6%. If a roof rises 9 feet in a horizontal distance (or run) of feet, then the roof has a 9 pitch. A roof with a 9 pitch is steeper than a roof with a 6 pitch. The grade of a road and the pitch of a roof are measurements of steepness. In each case the measurement is a ratio of rise (vertical change) to run (horizontal change). 6% GRADE 6 00 SLOW VEHICLES KEEP RIGHT ft run 9 ft rise helpful hint Since the amount of run is arbitrar, we can choose the run to be. In this case slope ri se rise. So the slope is the amount of change in for a change of in the coordinate.this is wh rates like 0 miles per hour (mph), 8 hours per da, and two people per car are all slopes. FIGURE.0 9 pitch We measure the steepness of a line in the same wa that we measure steepness of a road or a roof. The slope of a line is the ratio of the change in coordinate, or the rise, to the change in coordinate, or the run, between two points on the line. Slope Slope change in coordinate change in coordinate Consider the line in Fig..(a) on the net page. In going from (0, ) to (, ), there is a change of in the coordinate and a change of in the coordinate, rise run
2 () Chapter Graphs and Functions in the Cartesian Coordinate Sstem (, ) + (0, ) + (, ) (0, ) (a) FIGURE. (b) or a run of and a rise of. So the slope is or. If we move from (, ) to (0, ) as in Fig..(b) the rise is and the run is. So the slope is or. If we start at either point and move to the other point, we get the same slope. E X A M P L E Finding the slope from a graph Find the slope of each line b going from point A to point B. a) b) c) A B A B 6 6 B A a) A is located at (0, ) and B at (, 0). In going from A to B, the change in is and the change in is. So slope. b) In going from A(, ) to B(6, ), we must rise and run. So slope. c) In going from A(0, 0) to B(6, ), we find that the rise is and the run is 6. So slope 6.
3 . Slope of a Line () Note that in Eample (c) we found the slope of the line of Eample (b) b using two different points. The slope is the ratio of the lengths of the two legs of a right triangle whose hpotenuse is on the line. See Fig... As long as one leg is vertical and the other leg is horizontal, all such triangles for a given line have the same shape: The are similar triangles. Because ratios of corresponding sides in similar triangles are equal, the slope has the same value no matter which two points of the line are used to find it. Hpotenuse Run Hpotenuse Rise Run Rise (, ) (, ) Run Rise (, ) FIGURE. FIGURE. Using Coordinates to Find Slope We can obtain the rise and run from a graph, or we can get them without a graph b subtracting the coordinates to get the rise and the coordinates to get the run for two points on the line. See Fig... Slope Using Coordinates The slope m of the line containing the points (, ) and (, ) is given b m, provided that 0. E X A M P L E stud tip Don t epect to understand a new topic the first time that ou see it. Learning mathematics takes time, patience, and repetition. Keep reading the tet, asking questions, and working problems. Someone once said, All mathematics is eas once ou understand it. Finding slope from coordinates Find the slope of each line. a) The line through (, ) and (6, ) b) The line through (, ) and (, ) c) The line through (6, ) and the origin a) Let (, ) (, ) and (, ) (6, ). The assignment of (, ) and (, )is arbitrar. m 6 b) Let (, ) (, ) and (, ) (, ): () m ()
4 () Chapter Graphs and Functions in the Cartesian Coordinate Sstem c) Let (, ) (0, 0) and (, ) (6, ): 0 m CAUTION Do not reverse the order of subtraction from numerator to denominator when finding the slope. If ou divide b, ou will get the wrong sign for the slope. E X A M P L E Slope for horizontal and vertical lines Find the slope of each line. a) b) helpful hint Think about what slope means to skiers. No one skis on cliffs or even refers to them as slopes. Zero slope (, ) (, ) a) Using (, ) and (, ) to find the slope of the horizontal line, we get m (, ) (, ) Small slope Larger slope b) Using (, ) and (, ) to find the slope of the vertical line, we get 0. Because the definition of slope using coordinates sas that must be nonzero, the slope is undefined for this line. Since the coordinates are equal for an two points on a horizontal line, 0 and the slope is 0. Since the coordinates are equal for an two points on a vertical line, 0 and the slope is undefined. Horizontal and Vertical Lines The slope of an horizontal line is 0. Slope is undefined for an vertical line. CAUTION Do not sa that a vertical line has no slope because no slope could be confused with 0 slope, the slope of a horizontal line. Undefined slope As ou move the tip of our pencil from left to right along a line with positive slope, the coordinates are increasing. As ou move the tip of our pencil from
5 . Slope of a Line () left to right along a line with negative slope, the coordinates are decreasing. See Fig... Increasing coordinates Positive slope Negative slope Decreasing coordinates Slope Slope FIGURE. FIGURE. Parallel Lines Consider the two lines shown in Fig... Each of these lines has a slope of, and these lines are parallel. In general, we have the following fact. Parallel Lines Nonvertical parallel lines have equal slopes. Of course, an two vertical lines are parallel, but we cannot sa that the have equal slopes because slope is not defined for vertical lines. E X A M P L E (, ) Slope Slope FIGURE.6 Parallel lines Line l goes through the origin and is parallel to the line through (, ) and (, ). Find the slope of line l. The line through (, ) and (, ) has slope m 8 () 6. Because line l is parallel to a line with slope, the slope of line l is also. Perpendicular Lines The lines shown in Fig..6 have slopes and. These two lines appear to be perpendicular to each other. It can be shown that a line is perpendicular to another line if its slope is the negative of the reciprocal of the slope of the other.
6 6 (6) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Perpendicular Lines Two lines with slopes m and m are perpendicular if and onl if m. m Of course, an vertical line and an horizontal line are perpendicular, but we cannot give a relationship between their slopes because slope is undefined for vertical lines. E X A M P L E Perpendicular lines Line l contains the point (, 6) and is perpendicular to the line through (, ) and (, ). Find the slope of line l. The line through (, ) and (, ) has slope () m 7. 7 Because line l is perpendicular to a line with slope 7, the slope of line l is 7. E X A M P L E 6 D (, ) A (, ) C (, ) B (, ) FIGURE.7 Applications of Slope When a geometric figure is located in a coordinate sstem, we can use slope to determine whether it has an parallel or perpendicular sides. Using slope with geometric figures Determine whether (, ), (, ), (, ), and (, ) are the vertices of a rectangle. Figure.7 shows the quadrilateral determined b these points. If a parallelogram has at least one right angle, then it is a rectangle. Calculate the slope of each side. () m AB () m CD m BC 6 m AD 6 Because the opposite sides have the same slope, the are parallel, and the figure is a parallelogram. Because is the opposite of the reciprocal of, the intersecting sides are perpendicular. Therefore the figure is a rectangle. The slope of a line is a rate. The slope tells us how much the dependent variable changes for a change of in the independent variable. For eample, if the horizontal ais is hours and the vertical ais is miles, then the slope is miles per hour (mph).
7 . Slope of a Line (7) 7 If the horizontal ais is das and the vertical ais is dollars, then the slope is dollars per da. E X A M P L E 7 Slope as a rate Worldwide carbon dioide (CO ) emissions have increased from billion tons in 970 to billion tons in 99 (World Resources Institute, CO emission (in billions of tons) Year FIGURE FOR EXAMPLE 7 stud tip Finding out what happened in class and attending class are not the same. Attend ever class and be attentive. Don t just take notes and let our mind wander. Use class time as a learning time. a) Find and interpret the slope of the line in the accompaning figure. b) Predict the amount of worldwide CO emissions in 00. a) Find the slope of the line through (970, ) and (99, ): m The slope of the line is 0. billion tons per ear. b) If the (CO ) emissions keep increasing at 0. billion tons per ear, then in 0 ears the level will go up 0(0.) or billion tons. So in 00 CO emissions will be 8 billion tons. WARMUPS True or false? Eplain our answer.. Slope is a measurement of the steepness of a line. True. Slope is run divided b rise. False. The line through (, ) and (, ) has undefined slope. False. The line through (, 6) and (, ) has undefined slope. True. Slope cannot be negative. False 6. The slope of the line through (0, ) and (, 0) is. False 7. The line through (, ) and (, ) has slope. False 8. If a line contains points in quadrants I and III, then its slope is positive. True 9. Lines with slope and are perpendicular to each other. False 0. An two parallel lines have equal slopes. False
8 8 (8) Chapter Graphs and Functions in the Cartesian Coordinate Sstem. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What does slope measure? Slope measures the steepness of a line.. What is the rise and what is the run? The rise is the change in coordinates and run is the change in coordinates.. Wh does a horizontal line have zero slope? A horizontal line has zero slope because it has no rise. Determine the slope of each line. See Eample Wh is slope undefined for vertical lines? Slope is undefined for vertical lines because the run is zero and division b zero is undefined.. What is the relationship between the slopes of perpendicular lines? If m and m are the slopes of perpendicular lines, then m. m 6. What is the relationship between the slopes of parallel lines? If m and m are the slopes of parallel lines, then m m. Undefined
9 . Slope of a Line (9) 9 6. Find the slope of the line that contains each of the following pairs of points. See Eamples and. 7. (, 6), (, ) 8. (, ), (6, 0) 9. (, ), (, ) 0. (, ), (, ) 7. (, ), (, 7). (, ), (, 6). (, ), (0, 0). (0, 0), (, ). (0, ), (, 0) 6. (, 0), (0, 0) 0 7.,,, 8.,,, 6 9. (6, ), (7, 09) 0. (988, 06), (990, ) 9. (, 7), (, 7) 0. (, ), (9, ) 0. (, 6), (, 6) Undefined. (, ), (, 0) Undefined. (.,.9), (.7, 8.) (.7, 9.), (.6,.8) ,,, ,, 6, 0.90 In each case, make a sketch and find the slope of line l. See Eamples and. 9. Line l contains the point (, ) and is perpendicular to the line through (, ) and (, ) Line l goes through (, ) and is perpendicular to the line through (, 6) and (, ). 7. Line l goes through (, ) and is parallel to the line through (, ) and (, ). 7. Line l goes through the origin and is parallel to the line through (, ) and (, ). 7. Line l is perpendicular to a line with slope. Both lines contain the origin.. Line l is perpendicular to a line with slope. Both lines contain the origin. Solve each geometric figure problem. See Eample 6.. If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. Use slope to determine whether the points (6, ), (, ), (0, ), and (, ) are the vertices of a parallelogram. Yes 6. Use slope to determine whether the points (7, 0), (, 6), (, ), and (6, ) are the vertices of a parallelogram. See Eercise. No 7. A trapezoid is a quadrilateral with one pair of parallel sides. Use slope to determine whether the points (, ), (, ), (, 6), and (6, ) are the vertices of a trapezoid. No 8. A parallelogram with at least one right angle is a rectangle. Determine whether the points (, ), (, ), (0, 6), and (, 0) are the vertices of a rectangle. Yes 9. If a triangle has one right angle, then it is a right triangle. Use slope to determine whether the points (, ), (, 6), and (0, 0) are the vertices of a right triangle. No 0. Use slope to determine whether the points (0, ), (, ), and (, ) are the vertices of a right triangle. See Eercise 9. Yes Solve each problem. See Eample 7.. Pricing the Crown Victoria. The list price of a new Ford Crown Victoria fourdoor sedan was $0, in 99 and $, in 998 (Edmund s New Car Prices, a) Find the slope of the line shown in the figure. 0 b) Use the graph to predict the price in 00. $,00 c) Use the slope to predict the price of a new Crown Victoria in 00. $,6 List price (in thousands of dollars) 0 (99, 0,) (998,,) Year FIGURE FOR EXERCISE. Depreciating Monte Carlo. In 998 the average retail price of a oneearold Chevrolet Monte Carlo was $,9, whereas the average retail price of a earold Monte Carlo was $,09 (Edmund s Used Car Prices).
10 0 (0) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Selling price (in thousands of dollars) 0 a) Use the graph on the net page to estimate the average retail price of a earold car in 998. $,000 b) Find the slope of the line shown in the figure. 0 c) Use the slope to predict the price of a earold car. $, (,,9) (,,09) Age (in ears) FIGURE FOR EXERCISE MISCELLANEOUS. The points (, ) and (,7) are on the line that passes through (, ) and has slope. Find the missing coordinates of the points. (, ), (0, 7). If a line passes through (, ) and has slope, then what is the value of on this line when 8,, and?, 6, 6. Find k so that the line through (, k) and (, ) has slope. 6. Find k so that the line through (k,)and(, 0) has slope. or 7. What is the slope of a line that is perpendicular to a line with slope 0.7? What is the slope of a line that is perpendicular to the line through (.7,.6) and (.8,.6)?.76 GETTING MORE INVOLVED 9. Writing. What is the difference between zero slope and undefined slope? A horizontal line has a zero slope and a vertical line has undefined slope. 60. Writing. Is it possible for a line to be in onl one quadrant? Two quadrants? Write a rule for determining whether a line has positive, negative, zero, or undefined slope from knowing in which quadrants the line is found. Ever line goes through at least two quadrants. A nonhorizontal, nonvertical line that misses quadrant II or IV or both has a positive slope. A nonhorizontal, nonvertical line that misses quadrant I or III or both has a negative slope. 6. Eploration. A rhombus is a quadrilateral with four equal sides. Draw a rhombus with vertices (, ), (0, ), (, ), and (, ). Find the slopes of the diagonals of the rhombus. What can ou conclude about the diagonals of this rhombus?,, perpendicular 6. Eploration. Draw a square with vertices (, ), (, ), (, ), and (, ). Find the slopes of the diagonals of the square. What can ou conclude about the diagonals of this square?,, perpendicular GRAPHING CALCULATOR EXERCISES 6. Graph,,, and together in the standard viewing window. These equations are all of the form m. What effect does increasing m have on the graph of the equation? What are the slopes of these four lines? Increasing m makes the graph increase faster. The slopes of these lines are,,, and. 6. Graph,,, and together in the standard viewing window. These equations are all of the form m. What effect does decreasing m have on the graph of the equation? What are the slopes of these four lines? Decreasing m makes the graph decrease faster. The slopes of these lines are,,, and. In this section PointSlope Form SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Linear Functions. THREE FORMS FOR THE EQUATION OF A LINE In Section. ou learned how to graph a straight line corresponding to a linear equation. The line contains all of the points that satisf the equation. In this section we start with a line or a description of a line and write an equation corresponding to the line. PointSlope Form Figure.8 shows the line that has slope and contains the point (, ). In Section. ou learned that the slope is the same no matter which two points of the line
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. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationTHE POINTSLOPE FORM
. The PointSlope Form () 67. THE POINTSLOPE 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 informationLesson 8.3 Exercises, pages
Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5  b) = 5 c) =  d) =. Sketch broken lines to represent the vertical and horizontal
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
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
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.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 informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More information2.3 Writing Equations of Lines
. Writing Equations of Lines In this section ou will learn to use pointslope form to write an equation of a line use slopeintercept form to write an equation of a line graph linear equations using the
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationSECTION 22 Straight Lines
 Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
More informationLINEAR 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 informationCoordinate 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 informationChapter 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 informationREVIEW 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 informationThe SlopeIntercept Form
7.1 The SlopeIntercept 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 informationGRAPHING SYSTEMS OF LINEAR INEQUALITIES
444 (8 5) Chapter 8 Sstems of Linear Equations and Inequalities GETTING MORE INVOLVED 5. Discussion. When asked to graph the inequalit, a student found that (0, 5) and (8, 0) both satisfied. The student
More informationMath 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas
Math 0 Chapter Lecture Notes Professor Miguel Ornelas M. Ornelas Math 0 Lecture Notes Section. Section. The Rectangular Coordinate Sstem Plot each ordered pair on a Rectangular Coordinate Sstem and name
More informationAlex 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 information5. Equations of Lines: slope intercept & point slope
5. Equations of Lines: slope intercept & point slope Slope of the line m rise run SlopeIntercept Form m + b m is slope; b is intercept PointSlope Form m( + or m( Slope of parallel lines m m (slopes
More informationGRAPHS 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 information4 Writing Linear Functions
Writing Linear Functions.1 Writing Equations in SlopeIntercept Form. Writing Equations in PointSlope Form.3 Writing Equations in Standard Form. Writing Equations of Parallel and Perpendicular Lines.5
More information3.4 The PointSlope Form of a Line
Section 3.4 The PointSlope Form of a Line 293 3.4 The PointSlope Form of a Line In the last section, we developed the slopeintercept form of a line ( = m + b). The slopeintercept form of a line is
More informationCOORDINATE PLANES, LINES, AND LINEAR FUNCTIONS
G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs
More informationTHE PARABOLA section. Developing the Equation
80 (0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or
More informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
More informationCOORDINATE PLANES, LINES, AND LINEAR FUNCTIONS
a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated
More informationLinear Inequality in Two Variables
90 (7) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationSLOPES AND EQUATIONS OF LINES CHAPTER
CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8 The Slope of a Line 8 The Equation of a Line 83 Midpoint of a Line Segment 84 The Slopes of Perpendicular Lines 85 Coordinate Proof 86 Concurrence of the
More informationThe PointSlope Form
7. The PointSlope 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
More information2.1 Equations of Lines
Section 2.1 Equations of Lines 1 2.1 Equations of Lines The SlopeIntercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationTHE POWER RULES. Raising an Exponential Expression to a Power
8 (5) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar
More informationSection 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 information3 Rectangular Coordinate System and Graphs
060_CH03_13154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
More information25. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math
Chapter 2 Lesson 25 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 informationTranslating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More informationSolution: 2. Sketch the graph of 2 given the vectors and shown below.
7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit
More informationThe Rectangular Coordinate System
3.2 The Rectangular Coordinate Sstem 3.2 OBJECTIVES 1. Graph a set of ordered pairs 2. Identif plotted points 3. Scale the aes NOTE In the eighteenth centur, René Descartes, a French philosopher and mathematician,
More informationEssential Question How can you solve a system of linear equations? $15 per night. Cost, C (in dollars) $75 per Number of. Revenue, R (in dollars)
5.1 Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bedandbreakfast.
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationMath 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 informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationSlopeIntercept Form and PointSlope Form
SlopeIntercept Form and PointSlope Form In this section we will be discussing SlopeIntercept Form and the PointSlope Form of a line. We will also discuss how to graph using the SlopeIntercept Form.
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More informationChapter 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
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationIn 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:
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More information5.1. 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 informationx 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
More informationYears 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 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More informationChapter 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 informationTHIS CHAPTER INTRODUCES the Cartesian coordinate
87533_01_ch1_p001066 1/30/08 9:36 AM Page 1 STRAIGHT LINES AND LINEAR FUNCTIONS 1 THIS CHAPTER INTRODUCES the Cartesian coordinate sstem, a sstem that allows us to represent points in the plane in terms
More informationSTRETCHING, 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 information13 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 informationSlopeIntercept 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 yintercept. Determine
More informationLinear Equations and Graphs
2.12.4 Linear Equations and Graphs Coordinate Plane Quadrants  The xaxis and yaxis form 4 "areas" known as quadrants. 1. I  The first quadrant has positive x and positive y points. 2. II  The second
More informationLinear Equations Review
Linear Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The yintercept of the line y = 4x 7 is a. 7 c. 4 b. 4 d. 7 2. What is the yintercept
More informationGraphing Linear Inequalities in Two Variables
5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when
More informationReteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
More informationLesson 6: Linear Functions and their Slope
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
More informationMathematics 1. Lecture 5. Pattarawit Polpinit
Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance
More informationSimplification of Rational Expressions and Functions
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
More informationCHAPTER 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 xaxis, and the vertical axis or
More informationSolving 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
More informationThe slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6
Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means
More informationLines and Linear Equations. Slopes
Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means
More informationWhy should we learn this? One realworld 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 realworld connection is to find the rate
More informationSection 3.4 The Slope Intercept Form: y = mx + b
SlopeIntercept Form: y = mx + b, where m is the slope and b is the yintercept 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
More informationP1. 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 informationChapter 3 & 8.18.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x  4y = 10 16x + 8y = 10
Chapter 3 & 8.18.3 These are meant for practice. The actual test is different. Determine whether the pair of equations represents parallel lines. 1) 9 + 3 = 12 27 + 9 = 39 1) Determine whether the pair
More informationReview Exercises. Review Exercises 83
Review Eercises 83 Review Eercises 1.1 In Eercises 1 and, sketch the lines with the indicated slopes through the point on the same set of the coordinate aes. Slope 1. 1, 1 (a) (b) 0 (c) 1 (d) Undefined.,
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
More informationSection 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
More informationSection 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
More informationSOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses
CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning
More informationThe 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 informationSolving Special Systems of Linear Equations
5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More informationGraph each function. Compare to the parent graph. State the domain and range. 1. SOLUTION:
 Root Functions Graph each function. Compare to the parent graph. State the domain and range...5.. 5. 6 is multiplied b a value greater than, so the graph is a vertical stretch of. Another wa to identif
More informationhttp://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4
of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) m = y 2  y 1 x1  x 2
4.4.28 GraphingEquations of LinesSlope Interecpt MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) What is the
More informationEssential Question How can you graph a system of linear inequalities?
5.7 Sstems of Linear Inequalities Essential Question How can ou graph a sstem of linear inequalities? Graphing Linear Inequalities Work with a partner. Match each linear inequalit with its graph. Eplain
More information5 $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
More informationProperties of Special Parallelograms
Properties of Special Parallelograms Lab Summary: This lab consists of four activities that lead students through the construction of a parallelogram, a rectangle, a square, and a rhombus. Students then
More informationEquation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1
Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows : gradient = vertical horizontal horizontal A B vertical
More informationChapter 2: Concurrent force systems. Department of Mechanical Engineering
Chapter : Concurrent force sstems Objectives To understand the basic characteristics of forces To understand the classification of force sstems To understand some force principles To know how to obtain
More informationGraphing and transforming functions
Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6
More informationModifying Functions  Families of Graphs
Worksheet 47 Modifing Functions  Families of Graphs Section Domain, range and functions We first met functions in Sections and We will now look at functions in more depth and discuss their domain and
More information4.1 & Linear Equations in SlopeIntercept Form
4.1 & 4.2  Linear Equations in SlopeIntercept Form SlopeIntercept Form: y = mx + b Ex 1: Write the equation of a line with a slope of 2 and a yintercept of 5. Ex 2:Write an equation of the line shown
More informationSlope. SAFETY A ladder truck uses a moveable ladder to reach upper levels of houses and buildings.
9 MAIN IDEA Find the slope of a line. New Vocabular Slope SAFETY A ladder truck uses a moveable ladder to reach upper levels of houses and buildings. 1. The rate of change of the slope Math nline glencoe.com
More information3.1 Graphically Solving Systems of Two Equations
3.1 Graphicall Solving Sstems of Two Equations (Page 1 of 24) 3.1 Graphicall Solving Sstems of Two Equations Definitions The plot of all points that satisf an equation forms the graph of the equation.
More informationIdentify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
More informationUNIVERSITY OF WISCONSIN SYSTEM
Name UNIVERSITY OF WISCONSIN SYSTEM MATHEMATICS PRACTICE EXAM Check us out at our website: http://www.testing.wisc.edu/center.html GENERAL INSTRUCTIONS: You will have 90 minutes to complete the mathematics
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More informationGraphing Linear Equations in SlopeIntercept Form
4.4. Graphing Linear Equations in SlopeIntercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Analzing Graphs of Lines Work with a partner. Graph each equation. Find the slope
More information