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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 points find a point on a line given a known point on the line and the slope In Chapter 4, ou saw that the rate of change of a line can be a numerical and graphical representation of a real-world change like a car s speed. Look at the lines and equation shown on page 51 of our book. Because the coefficient of represents the rate of change of the line, ou can match the equations to the lines b looking at the coefficients the greater the coefficient, the steeper the line. The rate of change of a line is often referred to as its slope. You can find the slope of a line if ou know the coordinates of two points on the line. Investigation: Points and Slope Hector pas a flat monthl charge plus an hourl rate for Internet service. The table on page 51 of our book shows Hector s total monthl fee for three months. Because the hourl rate is constant, this is a linear relationship. To find the rate in dollars per hour, divide the change in fee b the change in time for two months. For eample, using October and November, ou get So the rate is \$.95 per hour. Verif that ou get this same result if ou use the data for September and October or September and November. Here is a graph of the Internet data with a line drawn through the points. The arrows show how ou can move from (5, 19.7) to (8, 8.55) using one vertical move and one horizontal move. The length of the vertical arrow is or 8.85 units, which is the change in total fees from October to November. The length of the horizontal arrow is 8 5 or 3 units, which is the change in the number of hours from October to November. Notice that the lengths are the two quantities we divided to find the hourl rate. This hourl rate,.95, is the slope of the line. The right triangle created b drawing arrows to show vertical and horizontal change is called a slope triangle. Total fee (\$) 3 (5, 19.7) (4, 1.75) 5 Time (hrs) (8, 8.55) (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 1

2 Previous Lesson 5.1 A Formula for Slope (continued) At right is the same graph with arrows drawn between the points for September and October. Notice that the slope or the vertical change divided b the horizontal change is the same. An pair of points on a line will give the same slope. You can find the vertical change and the horizontal change b subtracting corresponding coordinates. If one of the points is ( 1, 1 ) and the other is (, ), then the slope is 1 or Total fee (\$) 3 (5, 19.7) (4, 1.75) (8, 8.55) vertical change horizontal change Time (hrs) In the investigation the slope of the line is positive. The eample in our book involves a line with a negative slope. Read this eample carefull. Here is another eample. EXAMPLE Consider the line through (, 3) and (4, 1). Solution a. Find the slope of the line. b. Without graphing, verif that (3, 1 ) 3 is on the line. c. Find the coordinates of another point on the line. 1 3 a. Slope 4 ( ) 4 3 b. The slope between an two points will be the same. So if the slope between (3, 1 3 ) and either of the original points is, 3 then the point is on the line. The slope between (3, 1 ) 3 and (4, 1) is So (3, 1 ) 3 is on the line. c. Start with one of the original points. Add the change in to the -coordinate and the change in to the -coordinate. Let s start with (, 3). ( change in, 3 change in ) (, 3 (4)) (4, 1) So (4, 1) is on the line. (, 3) 4 (4, 1) Read the rest of Lesson 5.1 in our book. Make sure ou understand how ou can tell b looking at a line whether the slope is positive, negative, zero, or undefined. Note that when the equation for a line is written in the form a b, the letter b represents the slope. Discovering Algebra Condensed Lessons Ke Curriculum Press

3 CONDENSED L E S S O N 5. Writing a Linear Equation to Fit Data Previous In this lesson ou will find a line of fit for a set of data use a linear model to make predictions learn about the slope-intercept form of an equation Real-world data rarel fall eactl on a line. However, if the data show a linear pattern, ou can find a line to model the data. Such a line is called a line of fit for the data. Read the information about lines of fit on page 1 of our book. Investigation: Beam Strength Steps 1 3 In this investigation, students make beams from different numbers of spaghetti strands. Then the test each beam to see how man pennies it can hold before breaking. Here is the data collected b one group of students. Steps 4 8 You can make a scatter plot of the data on a graphing calculator or on paper. [, 7, 1,, 5, 5] Pennies Strands Number Number of strands of pennies B moving a strand of spaghetti around on our sketch, find a line ou think fits the data. Make our line go through two points. Use the coordinates of the two points to find the slope of the line. In the eample below, the line goes through (1, ) and (5, 41). The slope is or Pennies Strands 5 (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 3

4 Previous Lesson 5. Writing a Linear Equation to Fit Data (continued) Use the slope, b, to graph the equation b on our calculator. For the line above, this is Notice that this line is a little too low to fit the data. Using the sketch above, ou can estimate that the -intercept, a, of the line of fit is about. So the equation of the line of fit in intercept form is about Graph this equation on our calculator. This line appears to be a good fit. In some cases ou ll need to adjust the intercept value several times until ou are happ with the fit. Note that the equation ou end up with depends on the two data points our line passes through and our estimate of the -intercept. Different people will find different equations. Steps 9 1 The line of fit found above, 7.75, is a model for the relationship between the number of strands in the beam,, and the number of pennies the beam supports,. The slope, 7.75, represents the increase in the number of pennies each time ou add one strand to the beam. You can use the line of fit to make predictions. To predict the number of strands needed to support \$5 worth of pennies (5 pennies), trace the graph to find the -value corresponding to a -value of 5, or solve It would take about 4 strands to support this much weight. The model predicts that a -strand beam can hold 7.75() or about 8 pennies. A 17-strand beam can hold 7.75(17) or about 134 pennies. Now, follow along with the eample in our book. It shows ou how to fit a line to a different data set. To find the line of fit in the investigation, we started with the slope and then found the -intercept. Because of the importance of slope, man people use the slope-intercept form of a linear equation, which shows the slope first. In this form, m represents the slope and b represents the -intercept. The slope-intercept form is m b. 4 Discovering Algebra Condensed Lessons Ke Curriculum Press

5 CONDENSED L E S S O N 5.3 Point-Slope Form of a Linear Equation Previous In this lesson ou will write equations in point-slope form find the equation of a line given one point on the line and the slope find the equation of a line given two points on the line If ou are given the slope and the -intercept of a line, it is eas to write an equation for the line. The eample in our book shows ou how to find an equation when ou know one point and the slope. Here is an additional eample. EXAMPLE When Rosi bought her computer, she made a down pament and then made paments of \$5 per month. After 5 months, she had paid \$45. After 18 months, her computer was paid for. What is the total amount Rosi paid for her computer? Solution Because the rate of change is constant (\$5 per month), ou can model this relationship with a linear equation. Let represent time in months, and let represent the amount paid. The problem gives the slope, 5, and one point, (5, 45). Let (, ) represent a second point on the line, and use the slope formula, b, to find a linear equation Substitute the coordinates of the points. ( 5) ( 5) Multipl both sides b ( 5). Amount paid (dollars) Slope 5 point 1 (5, 45) point (, ) Time (months) 45 5( 5) Reduce. 45 5( 5) Add 45 to both sides. The equation 45 5( 5) gives the total amount paid,, in months. To find the total amount Rosi paid in 18 months, substitute 18 for. 45 5( 5) 45 5(18 5) 45 5(13) Rosi paid a total of \$195 for her computer. The equation 45 5( 5) is in point-slope form. Read about point-slope form on page 71 of our book. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 5

6 Previous Lesson 5.3 Point-Slope Form of a Linear Equation (continued) Investigation: The Point-Slope Form for Linear Equations Steps 1 5 Jenn is moving at a constant rate. The table on page 71 of our book shows her distances from a fied point after 3 seconds and after seconds. The slope of the line that represents this situation is If ou use the point (3, 4.), the equation for this situation in point-slope form is 4..( 3). If ou use (,.8), the equation is.8.( ). or.. Enter both equations into our calculator and graph them. You will see onl one line, which indicates that the equations are equivalent. [,, 1, 1,, 1] Now, look at the table for the two equations. Notice that for ever -value, the Y1- and Y-values are the same. This also indicates that the equations 4..( 3) and.8.( ) are equivalent. Steps 9 The table on page 7 of our book shows how the temperature of a pot of water changed over time as it was heated. If ou plot the data on our calculator, ou ll see a linear pattern. Choose a pair of data points. For this eample, we ll use (49, 35) and (, 4). The slope of the line through these points is Using (49, 35), the equation for the line in point-slope form is 35.38( 49). If ou graph this equation, ou ll see that the line fits the data ver closel. Now, pick a different pair of data points. Find an equation for the line through the points, and graph the equation on our calculator. Does one of the equations fit the data better than the other? [,,,,, ] Discovering Algebra Condensed Lessons Ke Curriculum Press

7 CONDENSED L E S S O N 5.4 Equivalent Algebraic Equations Previous In this lesson ou will determine whether equations are equivalent rewrite linear equations in intercept form use mathematical properties to rewrite and solve equations You have seen that the same line can be represented b more than one equation. In this lesson ou ll learn how to recognize equivalent equations b using mathematical properties and the rules for order of operations. Investigation: Equivalent Equations Page 7 of our book gives si equations. Although these equations look ver different, the are all equivalent. To see this, use the distributive propert to rewrite each equation in intercept form. a. 3 ( 1) b. 3 5 c. 9 ( ) d e. 7 ( 1) f ( 5) 5 5 (.5) ( 7) All the equations are equivalent to 5. To check that each original equation is equivalent to 5, enter both equations into our calculator and graph them. You will get the same line, which indicates that the same values satisf both equations. Now, look at the equations a o on page 77 of our book. These equations represent onl four different lines. To find the equivalent equations, write each equation in intercept form. You should get these results. Equations a, m, i, and l are equivalent to 5. Equations b, d, g, and k are equivalent to. Equations e, j, and n are equivalent to 3. Equations c, f, h, and o are equivalent to 4. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 7

8 Previous Lesson 5.4 Equivalent Algebraic Equations (continued) In equations h and j, and are on the same side of the equation and the other side is a constant. These equations are in standard form. Here are the steps for rewriting equation j Original equation. Subtract 1 from both sides Divide both sides b. In the investigation ou saw that no matter what form a linear equation is given in, ou can rewrite it in intercept form. When equations are in intercept form, it is eas to see whether the are equivalent. Page 78 of our book reviews the properties that allow ou to rewrite (and solve) equations. Read these properties and the eamples that follow. Here are two more eamples. EXAMPLE C Is equivalent to 7( 1)? Solution Rewrite in intercept form Original equation. Subtract 1 from both sides The epression 1 1 is Divide both sides b Divide 1 b 3 and 1 b 3. Rewrite 7( 1) in intercept form. 7( 1) 7 7 Original equation. Distributive propert. 3 7 Add and 7. The equations are not equivalent. EXAMPLE D Solve 4( 3) 7 4. Identif the propert of equalit used in each step. 4( 3) 7 4 Original equation. 4( 3) 8 Multiplication propert (multipl both sides b 7). 3 7 Division propert (divide both sides b 4). Addition propert (add 3 to both sides). 5 Division propert (divide both sides b ). 8 Discovering Algebra Condensed Lessons Ke Curriculum Press

9 CONDENSED L E S S O N 5.5 Writing Point-Slope Equations to Fit Data Previous In this lesson ou will write point-slope equations to fit data use equations of lines of fit to make predictions compare two methods for fitting lines to data This lesson gives ou more practice using the point-slope form to model data. You ma find that using the point-slope form is more efficient than using the intercept form because ou don t have to first write a direct variation equation and then adjust it for the intercept. Investigation: Life Epectanc Steps 1 4 The table on page 84 of our book shows the relationship between the number of ears a person might be epected to live and the ear he or she was born. Plot the female life-epectanc data in the window [193,,, 55, 85, 5]. Look for two points on the graph so that the line through the points closel reflects the pattern of all the points. For this eample, we ll use (197, 74.7) and (199, 78.8). The slope of the line through these points is Using (197, 74.7), ou can write the equation ( 197). To predict the life epectanc of a female who will be born in, substitute for in the equation ( 197) ( 197) (5) The equation predicts that a female born in will have a life epectanc of 85.3 ears. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 9

10 Previous Lesson 5.5 Writing Point-Slope Equations to Fit Data (continued) Steps 5 8 If we had chosen a different pair of points, we would have found a different equation and made a different life-epectanc prediction. For eample, if we had used (195, 71.1) and (1995, 78.9), we would have gotten the slope and the equation ( 195). This equation gives a life-epectanc prediction of about 83.5 ears for a female born in. Now, find equations of lines of fit for the male data and combined data. Here are equations for all three sets of data, using the points for 197 and 199. Female: ( 197) Male: ( 197) Combined: 7.8.3( 197) Notice that the slopes of all three lines are close to., indicating that life epectanc increases b about. ear for each age increase of 1 ear, regardless of gender. The graph below shows all three sets of data and all three lines graphed in the same window. The data for females is plotted with boes, the data for males is plotted with plus signs, and the combined data is plotted with dots. Notice that the line for the combined data is between the other two lines. This is reasonable because the combined life epectanc for both males and females should be between the life epectanc for females and the life epectanc for males. You have used two methods for finding the equation of a line of fit. One method uses the intercept form, and the other uses the point-slope form. For the interceptform method (which ou used in the spaghetti beam investigation), ou found a line parallel to the line of fit and then adjusted it up or down, using estimation, to fit the points. With the point-slope method, ou get an equation without making an adjustments, but ou ma find the line does not fit the data as well as ou would like. 7 Discovering Algebra Condensed Lessons Ke Curriculum Press

11 CONDENSED L E S S O N 5. More on Modeling Previous In this lesson ou will use Q-points to fit a line to a set of data use a line of fit to make predictions Several times in this chapter ou have found the equation of a line of fit for a data set. You probabl found that ou and our classmates often wrote different equations even though ou were working with the same data. In this investigation ou will learn a sstematic method for finding the equation of a line of fit. This method alwas gives the same equation for a given set of data. Investigation: FIRE!!!! Steps 1 3 In this investigation, students form a line and record the time it takes to pass a bucket from one end of the line to the other. After each trial, some students sit down and the remaining students repeat the eperiment. Here are the data one class collected and the graph of the data. Number of people, Passing time (sec), Passing time (sec) Number of people Steps 4 13 You can use the quartiles of the - and -values to fit a line to the data. First, find the five-number summaries. The five-number summar for the -values is 5, 11, 14, 18,. The five-number summar for the -values is 4, 8, 11.5, 14, 18. Use these values to add horizontal and vertical bo plots to the graph. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 71

12 Previous Lesson 5. More on Modeling (continued), draw vertical lines from the Q1- and Q3-values on the -ais bo plot, and draw horizontal lines from the Q1- and Q3-values on the -ais bo plot. These lines form a rectangle. The vertices of this rectangle are called Q-points. Q-points ma or ma not be actual points in the data set. Note that anone who starts with this data will get the same Q-points. Passing time (sec) Q-point Q-point Q-point Q-point Number of people Finall, draw the diagonal of the rectangle that reflects the direction of the data. This is a line of fit for the data. Use the coordinates of the Q-points (11, 8) and (18, 14) to write an equation for this line. The slope 14 8 is or 7, so one equation is 8 ( 7 11). In intercept form, this is The slope represents the number of seconds it takes each person to pass the bucket. The -intercept represents the amount of time it takes people to pass the bucket. In this case, the -intercept is a negative number, which doesn t make sense in the real situation. This demonstrates that a model onl approimates what actuall happens. Now, use our calculator to plot the data points, draw vertical and horizontal lines through the quartile values, and find a line of fit. (See Calculator Note 5B for help on using the draw menu.) The eample in our book uses the Q-point method to fit a line to a different set of data. Read this eample and make sure ou understand it. Passing time (sec) Number of people 7 Discovering Algebra Condensed Lessons Ke Curriculum Press

13 CONDENSED L E S S O N 5.7 Applications of Modeling Previous In this lesson ou will use and compare three methods of finding a line of fit for a set of data use linear models to make predictions You now know several methods for fitting a line to data. In this lesson ou will practice and compare these methods. Investigation: What s M Line? The tables on page 9 of our book show how the diameter of a person s pupil changes with age. Table 1 shows dalight pupil diameters. Table shows nighttime pupil diameters. Here, we will work with data in Table. Steps 1 First, ou ll find a line of fit b eeballing the data. Plot the data on graph paper. Then, la a strand of spaghetti on the plot so that it crosses the -ais and follows the direction of the data. Here is an eample. Diameter (mm) Age (ears) This line crosses the -ais at about (, 9.8). The point (3, 7) is also on the line. The slope of the line through these points is or about.93. So the equation of this line in intercept form is Steps 3 4 Now, ou ll fit a line using representative data points. Use our calculator with the window [,,,,, 1] to make a scatter plot of the data. Choose two points ou think reflect the direction of the data. For eample, ou might use (3, 7.) and (, 4.1). The slope of the line through these points is or about.97. An equation for the line through these points in point-slope form is 7.97( 3). Rewriting this in intercept form gives (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 73

14 Previous Lesson 5.7 Applications of Modeling (continued) Steps 5, ou ll find a line of fit using Q-points. The five-number summar of the -values is, 3, 5, 7, 8, so Q1 and Q3 are 3 and 7. The five-number summar of the -values is.5, 3., 5., 7., 8., so Q1 and Q3 are 3. and 7.. Use these values to draw a rectangle on the plot. Then, draw the diagonal that reflects the direction of the data. The diagonal passes through (3, 7.) and (7, 3.). Diameter (mm) Age (ears). 7. The slope of this line is or.95. An equation for this line in point-slope form is 7.95( 3). Rewriting this in intercept form gives Steps 7 13 In the preceding linear models, the slope indicates the change in pupil diameter for each age increase of 1 ear. The three methods gave slightl different values for the slope (.93,.97, and.95). The -intercept of each model represents the pupil diameter at birth (age ). These values are also slightl different for the three models (9.8, 9.91, and 9.85.) You can use an of the models to predict the age when a person s nighttime pupil diameter is about 4.5 mm. For eample, to predict a person s age using the first model, , solve The solution is about 57, so a person s nighttime pupil diameter is 4.5 mm at about age 57. Changing the -intercept of an equation increases or decreases ever -coordinate b the same amount. For eample, changing the -intercept of the first model, , to 9.3 gives the equation The -coordinate of each point on this second line is.5 mm less than the -coordinate of that point on the original line. For a person of age 45, the first equation gives a pupil diameter of 5.15 mm and the second equation gives a pupil diameter of mm. A small change in slope has little effect on points near the data points, but the effect is magnified for points far out on the line, awa from the points. For this data, age values much greater than 8 don t make sense in the real world. However, ou might tr changing the slope value of one of the models slightl and then substituting age values of,, and so on to see the effect of the change. 74 Discovering Algebra Condensed Lessons Ke Curriculum Press

### 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

### Name Class Date. Additional Vocabulary Support

- Additional Vocabular Support Rate of Change and Slope Concept List negative slope positive slope rate of change rise run slope slope formula slope of horizontal line slope of vertical line Choose the

### 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.

### SECTION 5-1 Exponential Functions

354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

### 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

### NAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is

- NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the

### Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan

Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan I. Topic: Slope-Intercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.

### 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

### 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

### SECTION 2-2 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

### Worksheet A5: Slope Intercept Form

Name Date Worksheet A5: Slope Intercept Form Find the Slope of each line below 1 3 Y - - - - - - - - - - Graph the lines containing the point below, then find their slopes from counting on the graph!.

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### 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

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### The Big Picture. Correlation. Scatter Plots. Data

The Big Picture Correlation Bret Hanlon and Bret Larget Department of Statistics Universit of Wisconsin Madison December 6, We have just completed a length series of lectures on ANOVA where we considered

### Chapter 3 & 8.1-8.3. Determine whether the pair of equations represents parallel lines. Work must be shown. 2) 3x - 4y = 10 16x + 8y = 10

Chapter 3 & 8.1-8.3 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

### Linear Equations. 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber

Linear Equations 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Tools: Geometer s Sketchpad Software Overhead projector with TI- 83

### 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.

### FACTORING QUADRATICS 8.1.1 through 8.1.4

Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten

### Chapter 13 Introduction to Linear Regression and Correlation Analysis

Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### 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

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### 1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) =

Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

### 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

### 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

### Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

### Ellington High School Principal

Mr. Neil Rinaldi Ellington High School Principal 7 MAPLE STREET ELLINGTON, CT 0609 Mr. Dan Uriano (860) 896- Fa (860) 896-66 Assistant Principal Mr. Peter Corbett Lead Teacher Mrs. Suzanne Markowski Guidance

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### 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

### 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

### Answer Key Building Polynomial Functions

Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,

### G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam

G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d

### 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

Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

### 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

R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract

### 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

### THIS CHAPTER INTRODUCES the Cartesian coordinate

87533_01_ch1_p001-066 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

### Tennessee Department of Education. Task: Sally s Car Loan

Tennessee Department of Education Task: Sally s Car Loan Sally bought a new car. Her total cost including all fees and taxes was \$15,. She made a down payment of \$43. She financed the remaining amount

### M122 College Algebra Review for Final Exam

M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant

### Mathematics Curriculum Guide Precalculus 2015-16. Page 1 of 12

Mathematics Curriculum Guide Precalculus 2015-16 Page 1 of 12 Paramount Unified School District High School Math Curriculum Guides 2015 16 In 2015 16, PUSD will continue to implement the Standards by providing

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### 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

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

### that satisfies (2). Then (3) ax 0 + by 0 + cz 0 = d.

Planes.nb 1 Plotting Planes in Mathematica Copright 199, 1997, 1 b James F. Hurle, Universit of Connecticut, Department of Mathematics, Unit 39, Storrs CT 669-39. All rights reserved. This notebook discusses

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### 2.4. Factoring Quadratic Expressions. Goal. Explore 2.4. Launch 2.4

2.4 Factoring Quadratic Epressions Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form The area of a rectangle

### 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.

### Students will use various media (computer, graphing calculator, paper and pencil) to graph/sketch linear equations.

Title: Lines, Lines, Everywhere!! A discovery/exploration lesson investigating equations of the form y = mx + b to see how the values of b and m affects the graph. Link to Outcomes: Communication/ Cooperation

### ( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:

Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (

### is the degree of the polynomial and is the leading coefficient.

Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

### Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

### Chapter 9 Descriptive Statistics for Bivariate Data

9.1 Introduction 215 Chapter 9 Descriptive Statistics for Bivariate Data 9.1 Introduction We discussed univariate data description (methods used to eplore the distribution of the values of a single variable)

74 In the Herb Business, Part III Factoring and Quadratic Equations In the herbal medicine business, you and your partner sold 120 bottles of your best herbal medicine each week when you sold at your original

### Trigonometry Review Workshop 1

Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### Overview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: Slope-Intercept Form

Name Date Linear Functions: Slope-Intercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### ALGEBRA I (Created 2014) Amherst County Public Schools

ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies

### Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Homework #1 Solutions

Homework #1 Solutions Problems Section 1.1: 8, 10, 12, 14, 16 Section 1.2: 2, 8, 10, 12, 16, 24, 26 Extra Problems #1 and #2 1.1.8. Find f (5) if f (x) = 10x x 2. Solution: Setting x = 5, f (5) = 10(5)

### Indiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A.

STUDENT S SOLUTIONS MANUAL JUDITH A. PENNA Indiana Universit Purdue Universit Indianapolis COLLEGE ALGEBRA: GRAPHS AND MODELS FIFTH EDITION Marvin L. Bittinger Indiana Universit Purdue Universit Indianapolis

### Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### 2-5 Rational Functions

-5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4

### To represent mathematical relationships using graphs. 4-1 Activity: Relating Quantities See Department Chair for File 1 day A.REI.10, F.IF.

CCSS for Mathematical Practice: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model

### Skills Practice Skills Practice for Lesson 1.1

Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabular Define each term in our own words.. function A function is a relation that maps each value of

### Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

### Unit 5: Coordinate Geometry Practice Test

Unit 5: Coordinate Geometry Practice Test Math 10 Common Name: Block: Please initial this box to indicate you carefully read over your test and checked your work for simple mistakes. What I can do in this

### Ministry of Education. The Ontario Curriculum. Mathematics. Mathematics Transfer Course, Grade 9, Applied to Academic

Ministry of Education The Ontario Curriculum Mathematics Mathematics Transfer Course, Grade 9, Applied to Academic 2 0 0 6 Contents Introduction....................................................... 2

Factoring Quadratic Trinomials Student Probe Factor x x 3 10. Answer: x 5 x Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials. Part 1 of the lesson consists

### As evident above, you will need strong factoring skills to find the domain of many rational expressions.

You may write in this packet, but answer all questions on separate sheets of paper. A "rational epression" is a polynomial fraction. In general, what holds true for simple fractions is true for rational

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### Teacher Resource Sampler. Includes Spanish Resources

Teacher Resource Sampler Includes Spanish Resources -6-5 9 0 0 0 0 online access to 0-13-31859-. S LEARNING Go beyond the tetbook with Pearson Algebra 1 Pearson Algebra 1 Common Core Edition 015 provides

Factoring Quadratic Trinomials Student Probe Factor Answer: Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials Part 1 of the lesson consists of circle puzzles

### Answers Teacher Copy. Systems of Linear Equations Monetary Systems Overload. Activity 3. Solving Systems of Two Equations in Two Variables

of 26 8/20/2014 2:00 PM Answers Teacher Copy Activity 3 Lesson 3-1 Systems of Linear Equations Monetary Systems Overload Solving Systems of Two Equations in Two Variables Plan Pacing: 1 class period Chunking

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second

### 15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

### 2-1 Position, Displacement, and Distance

2-1 Position, Displacement, and Distance In describing an object s motion, we should first talk about position where is the object? A position is a vector because it has both a magnitude and a direction:

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

### Pre Calculus Math 40S: Explained!

Pre Calculus Math 0S: Eplained! www.math0s.com 0 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph

### CHAPTER 7: FACTORING POLYNOMIALS

CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor

### The program also provides supplemental modules on topics in geometry and probability and statistics.

Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students

### American Diploma Project

Student Name: American Diploma Project ALGEBRA l End-of-Course Eam PRACTICE TEST General Directions Today you will be taking an ADP Algebra I End-of-Course Practice Test. To complete this test, you will

### Representing Polynomials

CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Polnomials Mathematics Assessment Resource Service Universit of Nottingham & UC Berkele

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Slope and Rate of Change

Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the

### LINEAR FUNCTIONS AND CHANGE

Chapter One LINEAR FUNCTIONS AND CHANGE A function describes how the value of one quantit depends on the value of another. A function can be represented b words, a graph, a formula, or a table of numbers.