4.1 The Coordinate Plane

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1 4. The Coordinate Plane Goal Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin -ais -ais Ordered pair -coordinate -coordinate Quadrant Scatter plot Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 77

2 Eample Plot Points in a Coordinate Plane Plot the points A(2, ), B(, 4), and C(0, 2) in a coordinate plane. To plot the point A(2, ), start at the. Move 2 units to the and units. To plot the point B(, 4), start at the. Move units to the and 4 units. To plot the point C(0, 2), start at the. Move 0 units to the and 2 units. Eample 2 Identif Quadrants Name the quadrants the points D(2, 9) and E(2, 4) are in. The point D(2, 9) is in Quadrant -coordinates are both. The point E(2, 4) is in Quadrant -coordinates are both. because its - and because its - and Checkpoint Plot the points in the same coordinate plane.. A(, 2) 2. B(4, 0). C(, 4) 4. D(, 2) Name the quadrant the point is in. 5. (6, 7) 6. (6, 7) 7. (6, 7) 8. (6, 7) Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 78

3 Eample Make a Scatter Plot The number of NCAA men s college basketball teams is shown in the table. Year Men s teams a. Make a scatter plot of the data. b. Describe the pattern of the number of men s basketball teams. a. Let M represent. Let t represent. Because ou want to see how the number of teams changed over time, put t on the ais and M on the ais. Choose a scale. Use a break in the scale for the number of teams to focus on the values between and. NCAA Men s Basketball Teams b. From the scatter plot, ou can see that the number of men s teams in the NCAA was for three ears and then began to. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 79

4 4.2 Graphing Linear Equations Goal Graph a linear equation using a table of values. VOCABULARY Linear equation of an equation Function form Graph of an equation Eample Check s of Linear Equations Determine whether the ordered pair is a solution of 2 6. a. (, 4) b. (4, ) a. 2 6 Write original equation. 2( ) ( ) 6 Substitute for and for. 6 Simplif. statement. Answer (, 4) a solution of the equation 2 6. b. 2 6 Write original equation. 2( ) ( ) 6 Substitute for and for. 6 Simplif. statement. Answer (4, ) a solution of the equation 2 6. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 80

5 Checkpoint Determine whether the ordered pair is a solution of 22.. (0, ) 2. (, ). (, 5) Eample 2 Find s of Linear Equations Find three ordered pairs that are solutions of Rewrite the equation in function form to make it easier to substitute values into the equation Write original equation. Add to each side. 2. Choose an value for and substitute it into the equation to find the corresponding -value. The easiest -value is. 5( ) 2 Substitute for. Simplif. The solution is.. Select a few more values of and make a table to record the solutions Answer,, and are three solutions of Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 8

6 GRAPHING A LINEAR EQUATION Step Rewrite the equation in form, if necessar. Step 2 Choose a few values of and make a. Step Plot the points from the table of values. A line through these points is the of the equation. Eample Graph a Linear Equation Use a table of values to graph the equation Rewrite the equation in function form b solving for. 4 4 Write original equation. 4 4 Subtract from each side. Divide each side b. When graphing a linear equation, tr choosing values of that include negative values, zero, and positive values to see how the graph behaves to the left and right of the -ais. 2. Choose a few values of and make a table of values You have found three solutions. (4, ), (0, ), (4, ). Plot the points and draw a line through them. Checkpoint Complete the following eercise. 4. Use a table of values to graph the equation 2. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 82

7 4. Graphing Horizontal and Vertical Lines Goal Graph horizontal and vertical lines. VOCABULARY Constant function EQUATIONS OF HORIZONTAL AND VERTICAL LINES b a In the coordinate plane, the graph of b is a line. In the coordinate plane, the graph of a is a line. Eample Graph the Equation b Graph the equation. The equation does not have as a variable. The -coordinate is alwas, regardless of the value of. Some points that are solutions of the equation are: (, ), (0, ), and (, ) The graph of is a line units the. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 8

8 Eample 2 Graph the Equation a Graph the equation 2. The equation does not have as a variable. The -coordinate is alwas, regardless of the value of. Some points that are solutions of the equation are: (, ), (, 0), and (,) Answer The graph of 2 is a line units to the of the. Eample Write an Equation of a Line Write the equation of the line in the graph. a. b. a. The graph is a line. The -coordinate is alwas equation of the line is.. The b. The graph is a line. The -coordinate is alwas. The equation of the line is. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 84

9 Checkpoint Complete the following eercises.. Graph the equation. 2. Write the equation of the 2 line in the graph. Eample 4 Write a Constant Function Tree Trunks The graph shows the diameter of a tree trunk over a 6-week period. Write an equation to represent the diameter of the tree trunk for this period. What is the domain of the function? What is the range? Diameter of a Tree Trunk Diameter (in.) D Number of Weeks t From the graph, ou can see that the diameter was about inches throughout the 6-week period. Therefore, the diameter D during this time t is D. The domain is. The range is. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 85

10 4.4 Graphing Lines Using Intercepts Goal Find the intercepts of the graph of a linear equation and then use them to make a quick graph of the equation. VOCABULARY -intercept -intercept Eample Find - and -Intercepts Find the - and -intercepts of the graph of the equation 4 2. To find the -intercept, substitute for and solve for. 4 2 Write original equation. 4( ) 2 Substitute for. 2 Simplif. Solve for. To find the -intercept, substitute for and solve for. 4 2 Write original equation. ( ) 4 2 Substitute for. 2 Answer The -intercept is Simplif. Solve for.. The -intercept is. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 86

11 Checkpoint Complete the following eercise.. Find the -intercept and the equation intercept of the graph of the The Quick Graph process works because onl two points are needed to determine a line. MAKING A QUICK GRAPH Step Findthe. Step 2Drawa coordinate plane that includes the. Step Plot the and draw a line through them. Eample 2 Make a Quick Graph Graph the equation Find the intercepts Write original equation. 9 6( ) 8 Substitute for. The -intercept is Write original equation. 9( ) 6 8 Substitute for. The -intercept is. 2. Draw a coordinate plane that includes the points (, ) and (, ).. Plot the points (, ) and (, ) and draw a line through them. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 87

12 Checkpoint Complete the following eercise. 2. Graph the equation Eample Choose Appropriate Scales When ou make a quick graph, find the intercepts before ou draw the coordinate plane. This will help ou find an appropriate scale on each ais. Graph the equation Find the intercepts. 5 5 Write original equation. 5 5 Substitute for. 5 Subtract from each side. Divide each side b. The -intercept is. 5 5 Write original equation. 5( ) 5 Substitute for. Simplif. The -intercept is. 2. Draw a coordinate plane that includes the points (, ) and (, ). With these values, it is reasonable to use tick marks at intervals.. Plot the points (, ) and (, ) and draw a line through them Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 88

13 4.5 The Slope of a Line Goal Find the slope of a line. VOCABULARY Slope Eample The Slope Ratio Find the slope of a ramp that has a vertical rise of feet and a horizontal run of 8 feet. Let m represent the slope. Horizontal Run 8 feet Vertical Rise feet m Answer The slope of the ramp is. In the slope formula, is read as sub one and 2 as sub two. THE SLOPE OF A LINE The slope m of a line that passes through the points (, ) and ( 2, 2 ) is m r ise change in run change in ( 2, 2 ) ( 2, ) 2. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 89

14 Eample 2 Positive Slope Find the slope of the line that passes through the points (, 2) and (2, ). Let (, ) (, 2) and ( 2, 2 ) (2, ). m Formula for slope Substitute values. 5 (2, ) (, 2) Simplif. Slope is. Answer The slope of the line is. The line from left to right. The slope is. Eample Zero Slope Find the slope of the line passing through the points (2, ) and (4, ). Let (, ) (2, ) and ( 2, 2 ) (4, ). m Formula for slope Substitute values. Simplif. 5 (2, ) (4, ) 5 Slope is. Answer The slope of the line is. The line is. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 90

15 Eample 4 Undefined Slope Find the slope of the line passing through the points (, 4) and (, 2). Let (, ) (, 4) and ( 2, 2 ) (, 2). m Formula for slope (, 2) 2 Substitute values. (, 4) 4 5 Division b is. Answer Because division b is, the slope is. The line is. Checkpoint Find the slope of the line passing through the points. Then state whether the slope of the line is positive, negative, zero, or undefined.. (5, 2), (7, 2) 2. (0, 0), (9, 0). (7, 8), (7, 8) 4. (2, 4), (8, 6) Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 9

16 4.6 Direct Variation Goal Write and graph equations that represent direct variation. VOCABULARY Direct variation Constant of variation The model for direct variation k is read as varies directl with. Eample Write a Direct Variation Model The variables and var directl. One pair of values is 7 and 2. a. Write an equation that relates and. b. Find the value of when 4. a. Because and var, the equation is of the form. k Write model for direct variation. k( ) Substitute for and for. k Divide each side b. Answer An equation that relates and is. b. ( ) Substitute for. Simplif. Answer When 4,. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 92

17 PROPERTIES OF GRAPHS OF DIRECT VARIATION MODELS p The graph of k is a line through the. p The slope of the graph of k is. k is negative. k is positive. Eample 2 Graph a Direct Variation Model Graph the equation.. Plot a point at the. 2. Find a second point b choosing an value of and substituting it into the equation to find the corresponding -value. Use the value for. Write original equation. ( ) Substitute for. Simplif. The second point is (, ).. Plot the second point and draw a line through the and the second point. Checkpoint The variables and var directl. Use the given values to write an equation that relates and.. 6, , 20..6,.8 Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 9

18 4.7 Graphing Lines Using Slope-Intercept Form Goal Graph a linear equation in slope-intercept form. VOCABULARY Slope-intercept form Parallel lines SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE The linear equation m b is written in slope-intercept form, where is the slope and is the -intercept. slope -intercept Eample Find the Slope and -Intercept Find the slope and -intercept of 2. Rewrite the equation in slope-intercept form. 2 Write original equation. 2 Add to each side. Answer The slope is Divide each side b. m and b.. The -intercept is. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 94

19 Eample 2 Graph an Equation in Slope-Intercept Form Graph the equation 2.. Find the slope,, and the -intercept,. 2. Plot the point (0, b) when b is.. Use the slope to locate a second point on the line. m r ise run move move units up unit right 4. Draw a line through the two points. Checkpoint Find the slope and -intercept of the equation Graph the equation in slope-intercept form Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 95

20 Eample Identif Parallel Lines Which of the following lines are parallel? line a: 2 line b: 2 line c: 2. Rewrite each equation in slope-intercept form. line a: line b: line c: 2. Identif the slope of each equation. The slope of line a is. The slope of line b is. The slope of line c is.. Compare the slopes. Lines and are parallel because each has a slope of. Line is not parallel to either of the other two lines because it has a slope of. Check The graph gives ou a visual check. It shows that line b each of the two parallel lines. Answer Lines and are parallel. b a c Checkpoint Which of the following lines are parallel? 6. line a: 4 6 line b: line c: 4 8 Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 96

21 4.8 Functions and Relations Goal Decide whether a relation is a function and use function notation. VOCABULARY Relation Function Function notation Linear function Eample Identif Functions Decide whether the relation is a function. If it is a function, give the domain and the range. a. p p b Input 2 4 Output 5 9 a. The relation a function because. b. The relation a function. For each there is. The domain of the function is. The range is. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 97

22 VERTICAL LINE TEST FOR FUNCTIONS A graph is a function if no line intersects the graph at point. Eample 2 Use the Vertical Line Test Use the vertical line test to determine whether the graph represents a function. a. b. a. vertical line can be drawn to intersect the graph more than once. The graph. b. vertical line can be drawn to intersect the graph more than once. The graph. Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 98

23 Checkpoint Decide whether the relation is a function. If it is a function, give the domain and the range.. p O p Input 2 4 Output Use the vertical line test to determine whether the graph represents a function.. 4. Eample Evaluate a Function You don t have to use f to name a function. Just as ou can use an letter as a variable, ou can use an letter to name a function. Evaluate g(( ) 2 when 4. g(( ) 2 Write original function. g( ) 2( ) Substitute for. Simplif. Answer When 4, g(( ). Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 99

24 Eample 4 Graph a Linear Function Graph f () Rewrite the function as. 2. Find the slope and -intercept. m and b. Use the to locate a second point. 4. Draw a line through the two points. Checkpoint Evaluate the function for the given value of the variable. 5. f(( ) 7 when 6. f(( ) 2 5 when 2 Graph the linear function. 7. f(( ) 2 8. f(( ) 4 Copright McDougal Littell, Chapter 4 Algebra, Concepts and Skills a division of Houghton Mifflin Compan. Notetaking Guide 00

o Checkpoint Plot the points in the same coordinate plane.

o Checkpoint Plot the points in the same coordinate plane. The ~ al Coordinate Plane Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin x-axls -axis Ordered pair x-coordinate -coordinate Quadrant Scatter plot ~. Lesson 4.. Algebra. Concepts