Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Introduction to Graphing The Rectangular Coordinate System Scatterplots and Line Graphs 3.1

Slide 3 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Rectangular Coordinate System One common way to graph data is to use the rectangular coordinate system, or xy-plane. In the xy-plane the horizontal axis is the x-axis, and the vertical axis is the y-axis. The axes intersect at the origin. The axes divide the xy-plane into four regions called quadrants, which are numbered I, II, III, and IV counterclockwise.

Slide 4 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Plotting points Plot the following ordered pairs on the same xy-plane. State the quadrant in which each point is located, if possible. a. (4, 3) b. ( 3, 4)c. ( 1, 0) Solution a. (4, 3) Move 4 units to the right of the origin and 3 units up. b. ( 3, 4) Move 3 units to the left of the origin and 4 units down. c. ( 1, 0) Move 1 unit to the left of the origin. Quadrant I Quadrant III Not in any quadrant

Slide 5 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Reading a graph Frozen pizza makers have improved their pizzas to taste more like homemade. Use the graph to estimate frozen pizza sales in 1994 and Solution a. To estimate sales in 1994, locate 1994 on the x-axis. Then move upward to the data point and approximate its y- coordinate. b. To estimate sales in 2000, locate 2000 on the x-axis. Then move upward to the data point and approximate its y- coordinate. a. about $2.1 billion in sales b. about $3.0 billion in sales

If distinct points are plotted in the xy-plane, then the resulting graph is called a scatterplot. Slide 6 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Scatterplots and Line Graphs

Slide 7 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Making a scatterplot of gasoline prices The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These price have not been adjusted for inflation. Year Cost (per gal in cents) The data point (1975, 56.7) can be used to indicate the average cost of a gallon of gasoline in 1975 was 56.7 cents. Plot the data points in the xy-plane.

Slide 8 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Making a scatterplot of gasoline prices The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These prices have not been adjusted for inflation. Year Cost (per gal in cents)

Line Graphs Slide 9 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sometimes it is helpful to connect consecutive data points in a scatterplot with line segments. This creates a line graph.

Slide 10 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Making a line graph Use the data in the table to make a line graph. x y Plot the points and then connect consecutive points with line segments.

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Linear Equations in Two Variables Basic Concepts Tables of Solutions Graphing Linear Equations in Two Variables 3.2

Slide 13 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Basic Concepts Equations can have any number of variables. A solution to an equation with one variable is one number that makes the statement true.

Slide 14 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Testing solutions to equations Determine whether the given ordered pair is a solution to the given equation. a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4) Solution a. y = x + 5 b. 2x + 3y = 18 7 = = 7 True The ordered pair (2, 7) is a solution. 2(3) + 3( 4) = = The ordered pair (3, 4) is NOT a solution.

Tables of Solutions Slide 15 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A table can be used to list solutions to an equation. A table that lists a few solutions is helpful when graphing an equation.

Slide 16 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Completing a table of solutions Complete the table for the equation y = 3x – 1. Solution x y x y

Slide 17 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Graphing an equation with two variables Make a table of values for the equation y = 3x, and then use the table to graph this equation. Solution Start by selecting a few convenient values for x such as –1, 0, 1, and 2. Then complete the table. xy –1– Plot the points and connect the points with a straight line.

Slide 18 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 19 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Graphing linear equations Graph the linear equation. Solution Because this equation can be written in standard form, it is a linear equation. Choose any three values for x. xy – Plot the points and connect the points with a straight line.

Slide 20 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Graphing linear equations Graph the linear equation. Solution Because this equation can be written in standard form, it is a linear equation. Choose any three values for x. xy Plot the points and connect the points with a straight line.

Slide 21 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solve for y and then graphing Graph the linear equation by solving for y first. Solution Solve for y. xy – Plot the points and connect the points with a straight line.

Slide 22 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

More Graphing of Lines Finding Intercepts Horizontal Lines Vertical Lines 3.3

Slide 24 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding Intercepts The y-intercept is where the graph intersects the y- axis. The x-intercept is where the graph intersects the x- axis.

Slide 25 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Using intercepts to graph a line Use intercepts to graph 3x – 4y = 12. Solution The x-intercept is found by letting y = 0. The graph passes through the two points (4, 0) and (0, –3). The y-intercept is found by letting x = 0.

Slide 26 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Using a table to find intercepts Complete the table. Then determine the x-intercept and y-intercept for the graph of the equation x – y = 3. Solution Find corresponding values of y for the given values of x. x y x y The x-intercept is (3, 0). The y-intercept is (0, –3).

Slide 27 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Modeling the velocity of a toy rocket A toy rocket is shot vertically into the air. Its velocity v in feet per second after t seconds is given by v = 320 – 32t. Assume that t 0 and t 10. a. Graph the equation by finding the intercepts. b. Interpret each intercept. Solution a. Find the intercepts. b. The rocket had velocity of 0 feet per second after 10 seconds. The v-intercept indicates that the rockets initial velocity was 320 feet per second.

Slide 28 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Horizontal Lines

Slide 29 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Graphing a horizontal line Graph the equation y = 2 and identify its y-intercept. Solution The graph of y = 2 is a horizontal line passing through the point (0, 2), as shown below. The y-intercept is 2.

Slide 30 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Vertical Lines

Slide 31 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Graphing a vertical line Graph the equation x = 2, and identify its x-intercept. Solution The graph of x = 2 is a vertical line passing through the point (2, 0), as shown below. The x-intercept is 2.

Slide 32 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Writing equations of horizontal and vertical lines Write the equation of the line shown in each graph. a.b. Solution a. The graph is a horizontal line. The equation is y = –1. b. The graph is a vertical line. The equation is x = –1.

Slide 33 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Writing equations of horizontal and vertical lines Find an equation for a line satisfying the given conditions. a. Vertical, passing through (3, 4). b. Horizontal, passing through (1, 2). c. Perpendicular to x = 2, passing through (1, 2). Solution a. The x-intercept is 3. The equation is x = 3. b. The y-intercept is 2. The equation is y = 2. c. A line perpendicular to x = 2 is a horizontal line with y-intercept –2. The equation is y = 2.

Slide 34 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slope and Rates of Change Finding Slopes of Lines Slope as a Rate of Change 3.4

Slide 36 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slope The rise, or change in y, is y 2 y 1, and the run, or change in x, is x 2 – x 1.

Slide 37 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Calculating the slope of a line Use the two points to find the slope of the line. Interpret the slope in terms of rise and run. Solution The rise is 3 units and the run is –4 units. (–4, 1) (0, –2)

Slide 38 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Calculating the slope of a line Calculate the slope of the line passing through each pair of points. a. ( 3, 3), (0, 4)b. ( 3, 4), (3, 2) c. ( 2, 4), (2, 4)d. (4, 5), (4, 2) Solution

Slide 39 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Calculating the slope of a line Calculate the slope of the line passing through each pair of points. a. ( 3, 3), (0, 4)b. ( 3, 4), (3, 2) c. ( 2, 4), (2, 4)d. (4, 5), (4, 2) Solution

Slide 40 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Calculating the slope of a line Calculate the slope of the line passing through each pair of points. a. ( 3, 3), (0, 4)b. ( 3, 4), (3, 2) c. ( 2, 4), (2, 4)d. (4, 5), (4, 2) Solution

Slide 41 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Calculating the slope of a line Calculate the slope of the line passing through each pair of points. a. ( 3, 3), (0, 4)b. ( 3, 4), (3, 2) c. ( 2, 4), (2, 4)d. (4, 5), (4, 2) Solution

Slide 42 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slope Positive slope: rises from left to right Negative slope: falls from left to right

Slide 43 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slope Zero slope: horizontal line Undefined slope: vertical line

Slide 44 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Finding slope from a graph Find the slope of each line. a. b. Solution a. The graph rises 2 units for each unit of run m = 2/1 = 2. b. The line is vertical, so the slope is undefined.

Slide 45 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Sketching a line with a given slope Sketch a line passing through the point (1, 2) and having slope ¾. Solution Start by plotting (1, 2). The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4. The line passes through the point (1 + 4, 2 + 3) = (5, 5).

Slide 46 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slope as a Rate of Change When lines are used to model physical quantities in applications, their slopes provide important information. Slope measures the rate of change in a quantity.

Slide 47 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Interpreting slope The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. a. Find the y-intercept. What does the y-intercept represent? b. The graph passes through the point (4, 15). Discuss the meaning of this point. c. Find the slope of the line. Interpret the slope as a rate of change. Solution a. The y-intercept is 35, so the boat is initially 35 miles from the dock.

Slide 48 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Interpreting slope The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. a. Find the y-intercept. What does the y-intercept represent? b. The graph passes through the point (4, 15). Discuss the meaning of this point. c. Find the slope of the line. Interpret the slope as a rate of change. Solution b. The point (4, 15) means that after 4 hours the boat is 15 miles from the dock.

Slide 49 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Interpreting slope The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. a. Find the y-intercept. What does the y-intercept represent? b. The graph passes through the point (4, 15). Discuss the meaning of this point. c. Find the slope of the line. Interpret the slope as a rate of change. Solution c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.

Slide 50 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slope-Intercept Form Finding Slope-Intercept Form Parallel and Perpendicular Lines 3.5

Slide 52 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding Slope-Intercept Form

Slide 53 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Using a graph to write the slope-intercept form For the graph write the slope-intercept form of the line. Solution The graph intersects the y-axis at 0, so the y-intercept is 0. The graph falls 3 units for each 1 unit increase in x, the slope is –3. The slope intercept-form of the line is y = –3x.

Slide 54 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Sketching a line Sketch a line with slope ¾ and y-intercept 2. Write its slope-intercept form. Solution The y-intercept is (0, 2). Slope ¾ indicates that the graph rises 3 units for each 4 units run in x. The line passes through the point (4, 1).

Slide 55 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Graphing an equation in slope-intercept form Write the y = 4 – 3x equation in slope-intercept form and then graph it. Solution Plot the point (0, 4). The line falls 3 units for each 1 unit increase in x.

Slide 56 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parallel and Perpendicular Lines

Slide 57 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Finding parallel lines Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2, 1). Sketch a graph of each line. Solution The line has a slope of 3 any parallel line also has slope 3. Slope-intercept form: y = 3x + b. The value of b can be found by substituting the point (2, 1) into the equation.

Slide 58 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Finding perpendicular lines Find the slope-intercept form of a line passing through the origin that is perpendicular to each line. a. y = 4xb. Solution a. The y-intercept is 0. Perpendicular line has a slope of b. The y-intercept is 0. Perpendicular line has a slope of

Slide 59 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Point-Slope Form Derivation of Point-Slope Form Finding Point-Slope Form Applications 3.6

Slide 61 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The line with slope m passing through the point (x 1, y 1 ) is given by y – y 1 = m(x – x 1 ), or equivalently, y = m(x – x 1 ) + y 1 the point-slope form of a line. POINT-SLOPE FORM

Slide 62 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding a point-slope form Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line? Let m = 2 and (x 1, y 1 ) = (3,1) in the point-slope form. To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y. y – y 1 = m(x – x 1 ) y 1 = 2(x – 3) 3 – 1 ? 2(4 – 3) 2 = 2 The point (4, 3) lies on the line because it satisfies the point-slope form.

Slide 63 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding an equation of a line Use the point-slope form to find an equation of the line passing through the points (2, 3) and (2, 5). Before we can apply the point-slope form, we must find the slope.

Slide 64 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE continued We can use either (2, 3) or (2, 5) for (x 1, y 1 ) in the point-slope form. If we choose (2, 3), the point-slope form becomes the following. y – y 1 = m(x – x 1 ) If we choose (2, 5), the point-slope form with x 1 = 2 and y 1 = 5 becomes

Slide 65 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Finding equations of lines Find the slope-intercept form of the line perpendicular to passing through the point (4, 6). The line has slope m 1 = 1. The slope of the perpendicular line is m 2 = 1. The slope-intercept form of a line having slope 1 and passing through (4, 6) can be found as follows.

Slide 66 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Modeling water in a pool A swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8000 gallons and after 4 hours it contains 2000 gallons. a.How fast is the pump removing water? b.Find the slope-intercept form of a line that models the amount of water in the pool. Interpret the slope. c.Find the y-intercept and the x-intercept. Interpret each. d.Sketch the graph of the amount of water in the pool during the first 5 hours. e.The point (2, 6000) lies on the graph. Explain its meaning.

Slide 67 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE continued a.The pump removes gallons of water in 3 hours, or 2000 gallons per hour. b.The line passes through the points (1,8000) and (4, 2000), so the slope is Solution Use the point-slope form to find the slope-intercept form. y – y 1 = m(x – x 1 ) y – 8000 = 2000(x – 1) y – 8000 = 2000x y = 2000x + 10,000 A slope of 2000, means that the pump is removing 2000 gallons per hour.

Slide 68 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE continued c.The y-intercept is 10,000 and indicates that the pool initially contained 10,000 gallons. To find the x-intercept let y = 0 in the slope-intercept form. The x-intercept of 5 indicates that the pool is emptied after 5 hours.

Slide 69 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE continued d.The x-intercept is 5 and the y-intercept is 10,000. Sketch a line passing through (5, 0) and (0, 10,000). e.The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water.