5.1 Study Guide For use with pages 282 291 GOAL Write equations of lines. EXAMPLE 1 Use slope and y-intercept to write an equation Write an equation of the line with a slope of 1 } 2 and a y-intercept of 27. y 5 mx 1 b Write slope-intercept form. y 5 1 } 2 x 2 7 Substitute 1 } 2 for m and 27 for b. Exercises for Example 1 Write an equation of the line with the given slope and y-intercept. 1. slope: 7 y-intercept: 211 5.1 2. slope: 2 } 3 y-intercept: 5 EXAMPLE 2 3. slope: 2 7 } 5 y-intercept: 22 Write an equation of a line given two points Write an equation of the line shown. STEP 1 Calculate the slope. m 5 y 2 2 y 1 } x2 2 x 1 5 2 2(21) } 0 2 2 5 2 3 } 2 STEP 2 Write an equation of the line. The line crosses the y-axis at (0, 2). So, the y-intercept is 2. y 5 mx 1 b Write slope-intercept form. 5 4 1 y (0, 2) 2 2322 O 1 3 4 5 (2, 21) 22 23 x y 52 3 } 2 x 1 2 Substitute 2 3 } 2 for m and 2 for b. 9
5.1 Study Guide continued For use with pages 282 291 Exercises for Example 2 Write an equation of the line that passes through the given points. 4. (10, 4), (0, 21) 5. (0, 8), (5, 21) 6. (26, 28), (0, 214) EXAMPLE 3 Write a linear function 5.1 Write an equation for the linear function f with the values f(0) 5 7 and f(12) 5 15. STEP 1 Write f(0) 5 7 as (0, 7) and f(12) 5 15 as (12, 15). STEP 2 Calculate the slope of the line that passes through (0, 7) and (12, 15). m 5 y 2 2 y 1 } x2 2 x 1 5 15 2 7 } 12 2 0 5 8 } 12 5 2 } 3 STEP 3 Write an equation of the line. The line crosses the y-axis at (0, 7). So, the y-intercept is 7. y 5 mx 1 b Write slope-intercept form. y 5 } 2 3 x 1 7 Substitute } 2 3 for m and 7 for b. The function is f(x) 5 } 2 3 x 1 7. Exercises for Example 3 Write an equation for the linear function f with the given values. 7. f(0) 5 21, f(4) 5 13 8. f(3) 5212, f(0) 5 6 10
5.2 Study Guide For use with pages 292 299 GOAL Write an equation of a line using points on the line. EXAMPLE 1 Write an equation given the slope and a point Write an equation of the line that passes through the point (2, 5) and has a slope of 3. STEP 1 Identify the slope. The slope is 3. STEP 2 Find the y-intercept. Substitute the slope and the coordinates of the given point into y 5 mx 1 b. Solve for b. y 5 mx 1 b Write slope-intercept form. 55 3(2) 1 b Substitute 3 for m, 2 for x, and 5 for y. 21 5 b Solve for b. STEP 3 Write an equation of the line. y 5 mx 1 b Write slope-intercept form. y 5 3x 2 1 Substitute 3 for m, and 21 for b. EXAMPLE 2 Write an equation given two points Write an equation of the line that passes through (3, 9) and (22, 21). STEP 1 Calculate the slope. m 5 y 2 2 y 1 } x2 2 x 1 5 21 2 9 } 22 2 3 5 210 } 25 5 2 STEP 2 Find the y-intercept. Use the slope and the point (3, 9). y 5 mx 1 b Write slope-intercept form. 9 5 2(3) 1 b Substitute 2 for m, 3 for x, and 9 for y. 3 5 b Solve for b. STEP 3 Write an equation of the line. y 5 mx 1 b Write slope-intercept form. y 5 2x 1 3 Substitute 2 for m and 3 for b. 5.2 19
5.2 Study Guide continued For use with pages 292 299 Exercises for Examples 1 and 2 Write an equation of the line that passes through the given point and has the given slope. 1. (7, 2); m 5 4 2. (9, 15); m 52 1 } 3 Write an equation of the line that passes through the two given points. 3. (5, 8), (13, 12) 4. (26, 27), (23, 5) EXAMPLE 3 Write a linear function Write an equation of the linear function with the values f(2) 5 3 and f(23) 5 8. STEP 1 Calculate the slope. Write f(2) 5 3 as (2, 3) and f(23) 5 8 as (23, 8). m 5 y 2 2 y 1 } x2 5 } 8 2 3 2 x 1 23 2 2 5 } 5 25 521 5.2 STEP 2 Find the y-intercept. Use the slope and the point (2, 3). y 5 mx 1 b Write slope-intercept form. 3 521(2) 1 b Substitute 21 for m, 2 for x, and 3 for y. 5 5 b Solve for b. STEP 3 Write an equation for the function. Use f(x) 5 mx 1 b. f(x) 52x15 Substitute 21 for m and 5 for b. Exercises for Example 3 Write an equation for a linear function f that has the given values. 5. f(2) 524 and f(24) 527 6. f(25) 5 17 and f(3) 5 9 20
5.3 Study Guide For use with pages 302 308 GOAL Write linear equations in point-slope form. Vocabulary The point-slope form of the equation of the nonvertical line through a given point (x 1, y 1 ) with a slope of m is y 2 y 1 5 m(x 2 x 1 ). EXAMPLE 1 Write an equation in point-slope form Write an equation in point-slope form of the line that passes through the point (5, 1) and has a slope of 23. y 2 y 1 5 m(x 2 x 1 ) Write point-slope form. y 2 1 5 23(x 2 5) Substitute 23 for m, 5 for x, and 1 for y. Exercises for Example 1 Write an equation in point-slope form of the line that passes through the given point and has the given slope. 1. (23, 22); m 5 5 2. (1, 4); m 524 EXAMPLE 2 3. (6, 28); m 52 4 } 9 EXAMP Graph an equation in point-slope form Graph the equation y 2 2 5 2 4 } 7 (x 1 4). Because the equation is in point-slope form, you know that the line has a slope of 2} 4 7 and passes through the point (24, 2). Plot the point (24, 2). Find a second point on the line using the slope. Draw a line through the two points. Exercise for Example 2 24 25 (24, 2) 23 22 4 3 2 1 O 7 24 y 1 2 3 (3, 2) x 5.3 4. Graph the equation y 1 3 5 4(x 1 2). 33
5.3 Study Guide continued For use with pages 302 308 EXAMPLE 3 Use point slope form to write an equation Write an equation in point-slope form of the line shown. STEP 1 Find the slope of the line. y 4 3 2 (3, 3) 1 2322 O 1 2 3 4 5 x m 5 y 2 2 y 1 } x2 2 x 1 5 3 2 (22) } 3 2 1 22 23 24 (1, 22) 5 5 } 2 STEP 2 Write the equation in point-slope form. You can use either point. Method 1 Use (3, 3). Method 2 Use (1, 22). y 2 y 1 5 m(x 2 x 1 ) y 2 y 1 5 m(x 2 x 1 ) y 2 3 5 5 } 2 (x 2 3) y 1 2 5 5 } 2 (x 2 1) 5.3 CHECK Check that the equations are equivalent by writing them in slope-intercept form. y 2 3 5 5 } 2 (x 2 3) y 1 2 5 5 } 2 (x 2 1) Exercises for Example 3 y 5 5 } 2 x 2 9 } 2 y 5 5 } 2 x 2 9 } 2 5. Write an equation in point-slope form of the line that passes through the points (23, 8) and (4, 213). 6. Write an equation in point-slope form of the line that passes through the points (10, 26), (26, 8). 34
5.4 Study Guide For use with pages 311 316 5.4 GOAL EXAMPLE 1 Write equations in standard form. Write equivalent equations in standard form Write two equations in standard form that are equivalent to 3x 2 9y 5 12. To write one equivalent equation, multiply each side by } 1 3. To write another equivalent equation, multiply each side by 2. x2 3y 5 4 6x 2 18y 5 24 EXAMPLE 2 Write an equation from a graph Write an equation in standard form of the line shown. STEP 1 Calculate the slope. m 5 y 2 2 y 1 } x2 2 x 1 5 3 1 3 } 1 1 2 5 6 } 3 5 2 STEP 2 Write an equation in point-slope form. Use (1, 3). y 2 y 1 5 m(x 2 x 1 ) Write point-slope form. 2524 23 22 O 1 2 3 x (22, 23) y 2 3 5 2(x 2 1) Substitute 2 for m, 1 for x, and 3 for y. STEP 3 Rewrite the equation in standard form. y 2 3 5 2x 2 2 Distributive property 22x 1 y 5 1 Exercises for Examples 1 and 2 4 3 2 1 23 24 Collect variable terms on one side, constants on the other. 1. Write two equations in standard form that are equivalent to 6x 1 2y 5 8. y (1, 3) Write an equation in standard form of the line that passes through the given points. 2. (4, 4), (8, 2) 3. (22, 3), (24, 25) 44
5.4 Study Guide continued For use with pages 311 316 EXAMPLE 3 Write an equation of a line Write equations of the horizontal and vertical lines that pass through the point (22, 8). The horizontal line has all the same y-coordinates. The y-coordinate of the given point is 8. So, an equation of the horizontal line is y 5 8. The vertical line has all the same x-coordinates. The x-coordinate of the given point is 22. So, an equation of the vertical line is x 522. Exercises for Example 3 5.4 Write equations of the horizontal and vertical lines that pass through the given point. 4. (7, 22) 5. (21, 5) EXAMPLE 4 Complete an equation in standard form The graph of 5x 1 By 5 6 is a line that passes through the point (2, 1). Find the missing coefficient and write the completed equation. STEP 1 Find the value of B. Substitute the coordinates of the given point for x and y in the equation. Solve for B. 5x 1 By 5 6 Write equation. 5(2) 1 B(1) 5 6 Substitute 2 for x and 1 for y. B 524 Simplify. STEP 2 Complete the equation. 5x 2 4y 5 6 Substitute 24 for B. The completed equation is 5x 2 4y 5 6. Exercises for Example 4 Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation. 6. Ax 1 5y 5 7, (4, 3) 7. 4x 1 By 5 6, (3, 22) 45
5.5 Study Guide For use with pages 318 324 GOAL Write equations of parallel and perpendicular lines. Vocabulary The converse of a conditional statement interchanges the hypothesis and conclusion. Parallel Lines If two nonvertical lines have the same slope, then they are parallel. If two nonvertical lines are parallel, then they have the same slope. Perpendicular Lines Two lines in a plane are perpendicular if they intersect to form a right angle. If two nonvertical lines have slopes that are negative reciprocals, then the lines are perpendicular. If two nonvertical lines are perpendicular, then their slopes are negative reciprocals. 5.5 EXAMPLE 1 Write an equation of a parallel line Write an equation of the line that passes through (2, 6) and is parallel to the line y 52x 1 2. STEP 1 Identify the slope. The graph of the given equation has a slope of 21. So, the parallel line through (2, 6) has a slope of 21. STEP 2 Find the y-intercept. Use the slope and the given point. y5 mx 1 b Write slope-intercept form. 6521(2) 1 b Substitute 21 for m, 2 for x, and 6 for y. 85 b Solve for b. STEP 3 Write the equation. Use y 5 mx 1 b. y 52x 1 8 Substitute 21 for m and 8 for b. Exercises for Example 1 Write an equation of the line that passes through the given point and is parallel to the given line. 1. (9, 2), y 5 2 } 3 x 1 1 2. (23, 24), y 522x 2 1 55
5.5 Study Guide continued For use with pages 318 324 EXAMPLE 2 Determine whether lines are parallel or perpendicular Determine which of the following lines, if any, are parallel or perpendicular: Line a: 4y 2 6y 5 28, Line b: y 5 2 } 2 3 x 1 1, Line c: 2x 1 3y 5 215. Find the slopes of the lines. Line b: The equation is in slope-intercept form. The slope is 2} 2 3. Write the equations for lines a and c in slope-intercept form. 5.5 Line a: 4y 2 6x 528 Line c: 2x 1 3y 5215 4y 5 6x 2 8 3y 522x 2 15 y 5 } 3 2 x 2 2 y 5 2} 2 3 x 2 5 Lines b and c have a slope of 2} 2 3, so they are parallel. Line a has a slope of } 3 2, the negative reciprocal of 2 2 } 3, so it is perpendicular to lines b and c. EXAMPLE 3 Write an equation of a perpendicular line Write an equation of the line that passes through (22, 1) and is perpendicular to the line y 5 2 1 } 3 x 1 2. STEP 1 Identify the slope. The graph of the given equation has a slope of 2} 1 3. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (22, 1) is 3. STEP 2 Find the y-intercept. Use the slope and the given point in y 5 mx 1 b. 1 5 3(22) 1 b Substitute 3 for m, 22 for x, and 1 for y. 7 5 b Solve for b. STEP 3 Write the equation. Use y 5 mx 1 b. y 5 3x 1 7 Substitute 3 for m and 7 for b. Exercises for Examples 2 and 3 3. Determine which of the following lines, if any, are parallel or perpendicular. Line a: 23x 2 12y 5 36 Line b: x 1 4y 5 2 Line c: y 5 4x 56 4. Write an equation of the line that passes through (5, 3) and is perpendicular to the line y 525x 1 3.
5.6 Study Guide For use with pages 325 333 GOAL Make scatter plots and write equations to model data. Vocabulary A scatter plot is a graph used to determine whether there is a relationship between paired data. Correlation If y tends to increase as x increases, the paired data are said to have a positive correlation. If y tends to decrease as x increases, the paired data are said to have a negative correlation. If x and y have no apparent relationship, the paired data are said to have relatively no correlation. When data show a positive or negative correlation, you can model the trend in the data using a line of fit. EXAMPLE 1 Describe the correlation of data Describe the correlation of the data graphed in the scatter plot. The data on the graph shows the results of a forest ranger s annual survey on rabbit and red-tail hawk populations. a. Rabbits (per square mile) Rabbit Population and Snow Depth y 40 35 30 25 20 15 0 0 5 101520253035x Snow depth (inches) b. Hawk and Rabbit Population Hawks sighted (per square mile) y 12 10 8 6 4 2 0 0 10 203040506070x Rabbits counted (per square mile) a. The scatter plot shows a negative b. The scatter plot shows a positive correlation between the depth of correlation between the number of snow cover during the winter and hawks sighted and the number of the number of rabbits counted in rabbits counted. This means that the spring. This means that as the as the number of rabbits increased, depth of the snow increased, the the number of hawks increased. number of rabbits counted decreased. 5.6 69
5.6 Study Guide continued For use with pages 325 333 EXAMPLE 2 Make a scatter plot Softball The table shows the number of girls signed up for a summer softball league each year for 5 years. Year 2001 2002 2003 2004 2005 Players 105 113 120 132 148 a. Make a scatter plot of the data. b. Describe the correlation of the data. c. Write an equation that models the number of girls signed up for a summer softball league as a function of the number of years since 2000. 5.6 a. Treat the data as ordered pairs. Let x represent the number of years since 2000. Let y represent the number of softball players. Plot the ordered pairs as points in a coordinate plane. b. The scatter plot shows a positive correlation, which means that more players have signed up each year since 2000. c. Draw a line that appears to fit the points in the scatter plot closely. Write an equation using two points on the line. Use (1, 100) and (5, 148). Find the slope of the line. m 5 y 2 y 2 1 148 2 100 } x2 5 } 2 x 1 5 2 1 5 } 48 4 5 12 Find the intercept of the line. Use the point (5, 148). Number of players Softball Sign-ups y 150 y 5 mx 1 b Write slope-intercept form. 148 5 12(5) 1 b Substitute 12 for m, 5 for x, and 148 for y. 88 5 b Solve for b. An equation of the line of fit is y 5 12x 1 88. Exercise for Example 1 and 2 1. a. Make a scatter plot of the data in the table. x 1 2 2 3 3 4 4 5 y 9 7 6 4 3 3 2 1 125 100 75 50 25 0 0 1 2 3 4 5 Years since 2000 x 70 b. Describe the correlation of the data. c. Use the data in the table to write an equation that models y as a function of x.
5.7 Study Guide For use with pages 334 342 GOAL Make predictions using best-fitting lines. Vocabulary The line that most closely follows a trend in data is called the bestfitting line. Using a line or its equation to approximate a value between two known values is called linear interpolation. Using a line or its equation to approximate a value outside the range of known values is called linear extrapolation. A zero of a function y 5 f(x) is an x-value for which f(x) 5 0 (or y 5 0). EXAMPLE 1 Interpolate using an equation Nesting Eagles The table shows the number of pairs of nesting Bald Eagles at a national wildlife reserve from 1998 to 2002. Year 1998 2000 2001 2002 Pairs of Nesting Eagles 5 12 18 25 5.7 a. Make a scatter plot of the data. b. Find an equation that models the number of pairs of nesting Bald Eagles as a function of the number of years since 1998. c. Approximate the number of nesting eagle pairs in 1999. a. Enter the data into lists on a graphing calculator. Make a scatter plot, letting the number of years since 1998 be the x-values (0, 2, 3, 4) and the number of nesting eagle pairs be the y-values. b. Perform a linear regression using the paired data. The equation of the bestfitting line is approximately y 5 5x 1 4. c. Graph the best-fitting line. Use the trace feature and the arrow keys to find the value of the equation when x 5 1. There were about 9 nesting pairs in 1999. X=1 Y=8.8563 84
5.7 Study Guide continued For use with pages 334 342 EXAMPLE 2 Extrapolate using an equation Nesting Eagles Look back at Example 1. a. Use the equation from Example 1 to approximate the number of nesting Bald Eagle pairs in 2003 and 2005. b. In 2003 there were actually 31 pairs of nesting eagles. In 2005 there were actually 34 pairs of nesting eagles. Describe the accuracy of the extrapolations made in part (a). a. Evaluate the equation of the best-fitting line from Example 1 for x 5 5 and x 5 7. The model predicts about 29 pairs in 2003 and about 39 pairs for 2005. b. The difference between the predicted number of pairs and the actual number of pairs are 2 and 5, respectively. The difference in actual and predicted numbers increased from 2003 to 2005. So, the equation of the best-fitting line gives a less accurate prediction for years farther from the given data. Exercise for Examples 1 and 2 1. Sales The table shows the sales (in thousands of dollars) of pet care items at one pet store during the period 1998 to 2003. Year 1998 2000 2002 2003 Dollar amount spent (thousands) 21 25 32 37 a. Find an equation that models the sales (in thousands of dollars) of pet care items as a function of the number of years since 1998. b. Predict the sales of pet care items for the years 2001 and 2004. Tell whether the prediction is an interpolation or an extrapolation. c. Which of the predictions from part (b) would you expect to be more accurate? Explain your reasoning. 5.7 85