Ultimate guide to coordinate plane
- 1. Review The Coordinate Plane Everything you want need to know about writing, graphing, & solving equations of lines. Includes Bonus Content: Your ultimate guide to Parallel & Perpendicular Lines
- 2. EXAMPLE 1 Use slope and y-intercept to write an equation Write an equation of the line with a slope of –2 and a y-intercept of 5. y = mx + b Write slope-intercept form. y = –2x + 5 Substitute –2 for m and 5 for b.
- 3. EXAMPLE 2 Standardized Test Practice Which equation represents the line shown? 2 5 A y=– x+3 B y=– x+3 5 2 2 2 C y=– x+1 D y = 3x + 5 5 rise –2 2 The slope of the line is = = – . run 5 5 The line crosses the y-axis at (0, 3). So, the y-intercept is 3. y = mx + b Write slope-intercept form. 2 2 y=– 5 x+3 Substitute – 5 for m and 3 for b.
- 4. EXAMPLE 2 Standardized Test Practice ANSWER The correct answer is A. A B C D
- 5. GUIDED PRACTICE for Examples 1 and 2 Write an equation of the line with the given slope and y-intercept. 1. Slope is 8; y-intercept is –7. ANSWER y = 8x – 7
- 6. GUIDED PRACTICE for Examples 1 and 2 Write an equation of the line with the given slope and y-intercept. 3 2. Slope is 4 ; y intercept is –3. 3 ANSWER y= x–3 4
- 7. EXAMPLE 3 Write an equation of a line given two points Write an equation of the line shown.
- 8. EXAMPLE 3 Write an equation of a line given two points SOLUTION STEP 1 Calculate the slope. y2 – y1 –1 – (–5) 4 m = = = x2 – x1 3–0 3 STEP 2 Write an equation of the line. The line crosses the y-axis at (0, – 5). So, the y-intercept is –5. y = mx + b Write slope-intercept form. y= 4 x–5 3 Substitute 4 for m and 5 for b. 3
- 9. EXAMPLE 4 Write a linear function Write an equation for the linear function f with the values f(0) = 5 and f(4) = 17. SOLUTION STEP 1 Write f(0) = 5 as (0, 5) and f (4) = 17 as (4, 17). STEP 2 Calculate the slope of the line that passes through (0, 5) and (4, 17). y2 – y1 17 – 5 12 m = = = = 3 x2 – x1 4–0 4
- 10. EXAMPLE 4 Write a linear function STEP 3 Write an equation of the line. The line crosses the y-axis at (0, 5). So, the y-intercept is 5. y = mx + b Write slope-intercept form. y = 3x + 5 Substitute 3 for m and 5 for b. ANSWER The function is f(x) = 3x + 5.
- 11. GUIDED PRACTICE for Examples 3 and 4 3. Write an equation of the line shown. 1 x+1 ANSWER y= – 2
- 12. GUIDED PRACTICE for Examples 3 and 4 4. Write an equation for the linear function f with the given values. f(0) = –2, f(8) = 4 ANSWER y =3 x – 2 4
- 13. GUIDED PRACTICE for Examples 3 and 4 5. Write an equation for the linear function f with the given values. f(–3) = 6, f(0) = 5 ANSWER y=–1 x+5 3
- 14. EXAMPLE 1 Write an equation given the slope and a point Write an equation of the line that passes through the point (–1, 3) and has a slope of –4. SOLUTION STEP 1 Identify the slope. The slope is – 4. STEP 2 Find the y-intercept. Substitute the slope and the coordinates of the given point in y = mx + b. Solve for b. y = mx + b Write slope-intercept form. 3 = –4(–1) + b Substitute –4 for m, –1 for x, and 3 for y.
- 15. EXAMPLE 1 Write an equation given the slope and a point –1 = b Solve for b. STEP 3 Write an equation of the line. y = mx + b Write slope-intercept form. y = –4x – 1 Substitute –4 for m and –1 for b.
- 16. GUIDED PRACTICE for Example 1 Write an equation of the line that passes through the point (6, 3) and has a slope of 2. ANSWER y = 2x – 9
- 17. EXAMPLE 2 Write an equation given two points Write an equation of the line that passes through (–2, 5) and (2, –1). SOLUTION STEP 1 Calculate the slope. y2 – y1 –1 – 5 –6 3 m= = = = – x2 – x1 2 – (–2) 4 2 STEP 2 Find the y-intercept. Use the slope and the point (–2, 5). y = mx + b Write slope-intercept form.
- 18. EXAMPLE 2 Write an equation given two points 3 5=– 2 (–2) +b Substitute – 3 for m, –2 for x, and 5 for y. 2 2=b Solve for b. STEP 3 Write an equation of the line. y = mx + b Write slope-intercept form. 3 y=– x +2 Substitute – 3 for m and 2 for b. 2 2
- 19. EXAMPLE 33 EXAMPLE Standardized Test Practice Which function has the values f(4) = 9 and f(–4) = –7? A f (x) = 2x + 10 B f (x) = 2x + 1 C f (x) = 2x – 13 D f (x) = 2x – 14 STEP 1 Calculate the slope. Write f (4) = 9 as (4, 9) and f (–4) = –7 as (–4, –7). y2 – y1 –7 – 9 m= = = –16 = 2 x2 – x1 –4 – 4 –8 STEP 2 Find the y-intercept. Use the slope and the point (4, 9). y = mx + b Write slope-intercept form.
- 20. EXAMPLE 3 Standardized Test Practice 9 = 2(4) + b Substitute 2 for m, 4 for x, and 9 for y. 1=b Solve for b. STEP 3 Write an equation for the function. Use function notation. f (x) = 2x + 1 Substitute 2 for m and 1 for b. ANSWER The answer is B. A B C D
- 21. GUIDED PRACTICE for Examples 2 and 3 2. Write an equation of the line that passes through (1, –2) and (–5, 4). ANSWER y = –x – 1
- 22. GUIDED PRACTICE for Examples 2 and 3 3. Write an equation for the linear function with values f(–2) = 10 and f(4) = –2? ANSWER y = –2x + 6
- 23. EXAMPLE 2 Graph an equation in point-slope form Graph the equation y + 2 = 2 (x – 3). 3 SOLUTION Because the equation is in point-slope form, you know that the line has a slope of 2 and passes through the 3 point (3, –2). Plot the point (3, –2). Find a second point on the line using the slope. Draw a line through both points.
- 24. EXAMPLE 2 GUIDED PRACTICE an for Examplepoint-slope form Graph equation in 2 2. Graph the equation y – 1 = –(x – 2). ANSWER
- 25. EXAMPLE 1 Write an equation of a parallel line Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1. SOLUTION STEP 1 Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3. STEP 2 Find the y-intercept. Use the slope and the given point.
- 26. EXAMPLE 1 Write an equation of a parallel line y = mx + b Write slope-intercept form. –5 = 3(–3) + b Substitute 3 for m, 3 for x, and 5 for y. 4=b Solve for b. STEP 3 Write an equation. Use y = mx + b. y = 3x + 4 Substitute 3 for m and 4 for b.
- 27. GUIDED PRACTICE for Example 1 1. Write an equation of the line that passes through (–2, 11) and is parallel to the line y = –x + 5. ANSWER y = –x + 9
- 28. EXAMPLE 3 Determine whether lines are perpendicular STATE FLAG The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they? Line a: 12y = –7x + 42 Line b: 11y = 16x – 52 SOLUTION Find the slopes of the lines. Write the equations in slope-intercept form.
- 29. EXAMPLE 3 Determine whether lines are perpendicular Line a: 12y = –7x + 42 7 42 y=– x + 12 12 Line b: 11y = 16x – 52 16 52 y= x – 11 11 ANSWER 7 16 The slope of line a is – . The slope of line b is 11 12 The two slopes are not negative reciprocals, so lines a and b are not perpendicular.
- 30. EXAMPLE 4 Write an equation of a perpendicular line Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2x + 3. SOLUTION STEP 1 Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is . –1 2
- 31. EXAMPLE 4 Write an equation of a perpendicular line STEP 2 Find the y-intercept. Use the slope and the given point. y = mx + b Write slope-intercept form. –5 = –1 (4) + b 2 Substitute – 1 for m, 4 for x, and –5 for y. 2 –3 = b Solve for b. STEP 3 Write an equation. y = mx + b Write slope-intercept form. 1 1 y= – x – 3 Substitute – for m and –3 for b. 2 2
- 32. GUIDED PRACTICE for Examples 3 and 4 3. Is line a perpendicular to line b? Justify your answer using slopes. Line a: 2y + x = –12 Line b: 2y = 3x – 8 ANSWER 1 3 No; the slope of line a is – 2 , the slope of line b is 2 . The slopes are not negative reciprocals so the lines are not perpendicular.
- 33. GUIDED PRACTICE for Examples 3 and 4 4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7. ANSWER y= – 1 x+4 4
- 34. EXAMPLE 2 Determine whether lines are parallel or perpendicular Determine which lines, if any, are parallel or perpendicular. Line a: y = 5x – 3 Line b: x + 5y = 2 Line c: –10y – 2x = 0 SOLUTION Find the slopes of the lines. Line a: The equation is in slope-intercept form. The slope is 5. Write the equations for lines b and c in slope-intercept form.
- 35. EXAMPLE 2 Determine whether lines are parallel or perpendicular Line b: x + 5y = 2 5y = – x + 2 1 2 y= – x + 5 5 Line c: –10y – 2x = 0 –10y = 2x 1 y= – x 5
- 36. EXAMPLE 2 Determine whether lines are parallel or perpendicular ANSWER Lines b and c have slopes of – 1 , so they are 5 parallel. Line a has a slope of 5, the negative reciprocal 1 of – 5 , so it is perpendicular to lines b and c.
- 37. GUIDED PRACTICE for Example 2 Determine which lines, if any, are parallel or perpendicular. Line a: 2x + 6y = –3 Line b: y = 3x – 8 Line c: –1.5y + 4.5x = 6 ANSWER parallel: b and c; perpendicular: a and b, a and c
- 38. EXAMPLE 5 Solve a multi-step problem LIBRARY Your class is taking a trip to the public library. You can travel in small and large vans. A small van holds 8 people and a large van holds 12 people. Your class could fill 15 small vans and 2 large vans. a. Write an equation in standard form that models the possible combinations of small vans and large vans that your class could fill. b. Graph the equation from part (a). c. List several possible combinations.
- 39. EXAMPLE 5 Solve a multi-step problem SOLUTION a. Write a verbal model. Then write an equation. 8 s + 12 l = p Because your class could fill 15 small vans and 2 large vans, use (15, 2) as the s- and l-values to substitute in the equation 8s + 12l = p to find the value of p. 8(15) + 12(2) = p Substitute 15 for s and 2 for l. 144 = p Simplify. Substitute 144 for p in the equation 8s + 12l = p.
- 40. EXAMPLE 5 Solve a multi-step problem ANSWER The equation 8s + 12l = 144 models the possible combinations. b. Find the intercepts of the graph. Substitute 0 for s. 8(0) + 12l = 144 l = 12 Substitute 0 for l. 8s + 12(0) = 144 s = 18
- 41. EXAMPLE 5 Solve a multi-step problem Plot the points (0, 12) and (18, 0). Connect them with a line segment. For this problem only nonnegative whole-number values of s and l make sense. c. The graph passes through (0, 12), (6, 8), (12, 4), and (18, 0). So, four possible combinations are 0 small and 12 large, 6 small and 8 large, 12 small and 4 large, 18 small and 0 large.
- 42. GUIDED PRACTICESolve a multi-step problem EXAMPLE 5 for Example 5 Solve a multi-step problem EXAMPLE 5 7. WHAT IF? In Example 5, suppose that 8 students decide not to go on the class trip. Write an equation that models the possible combinations of small and large vans that your class could fill. List several possible combinations. ANSWER 8s + 12l = 136; 17 small, 0 large; 14 small, 2 large; 11 small, 4 large; 8 small, 6 large; 5 small, 8 large; 2 small, 10 large