Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.

Transcription

1 Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown.

2 EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is m = 4 and the y-intercept is b = 2. Use slope-intercept form to write an equation of the line. y = mx + b y = x + ( 2) 4 y = x ( 2) 4 Use slope-intercept form. Substitute for m and 2 for b. 4 Simplify.

3 GUIDED PRACTICE for Example 1 Write an equation of the line that has the given slope and y - intercept. 1. m =, b = 1 Use slope intercept point form to write an equation of the line y = mx + b Use slope - intercept form. y = x + 1 Substitute for m and 1 for b. y = x + 1 Simplify.

4 GUIDED PRACTICE for Example 1 2. m = 2, b = 4 Use slope intercept point form to write an equation of the line y = mx + b Use slope - intercept form. y = 2x + ( 4 ) Substitute 2 for m and 4 for b. y = 2x 4 Simplify.

5 GUIDED PRACTICE for Example 1. m = b = Use slope intercept point form to write an equation of the line y = mx + b y = x + 4 y = x Use slope - intercept form. 7 Substitute for m and for b. 4 2 Simplify.

6 EXAMPLE 2 Write an equation given the slope and a point Write an equation of the line that passes through (5, 4) and has a slope of. Because you know the slope and a point on the line, use pointslope form to write an equation of the line. Let (x 1, y 1 ) = (5, 4) and m =. y y 1 = m(x x 1 ) Use point-slope form. y 4 = (x 5) Substitute for m, x 1, and y 1. y 4 = x + 15 y = x + 19 Distributive property Write in slope-intercept form.

7 EXAMPLE Write equations of parallel or perpendicular lines Write an equation of the line that passes through ( 2,) and is (a) parallel to, and (b) perpendicular to, the line y = 4x + 1. a. The given line has a slope of m 1 = 4. So, a line parallel to it has a slope of m 2 = m 1 = 4. You know the slope and a point on the line, so use the point-slope form with (x 1, y 1 ) = ( 2, ) to write an equation of the line.

8 EXAMPLE Write equations of parallel or perpendicular lines y y 1 = m 2 (x x1) y = 4(x ( 2)) y = 4(x + 2) y = 4x 8 y = 4x 5 Use point-slope form. Substitute for m 2, x 1, and y 1. Simplify. Distributive property Write in slope-intercept form.

9 EXAMPLE Write equations of parallel or perpendicular lines b. A line perpendicular to a line with slope m 1 = 4 has 1 a slope of m 2 = = 1 m. Use point-slope form with 1 4 (x 1, y 1 ) = ( 2, ) y y 1 = m 2 (x x1) 1 y = (x ( 2)) 4 1 y = (x +2) y = x y = x Use point-slope form. Substitute for m 2, x 1, and y 1. Simplify. Distributive property Write in slope-intercept form.

10 GUIDED PRACTICE for Examples 2 and 4. Write an equation of the line that passes through ( 1, 6) and has a slope of 4. Because you know the slope and a point on the line, use the point-slope form to write an equation of the line. Let (x 1, y 1 ) = ( 1, 6) and m = 4 y y 1 = m(x x1) y 6 = 4(x ( 1)) y 6 = 4x + 4 y = 4x + 10 Use point-slope form. Substitute for m, x 1, and y 1. Distributive property Write in slope-intercept form.

11 GUIDED PRACTICE for Examples 2 and 5. Write an equation of the line that passes through (4, 2) and is (a) parallel to, and (b) perpendicular to, the line y = x 1. The given line has a slope of m 1 =. So, a line parallel to it has a slope of m 2 = m 1 =. You know the slope and a point on the line, so use the point - slope form with (x 1, y 1 ) = (4, 2) to write an equation of the line. y y 1 = m 2 (x x 1 ) Use point-slope form. y ( 2) = (x 4) Substitute for m 2, x 1, and y 1. y + 2 = (x 4) Simplify. y + 2 = x 12 Distributive property y = x 14 Write in slope-intercept form.

12 GUIDED PRACTICE for Examples 2 and b. A line perpendicular to a line with slope m 1 = has a slope of m 1 2 = 1 = m 1 Use point - slope form with (x 1, y 1 ) = (4, 2) y y 1 = m 2 (x x 1 ) 1 y ( 2) = (x 4) 1 y + 2 = (x 4) 1 4 y + 2 = x 1 2 y = x Use point-slope form. Substitute for m 2, x 1, and y 1. Simplify. Distributive property Write in slope-intercept form.

13 EXAMPLE 4 Write an equation given two points Write an equation of the line that passes through (5, 2) and (2, 10). The line passes through (x 1, y 1 ) = (5, 2) and (x 2, y 2 ) = (2, 10). Find its slope. m = y 2 y 1 x 2 x 1 = 10 ( 2) = = 4

14 EXAMPLE 4 Write an equation given two points You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x 1, y 1 ) = (4, 7). y 2 y 1 = m(x x 1 ) Use point-slope form. y 10 = 4(x 2) Substitute for m, x 1, and y 1. y 10 = 4x + 8 y = 4x + 8 Distributive property Write in slope-intercept form.

15 GUIDED PRACTICE for Examples 4 and 5 Write an equation of the line that passes through the given points. 6. ( 2, 5), (4, 7) The line passes through (x 1, y 1 ) = ( 2, 5) and (x 2, y 2 ) = (4, 7). Find its slope. m = y 2 y 1 x 2 x 1 = ( 2) = 2

16 GUIDED PRACTICE for Examples 4 and 5 You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x 1, y 1 ) = (4, 7). y y 1 = m(x x 1 ) Use point-slope form. y 7 = 2(x 4) Substitute for m, x 1, and y 1. y 7 = 2 (x + 4) y + 7 = 2x + 8 Simplify Distributive property y = 2x + 1 Write in slope-intercept form.

17 GUIDED PRACTICE for Examples 4 and 5 7. (6, 1), (, 8) The line passes through (x 1, y 1 ) = (6, 1) and (x 2, y 2 ) = (, 8). Find its slope. m = y 2 y 1 x 2 x 1 = = = 1 9

18 GUIDED PRACTICE for Examples 4 and 5 You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x 1, y 1 ) = (, 8). y y 1 = m(x x 1 ) y ( 8)) = 1(x ( )) y + 8 = 1 (x + ) y + 8 = x + y = x 5 Use point-slope form. Substitute for m, x, and y 1. Simplify Distributive property Write in slope-intercept form.

19 GUIDED PRACTICE for Examples 4 and 5 8. ( 1, 2), (10, 0) The line passes through (x 1, y 1 ) = ( 1, 2) and (x 2, y 2 ) = (10, 0). Find its slope. m = y 2 y 1 x 2 x 1 = ( 1) = 2 11

20 GUIDED PRACTICE for Examples 4 and 5 You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x 1, y 1 ) = (10, 0). y y 1 = m(x x 1 ) y 0 = 2 (x 10) 11 y = 2 11 (x 10) 2 y = 11 x = 11 x Use point-slope form. Substitute for m, x, and y 1. Simplify Distributive property Write in slope-intercept form.