2-3 Linear Functions and Slope-Intercept Form

Quick Review

The graph of a linear function is a line. You can represent a linear function with a linear equation. Given two points on a line, the slope of the line is the ratio of the change in the y-coordinates to the change in the corresponding x-coordinates. The slope is the coefficient of x when you write a linear equation in slope-intercept form.

Example

What is the slope of the line that passes through (3, 5) and ( − 1 , − 2 ) ? negative 1 comma negative 2 close question mark

m = y 2 − y 1 x 2 − x 1 Find the difference between the coordinates .   = 5 − ( − 2 ) 3 − ( − 1 ) = 7 4 Simplify . table with 2 rows and 3 columns , row1 column 1 , m , column 2 equals . fraction y sub 2 , minus , y sub 1 , over x sub 2 , minus , x sub 1 end fraction , column 3 table with 2 rows and 1 column , row1 column 1 , cap findthedifferencebetweenthe , row2 column 1 , coordinates . . , end table , row2 column 1 , , column 2 equals . fraction 5 minus . open , negative 2 , close , over 3 minus . open , negative 1 , close end fraction . equals , 7 fourths , column 3 cap simplify , . , end table

Exercises

Identify the slope of the line that passes through the given points.

16. (1, 3) and (6, 1)
17. (4, 4) and ( − 2 , − 3 ) negative 2 comma negative 3 close
18. (3, 2) and ( − 3 , − 2 ) negative 3 comma negative 2 close
19. (5, 2) and ( − 4 , 6 ) negative 4 comma 6 close

Write an equation for each line in slope-intercept form.

20. slope = − 3 slope , equals negative 3  and the y-intercept is (0, 4)
21. slope = 1 2 slope , equals , 1 half  and the y-intercept is (0, 6)

Rewrite each equation in slope-intercept form. Graph each line.

22. 4 x − 2 y = 3 4 x minus 2 y equals 3
23. − 4 x + 6 y = 18 negative 4 x plus 6 y equals 18
24. 3 y + 3 x = 15 3 y plus 3 x equals 15
25. 3 y + x = 5 3 y plus x equals 5

2-4 More About Linear Equations

Quick Review

You write the equation of a line in point-slope form when you have a point and the slope or when you have two points. The standard form of an equation has both variables and no constants on the left side.

When two lines have the same slope, they are parallel. When two lines have slopes that are negative reciprocals of each other, they are perpendicular.

Example

Write an equation in standard form for the line with a slope of 2, going through (1, 6).

y − 6 = 2 ( x − 1 ) Write the equation in point-slope form  ,  substituting the given point and slope. y = 2 x − 2 + 6 Simplify. − 2 x + y = 4 Write in standard form. table with 3 rows and 3 columns , row1 column 1 , y minus , 6 , column 2 equals 2 open x minus 1 close , column 3 table with 2 rows and 1 column , row1 column 1 , cap writetheequationinpointminusslopeform . comma , row2 column 1 , substitutingthegivenpointandslope. , end table , row2 column 1 , y , column 2 equals 2 x minus 2 plus 6 , column 3 cap simplify. , row3 column 1 , negative 2 x plus y , column 2 equals 4 , column 3 cap writeinstandardform. , end table

Exercises

Write an equation for each line in point-slope form and then convert it to standard form.

26. slope = − 3 , equals negative 3 comma  through (4, 0)
27. slope = 5, through ( 1 , − 1 ) open 1 comma negative 1 close
28. through (0, 0) and ( 3 , − 7 ) open 3 comma negative 7 close
29. through (2, 3) and (3, 5)
30. 1. Write an equation of the line parallel to x + 2y = 6 through (8, 3).
    2. Write an equation of the line perpendicular to x + 2y = 6 through (8, 3).
    3. Graph the three lines on the same coordinate plane.

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