1. Objectives Determine whether a function is linear.
Graph a linear function given two points, a table, an equation, or a point and a slope.

2. Vocabulary linear function slope y-intercept x-intercept
slope-intercept form

3. Meteorologists begin tracking a hurricane's distance from land when it is 350 miles off the coast of Florida and moving steadily inland.
The meteorologists are interested in the rate at which the hurricane is approaching land.

4. +1
–25 +1
–25 +1
–25 +1
–25
Time (h) 1 2 3 4
Distance from Land (mi)
350
325
300
275
250
This rate can be expressed as
Notice that the rate of change is constant. The hurricane moves 25 miles closer each hour.

5. Functions with a constant rate of change are called linear functions
Functions with a constant rate of change are called linear functions. A linear function can be written in the form f(x) = mx + b, where x is the independent variable and m and b are constants. The graph of a linear function is a straight line made up of all points that satisfy y = f(x).

6. Example 1A: Recognizing Linear Functions
Determine whether the data set could represent a linear function. +2 –1 +2 –1 +2 –1 x –2 2 4
f(x) 1 –1
The rate of change, , is constant So the data set is linear.

7. Example 1B: Recognizing Linear Functions
Determine whether the data set could represent a linear function. +1 +2 +1 +4 +1 +8 x 2 3 4 5
f(x) 8 16
The rate of change, , is not constant ≠ 4 ≠ 8. So the data set is not linear.

8. The constant rate of change for a linear function is its slope
The constant rate of change for a linear function is its slope. The slope of a linear function is the ratio , or .
The slope of a line is the same between any two points on the line. You can graph lines by using the slope and a point.

9. Example 2A: Graphing Lines Using Slope and a Point
Graph the line with slope that passes through (–1, –3).
Plot the point (–1, –3).
The slope indicates a rise of 5 and a run of 2. Move up 5 and right 2 to find another point.
Then draw a line through the points.

10. Example 2B: Graphing Lines Using Slope and a Point
Graph the line with slope that passes through (0, 2).
Plot the point (0, 2).
The negative slope can be viewed as
You can move down 3 units and right 4 units, or move up 3 units and left 4 units.

11. Check It Out! Example 2
Graph the line with slope that passes through (3, 1).
Plot the point (3, 1).
The slope indicates a rise of 4 and a run of 3. Move up 4 and right 3 to find another point.
Then draw a line through the points.

12. Recall from geometry that two points determine a line
Recall from geometry that two points determine a line. Often the easiest points to find are the points where a line crosses the axes.
The y-intercept is the
y-coordinate of a point where the line crosses the x-axis.
The x-intercept is the
x-coordinate of a point where the line crosses the y-axis.

13. Example 3: Graphing Lines Using the Intercepts
Find the intercepts of 4x – 2y = 16, and graph the line.
Find the x-intercept: 4x – 2y = 16
4x – 2(0) = 16
Substitute 0 for y.
4x = 16
x-intercept
y-intercept
x = 4
The x-intercept is 4.
Find the y-intercept: 4x – 2y = 16
4(0) – 2y = 16
Substitute 0 for x.
–2y = 16
y = –8
The y-intercept is –8.

14. Linear functions can also be expressed as linear equations of the form y = mx + b. When a linear function is written in the form y = mx + b, the function is said to be in slope-intercept form because m is the slope of the graph and b is the
y-intercept. Notice that slope-intercept form is the equation solved for y.

15. Example 4A: Graphing Functions in Slope-Intercept Form
Write the function –4x + y = –1 in slope-intercept form. Then graph the function.
Solve for y first.
–4x + y = –1
+4x x
Add 4x to both sides.
y = 4x – 1
The line has y-intercept –1 and slope 4, which is . Plot the point (0, –1). Then move up 4 and right 1 to find other points.

16. Example 4B: Graphing Functions in Slope-Intercept Form
Write the function in slope-intercept form. Then graph the function.
Solve for y first.
Multiply both sides by
Distribute.
The line has y-intercept 8 and slope Plot the point (0, 8). Then move down 4 and right 3 to find other points.

17. An equation with only one variable can be represented by either a vertical or a horizontal line.

18. Vertical and Horizontal Lines
Vertical Lines
Horizontal Lines
The line x = a is a vertical line at a.
The line y = b is a vertical line at b.

19. The slope of a vertical line is undefined.
The slope of a horizontal line is zero.

20. Example 5: Graphing Vertical and Horizontal Lines
Determine if each line is vertical or horizontal.
A. x = 2
This is a vertical line located at the x-value 2. (Note that it is not a function.)
x = 2
y = –4
B. y = –4
This is a horizontal line located at the y-value –4.

21. Example 6: Application
A ski lift carries skiers from an altitude of 1800 feet to an altitude of 3000 feet over a horizontal distance of 2000 feet. Find the average slope of this part of the mountain. Graph the elevation against the distance.
Step 2 Graph the line.
Step 1 Find the slope.
The y-intercept is the original altitude, 1800 ft. Use (0, 1800) and (2000, 3000) as two points on the line. Select a scale for each axis that will fit the data, and graph the function.
The rise is 3000 – 1800, or 1200 ft.
The run is 2000 ft.
The slope is

22. Objectives
Use slope-intercept form and point-slope form to write linear functions.
Write linear functions to solve problems.

23. Vocabulary
Point-slope form

24. Example 1: Writing the Slope-Intercept Form of the Equation of a Line
Write the equation of the graphed line in slope-intercept form.
Step 1
Identify the y-intercept.
The y-intercept b is 1.

25. Example 1 Continued Step 2 Find the slope.
Choose any two convenient points on the line, such as (0, 1) and (4, –2). Count from (0, 1) to (4, –2) to find the rise and the run. The rise is –3 units and the run is 4 units. 3 –4 4 –3
Slope is = = – .
rise
run –3 4 3

26. Example 1 Continued
Step 3
Write the equation in slope-intercept form.
y = mx + b 3 4
y = – x + 1
m = – and b = 1. 3 4
The equation of the line is 3 4
y = – x + 1.

27. Notice that for two points on a line, the rise is the differences in the y-coordinates, and the run is the differences in the x-coordinates. Using this information, we can define the slope of a line by using a formula.

28. Example 2A: Finding the Slope of a Line Given Two or More Points
Find the slope of the line through (–1, 1) and (2, –5).
Let (x1, y1) be (–1, 1) and (x2, y2) be (2, –5).
Use the slope formula.
The slope of the line is –2.

29. Example 2B: Finding the Slope of a Line Given Two or More Points
Find the slope of the line. x 4 8 12 16 y 2 5 11
Choose any two points.
Let (x1, y1) be (4, 2) and (x2, y2) be (8, 5).
Use the slope formula.
The slope of the line is . 3 4

30. Example 2C: Finding the Slope of a Line Given Two or More Points
Find the slope of the line shown.
Let (x1, y1) be (0,–2) and (x2, y2) be (1, –2).
The slope of the line is 0.

31. Because the slope of line is constant, it is possible to use any point on a line and the slope of the line to write an equation of the line in point-slope form.

32. Example 3: Writing Equations of Lines
In slope-intercept form, write the equation of the line that contains the points in the table. x –8 –4 4 8 y –5
–3.5
–0.5 1
First, find the slope. Let (x1, y1) be (–8, –5) and (x2, y2) be (8, 1).
Next, choose a point, and use either form of the equation of a line.

33. Example 3 Continued
Method A Point-Slope Form
Rewrite in slope-intercept form.
Using (8, 1):
y – y1 = m(x – x1)
Distribute.
Substitute.
Solve for y.
Simplify.

34. Example 3 Continued
Method B Slope-intercept Form
Using (8, 1), solve for b.
Rewrite the equation using m and b.
y = mx + b
y = mx + b
Substitute.
1 = 3 + b
Simplify.
b = –2
Solve for b.
The equation of the line is

35. Example 4A: Entertainment Application
The table shows the rents and selling prices of properties from a game.
Selling Price ($)
Rent ($) 75 9 90 12
160 26
250 44
Express the rent as a function of the selling price.
Let x = selling price and y = rent.
Find the slope by choosing two points. Let (x1, y1) be (75, 9) and (x2, y2) be (90, 12).

36. Example 4A Continued
To find the equation for the rent function, use point-slope form.
y – y1 = m(x – x1)
Use the data in the first row of the table.
Simplify.

37. Example 4B: Entertainment Application
Graph the relationship between the selling price and the rent. How much is the rent for a property with a selling price of $230?
To find the rent for a property, use the graph or substitute its selling price of $230 into the function.
Substitute.
y = 46 – 6
y = 40
The rent for the property is $40.

38. By comparing slopes, you can determine if the lines are parallel or perpendicular. You can also write equations of lines that meet certain criteria.

40. Example 5A: Writing Equations of Parallel and Perpendicular Lines
Write the equation of the line in slope-intercept form.
parallel to y = 1.8x + 3 and through (5, 2)
m = 1.8
Parallel lines have equal slopes.
Use y – y1 = m(x – x1) with (x1, y1) = (5, 2).
y – 2 = 1.8(x – 5)
y – 2 = 1.8x – 9
Distributive property.
y = 1.8x – 7
Simplify.

41. Example 5B: Writing Equations of Parallel and Perpendicular Lines
Write the equation of the line in slope-intercept form.
perpendicular to and through (9, –2)
The slope of the given line is , so the slope of
the perpendicular line is the opposite reciprocal, .
Use y – y1 = m(x – x1). y + 2 is equivalent to y – (–2).
Distributive property.
Simplify.

42. Example 5B: Writing Equations of Parallel and Perpendicular Lines
Write the equation of the line in slope-intercept form.
perpendicular to and through (9, –2)
The slope of the given line is , so the slope of
the perpendicular line is the opposite reciprocal, .
Use y – y1 = m(x – x1). y + 2 is equivalent to y – (–2).
Distributive property.
Simplify.