1. 1.3 Linear Functions Linear Function A function f defined by where
a and b are real numbers, is called a linear
function.
Its graph is called a line.
Its solution is an ordered pair, (x,y), that makes the equation true.

2. Example 1.3 Linear Functions
The points (0,6) and (–1,3) are solutions of
since 6 = 3(0) + 6 and 3 = 3(–1) + 6.

3. 1.3 Graphing a Line Using Points
Graphing the line
Connect with a straight line. x y 2 1 3 6 1 9
(0,6) and (–2,0) are the y- and x-intercepts of the
line y = 3x + 6, and x = –2 is the zero of the function.

4. 1.3 Graphing a Line with the TI-83
Graph the line with the TI-83
Xmin=-10, Xmax=10, Xscl=1
Ymin=-10, Ymax=10,Yscl=1

5. Locating x- and y-Intercepts
To find the x-intercept of the graph of
y = ax + b, let y = 0 and solve for x.
To find the y-intercept of the graph of
y = ax + b, let x = 0 and solve for y.

6. 1.3 Zero of a Function Zero of a Function
Let f be a function. Then any number c
for which f (c) = 0 is called a zero of the
function f.

7. 1.3 Graphing a Line Using the Intercepts
Example: Graph the line x y
x-intercept 5
y-intercept
2.5

8. 1.3 Application of Linear Functions
A 100 gallon tank full of water is being drained at a rate of
5 gallons per minute.
a) Write a formula for a linear function f that models the number of
gallons of water in the tank after x minutes.
b) How much water is in the tank after 4 minutes?
c) Use the x- and y-intercepts to graph f. Interpret each intercept.

9. 1.3 Constant Function Constant Function b is a real number.
The graph is a horizontal line.
y-intercept: (0,b)
Domain
range
Example:

10. 1.3 Constant Function Constant Function
A function defined by f(x) = b, where b is a real number,
is called a constant function. Its graph is a horizontal
line with y-intercept b. For b not equal to 0, it has no
x-intercept. (Every constant function is also linear.)

11. 1.3 Graphing with the TI-83 Different views with the TI-83
Comprehensive graph shows all intercepts

12. 1.3 Slope
Slope of a Line
In 1984, the average annual cost for tuition and fees at private four-year colleges was $5991. By 2004, this cost had increased to $20,082. The line graphed to the right is actually somewhat misleading, since it indicates that the increase in cost was the same from year to year.
The average yearly cost was $705.

13. 1.3 Formula for Slope
The slope m of the line passing through the points
(x1, y1) and (x2, y2) is

14. 1.3 Example: Finding Slope Given Points
Determine the slope of a line passing through points (2, 1) and (5, 3).

15. 1.3 Graph a Line Using Slope and a Point
Example using the slope and a point to graph a line
Graph the line that passes through (2,1) with slope

16. 1.3 Slope of a Line Geometric Orientation Base on Slope
For a line with slope m,
If m > 0, the line rises from left to right.
If m < 0, the line falls from left to right.
If m = 0, the line is horizontal.

17. 1.3 Slope of Horizontal and Vertical Lines
The slope of a horizontal line is 0.
The slope of a vertical line is undefined.
The equation of a vertical line that passes through the point (a,b) is

18. 1.3 Vertical Line
Vertical Line
A vertical line with x-intercept a has an
equation of the form x = a. Its slope is
undefined.

19. 1.3 Slope-Intercept Form of a Line
The slope-intercept form of the equation of a
line is where m is the slope and
b is the y-intercept.

20. 1.3 Matching Examples
Solution: A. B. C.
1) C, 2) A, 3)B

21. Interpreting Slope 1.3 Application of Slope
In 1980, passengers traveled a total of 4.5 billion miles on Amtrak, and in 2007 they traveled 5.8 billion miles.
a) Find the slope m of the line passing through the points (1980, 4.5) and (2007, 5.8).
Solution:
b) Interpret the slope.
The average number of miles traveled on Amtrak increased by about 0.05 billion, or 50 million miles per year.