Define linear function Determine the slope of a linear function

1. Linear Functions
Chapter 2 Graphs and Functions

2. Concepts & Objectives
⚫ Linear Functions
⚫ Calculate the slope between two points
⚫ Graph a linear function

3. Linear Functions
⚫ A function f is a linear function if, for a and b  ,
⚫ If a ≠ 0, the domain and the range of a linear function are
both .
⚫ The slope of a linear function is defined as the rate of
change or the ratio of rise to run.
( )f x ax b= +
( ),− 
The slope m of the line through the
points and is( )1 1,x y ( )2 2,x y
2 1
2 1
rise
run
y y
m
x x
−
= =
−

4. Linear Functions (cont.)
⚫ A linear function can be written in one of the following
forms:
⚫ Standard form: Ax + By = C, where A, B, C  , A 0,
and A, B, and C are relatively prime
⚫ Point-slope form: y – y1 = m(x – x1), where m   and
(x1, y1) is a point on the graph
⚫ Slope-intercept form: y = mx + b, where m, b  
⚫ You should recall that in slope-intercept form, m is the
slope and b is the y-intercept (where the graph crosses
the y-axis).
⚫ If A = 0, then the graph is a horizontal line at y = b.

5. Linear Functions (cont.)
⚫ Let’s take another look at the standard form:
If B = 0 (the coefficient of the y term), we end up with
which is undefined. This is not good.
+ =
= − +
= − +
Ax By C
By Ax C
A C
y x
B B
= − + ,
0 0
A C
y x

6. Linear Functions (cont.)
⚫ Since we cannot divide by 0, we say that a line of the
form x = a has no slope, and is a vertical line.
⚫ Technically, a vertical line is not a function at all,
because one value of x has more than one y value
(actually an infinite number of y values), but since it is a
straight line, we include it along with the linear
functions.

7. Graphing a Linear Function
To graph a line:
⚫ If you are only given two points, plot them and draw a
line between them.
⚫ If you are given a point and a slope:
⚫ Plot the point.
⚫ From the point count the rise and the run of the slope
and mark your second point.
⚫ Connect the two points.
⚫ If the slope is negative, pick either the rise or the run
to go in a negative direction, but not both.

8. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.

9. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.
⚫ Plot the y-intercept at (0, 1).

10. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.
⚫ Plot the y-intercept at (0, 1).
⚫ Count down 2 and over 1.

11. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.
⚫ Plot the y-intercept at (0, 1).
⚫ Count down 2 and over 1.
⚫ Plot the second point at (1, –1).

12. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.
⚫ Plot the y-intercept at (0, 1).
⚫ Count down 2 and over 1.
⚫ Plot the second point at (1, –1).
⚫ Connect the points.

13. Finding the Slope
⚫ Using the slope formula:
⚫ Example: Find the slope of the line through the points
(–4, 8), (2, –3).
( )
3 8
2 4
m
− −
=
− −
x1 y1 x2 y2
–4 8 2 –3
11
6
−
=
11
6
= −

14. Finding the Slope (cont.)
⚫ From an equation: Convert the equation into slope-
intercept form (y = mx + b) if necessary. The slope is the
coefficient of x.
⚫ Example: What is the slope of the line y = –4x + 3?
The equation is already in slope intercept form, so the
slope is the coefficient of x, so m = –4.

15. Finding the Slope (cont.)
⚫ Example: What is the slope of the line 3x + 4y = 12?
The slope is .
3 4 12
4 3 12
x y
y x
+ =
= − +
3
3
4
y x= − +
3
4
−

16. Classwork
⚫ 2.4 Assignment (College Algebra)
⚫ Page 226: 8-28 (×4); page 214: 44-68 (×4);
page 200: 38-44 (even)
⚫ 2.4 Classwork Check
⚫ Quiz 2.3