Identify whether a relation is a function Calculate the slope between two points Graph a linear function

1. Linear Functions
Chapter 2 Graphs and Functions

2. Concepts & Objectives
⚫ Function Properties
⚫ Identify whether a relation is a function
⚫ Identify the domain and range of a given function
⚫ Linear Functions
⚫ Calculate the slope between two points
⚫ Graph a linear function

3. Functions
⚫ A relation is a set of ordered pairs.
⚫ A function is a relation in, for each distinct value of the
first component of the ordered pairs, there is exactly one
value of the second component.
⚫ More formally:
If A and B are sets, then a function f from A to B
(written f: A → B)
is a rule that assigns to each element of A
a unique element of set B.

4. Functions (cont.)
⚫ The set A is called the domain of the function f
⚫ Every element of A must be included in the function.
⚫ The set B is called the codomain of f
⚫ The subset of B consisting of those elements that are
images under the function f is called the range.
⚫ The range and the codomain may or may not be the
same.

5. Functions (cont.)
⚫ In terms of ordered pairs, a function is the set of ordered
pairs (A, f (A)).
⚫ Historical note: The notation f (x) for a function of a
variable quantity x was introduced in 1748 by Leonhard
Euler in his text Algebra, which was the forerunner of
today’s algebra texts. Many other mathematical symbols
in use today (such as e and ) were introduced by Euler
in his writings.

6. Functions (cont.)
⚫ For the most part, we use f (x) and y interchangeably to
denote a function of x, but there are some subtle
differences.
⚫ y is the output variable, while f (x) is the rule that
produces the output variable.
⚫ An equation with two variables, x and y, may not be a
function at all.
Example: is a circle, but not a function+ =2 2
4x y

7. 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
−
= =
−

8. 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.

9. 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

10. 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.

11. 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.

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

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

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

15. 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).

16. 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.

17. Classwork
⚫ College Algebra
⚫ Page 226: 26-32 (even), page 163: 28-40 (even),
page 155: 36-48 (even)