This document provides an overview of linear functions including: representing linear functions with equations in slope-intercept, point-slope, and standard form; determining if a function is increasing, decreasing, or constant based on its slope; interpreting slope as a rate of change; writing equations of lines from graphical or numerical information; finding x- and y-intercepts; and identifying parallel and perpendicular lines based on their slopes. Examples are provided for finding slope from graphs or equations, writing equations in different forms, graphing lines, and determining parallel/perpendicular relationships between lines. The document concludes with classwork and quiz assignments related to linear functions.

1. Linear Functions
Chapter 4 Linear Functions

2. Concepts & Objectives
⚫ Objectives for this section are
⚫ Represent a linear function.
⚫ Determine whether a linear function is increasing,
decreasing, or constant.
⚫ Interpret slope as a rate of change.
⚫ Write and interpret an equation for a linear function.
⚫ Determine whether lines are parallel or
perpendicular.
⚫ Write the equation of a line parallel or perpendicular
to a given line.

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. 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
= −

6. 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.

7. 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
−

8. Increasing, Decreasing, or Constant
⚫ Since linear functions have a constant rate of change,
they are increasing, decreasing, or constant across their
entire domain.
x
f(x)
x
f(x)
x
f(x)
increasing
m > 0
decreasing
m < 0
constant
m = 0

9. Writing a Linear Function
⚫ Recall that in section 2.2, we wrote equations of lines in
both slope-intercept (y = mx + b) and point-slope
( ) form. Also recall that we can write
these equations from a graph, a point and a slope, or two
points.
⚫ To write a linear function using function notation, just
substitute f(x) for y:
⚫ Slope-intercept becomes
⚫ Point-slope becomes (notice
how the sign of y1 changed!)
( )
1 1
y y m x x
− = −
( )
f x mx b
= +
( ) ( )
1 1
f x m x x y
= − +

10. 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.
⚫ If the slope is negative, pick either the rise or the run to
go in a negative direction, but not both.
⚫ Connect the two points.

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

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

13. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.
⚫ Plot the y-intercept at (0, 1).
⚫ The slope is ‒2, so from the y-intercept, count down 2
and over 1.

14. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.
⚫ Plot the y-intercept at (0, 1).
⚫ The slope is ‒2, so from the y-intercept, count down 2
and over 1.
⚫ Plot the second point at (1, –1).

15. Graphing a Linear Function
⚫ Example: Graph the line y = –2x + 1.
⚫ Plot the y-intercept at (0, 1).
⚫ The slope is ‒2, so from the y-intercept, count down 2
and over 1.
⚫ Plot the second point at (1, –1).
⚫ Connect the points.

16. Finding the x-intercept
⚫ So far we have been finding the y-intercepts of a
function: the point at which the graph of the function
crosses the y-axis (where the input value is 0).
⚫ Recall that a function may also have an x-intercept, i.e.,
the x-coordinate of the point where the graph of the
function crosses the x-axis (where the output value is 0).
⚫ To find the x-intercept, set a function f(x) equal to zero
and solve for the value of x.

17. Finding the x-intercept (cont.)
⚫ Example: Find the x-intercept of ( )
1
3
2
f x x
= −

18. Finding the x-intercept (cont.)
⚫ Example: Find the x-intercept of
The graph crosses the x-axis at the point (6, 0).
( )
1
3
2
f x x
= −
1
0 3
2
1
3
2
6
x
x
x
= −
=
=

19. Horizontal and Vertical Lines
⚫ There are two special cases of lines on a graph—
horizontal and vertical lines.
⚫ A horizontal line indicates a constant output, or y-value,
i.e., the slope is 0.
⚫ A vertical line indicates a constant input, or x-value.
⚫ Because the input value is mapped to more than one
output value, a vertical line does not represent a
function.
⚫ In the slope formula, the denominator will be zero, so
the slope is undefined.

20. Parallel and Perpendicular Lines
⚫ Recall (again) from section 2.2 that parallel lines have
the same slope and the slopes of perpendicular lines are
negative reciprocals.
⚫ Example: Identify the functions whose graphs are a pair
of parallel lines and a pair of perpendicular lines.
( ) 2 3
f x x
= +
( )
1
4
2
g x x
= −
( ) 2 2
h x x
= − +
( ) 2 6
j x x
= −

21. Parallel and Perpendicular Lines
⚫ Example: Identify the functions whose graphs are a pair
of parallel lines and a pair of perpendicular lines.
Parallel lines have the same slope. Because f and j each
have a slope of 2, they are parallel.
Because ‒2 and ½ are negative repciprocals (their
product is ‒1), g and h are perpendicular.
( ) 2 3
f x x
= +
( )
1
4
2
g x x
= −
( ) 2 2
h x x
= − +
( ) 2 6
j x x
= −

22. Parallel and Perpendicular Lines
⚫ To find the equation of a line parallel or perpendicular to
a given line or set of points through a given point
⚫ Find the slope of the given line or points
⚫ The slope of the new line will either be the same
(parallel) or a negative reciprocal (perpendicular)
⚫ Use the earlier procedures to write the equation of
the line from the point and the slope.

23. Classwork
⚫ College Algebra 2e
⚫ 4.1: 8-36 (4); 3.7: 20-32 (even); 3.5: 48-76 (4)
⚫ 4.1 Classwork Check
⚫ Quiz 3.7