Chapter 4
LINEAR FUNCTIONS
Section 4-1
LINEAR FUNCTION – A function whose
graph forms a straight line.
 Linear functions can describe many realworld situations, such as distances
traveled at a constant speed.
LINEAR EQUATION – Any equation that
can be written in standard form.
STANDARD FORM OF A LINEAR
EQUATION
Ax + By = C
(where A, B, and C are real numbers and A
and B are not both 0)
 X and y both have exponents of 1.
 X and y are not multiplied together.
 X and y do not appear in denominators,
exponents, or radical signs.
Lesson 4-2
 y-intercept: The y-coordinate of the point
where the graph intersects the y-axis. The
x-coordinate of this point is always 0.
 x-intercept: The x-coordinate of the point
where the graph intersects the x-axis. The
y-coordinate of this point is always 0.
Graphing Ax + By = C Using
Intercepts
 Find the x-intercept by _______________
 Find the y-intercept by________________
 Graph the line by____________________
Lesson 4-3
 Rate of change – a ratio that compares the
amount of change in a dependent variable
to the amount of change in an
independent variable.
change in dependent variable
change in independent variable
 Rise – the difference in the y-values of two
points on a line.
 Run – the difference in the x-values of two
points on a line.
 slope – The ratio of rise to run for any two
points on the line.
Slope = rise = change in y
run change in x
 The slope of a horizontal line is 0.
 The slope of a vertical line is undefined.
Lesson 4-4
 You can use the slope formula to find how
quickly a quantity is changing.
 Slope = change in y
change in x
Slope Formula
m=y –y
2
1
x2 – x1
Lesson 4-5
 Direct Variation – a special type of linear
relationship that can be written in the form
y = kx
 Constant of variation – “k” the ratio of y/x
in an equation with direct variation
Lesson 4-6
 Slope-intercept form
y = mx + b,
Where m is the slope, and b is the y
intercept.
Lesson 4-7
 Point-Slope Form .
y – y1 = m(x – x1)
This formula can be used when you are given a point (x1, y1) and the
slope (m), or two points (x1, y1), (x2,y2)
When simplified, point- slope form becomes
slope intercept form and can be used to
graph a line.
Forms of Linear Equations
STANDARD FORM OF A LINEAR
EQUATION
Ax + By = C
- Useful for identifying linear equations; finding x
and y intercepts; and graphing a line using a
table of values or x and y intercepts.
m = y2 – y
x 2 – x1
Slope Formula
- Used to find the slope of a line when given two
ordered pairs.
Direct Variation
y = kx (k = y/x is the constant of variation)
- Used to graph lines (the constant of variation is
the slope of the line)
Slope Intercept Form
y = mx + b (m = slope, b = y-intercept)
-Used to graph a line from an equation or write an
equation from two ordered pairs.
Point-Slope Form
y – y 1 = m(x – x1)
- Used to write an equation from two points.
Lesson 4-9 Slopes of Parallel
and Perpendicular Lines
 Two different nonvertical lines are Parallel
if and only if they have the same slope.
 All different vertical lines are parallel.
 Two different nonvertical lines are
Perpendicular if and only if the product of
their slopes is -1.
 Vertical lines are perpendicular to
horizontal lines.
Lesson 4-10
 Family of Functions – a set of functions whose
graphs have basic characteristics in common
(their graphs are the same basic shape).
 Parent Function – the most basic function in a
family (for linear functions, the parent function is
f(x) = x.
 Transformation – a change in position or
size of a figure
Translation of a Linear Function
 Translation (slide) – a type of
transformation that moves every point the
same distance in the same directions
When the y-intercept b is changed in the
function f(x) = mx + b the graph is
translated vertically.
Rotation of a Linear Function
 Rotation (turn) – a transformation about a
point (the y-intercepts are the same, but
the slope is different).
When the slope m is changed in the function
f(x) = mx + b it causes a rotation of the
graph about the point (0,b), which changes
the line’s steepness.
Reflection of a Linear Function
 Reflection (flip) – a transformation across
a line that produces a mirror image.
When the slope m is multiplied by -1 in
f(x) = mx + b, the graph is reflected across
the y-axis.