Download 5-1 Linear Functions

Identify Linear
Functions & Their
Graphs
Honors Math – Grade 8
Get Ready for the Lesson
It is recommended that no more than 30% of a
person’s daily caloric intake come from fat. Since
each gram of fat contains 9 Calories, the most
grams of fat f that you should have each day is
given by
C
f 
30
C is the total number of Calories that you
consume. The graph of this equation is shown to
the right.
This type of equation is known as a linear equation.
x6
A linear equation is the equation of a line.
KEY CONCEPT
y0
Standard Form of a Linear Equation
The standard form of a linear equation is
Ax  By  C
where A > 0, A and B are not both zero and A, B, and C are
integers with a greatest common factor of 1.
5 x  9 y  32
2 x  y  4
Determine whether each equation is a linear equation.
If so, write the equation in standard form.
Rewrite the equation so
that it is in the form:
Ax + By = C
Both variables need to be
on the same side of the
equation.
Add 2x to both sides of
the equation.
y  5  2x
 2x   2x
2x  y  5
This equation is in standard form where A = 2, B = 1 and C = 5
This equation is a linear equation.
Determine whether each equation is a linear equation.
If so, write the equation in standard form.
Rewrite the equation so
that it is in the form:
Ax + By = C
2 xy  5 y  6
Since the term 2xy has two
variables, the equation cannot be
written in the form Ax + By = C.
This equation is not
a linear equation.
5x  3 y  z  2
Rewrite the equation so
that it is in the form:
Ax + By = C
This equation is not
a linear equation.
Since the equation has a third
variable—z, it cannot be written in
the form Ax + By = C.
Determine whether each equation is a linear equation.
If so, write the equation in standard form.
Rewrite the equation so
that it is in the form:
Ax + By = C
Since the equation contains
an x2, it cannot be written in
the form Ax + By = C.
1  
y


1


3
3  
 
y  3


3


This equation is a linear equation.
y  x 3
2
This equation is not
a linear equation.
Eliminate the denominator by
multiplying both sides of the
equation by 3.
This equation is in
standard form where
A = 0, B = 1 and C = -3
The x-coordinate of the point at which
the graph of an equation crosses the
x-axis is an x-intercept.
The coordinates of an x-intercept are
(x, 0). An x-intercept occurs when
y = 0.
Values for x for which f(x) = 0
(AKA y = 0) are called zeros of
the function. The zero of a
linear function is its x-intercept.
The y-coordinate of the point at which
the graph of an equation crosses the
y-axis is a y-intercept.
The coordinates of a y-intercept are
(0, y). An y-intercept occurs when
x = 0.
Graph each equation
1
y  x3
2
The graph of an equation represents all of its solutions. So, every ordered
pair that satisfies the equation represents a point on the line. An order pair
that does not satisfy the equation represents a point not on the line.
To graph an equation,
make a function table.
Choose at
least three
numbers.
Evaluate for
each value
and write
the ordered
pair.
Plot the
points and
connect
them with a
line.
Choose values for the
domain (x).
x
-2
0
2
4
1
x3
2
1
( 2)  3
2
1
(0)  3
2
1
(2)  3
2
1
(4)  3
2
y
Ordered
Pair
-4 (-2,-4)
-3 (0,-3)
-2 (2,-2)
-1 (4,-1)
Graph each equation
y  2
Choose at least three
numbers.
Plot the points and
connect them with a line.
This equation states that y is always -2.
To graph an equation, make a function table.
Choose values for the domain (x). Remember
that no matter what you choose, y is always -2.
An equation
in the form
y = C is the
equation of
a horizontal
line.
Ordered
Pair
x
y
-1
-2 (-1,-2)
0
-2 (0,-2)
1
-2 (1,-2)
2
-2 (2,-2)
Graph each equation
Plot the ordered pairs and
connect them with a line.
3x  2 y  9
An equation can also be graphed using the xand y-intercepts of the equation.
(3,0)
To find the x-intercept let y = 0.
(0,4.5)
3x  2 y  9
3x  2(0)  9
3x  9
x 3
To find the y-intercept let x = 0.
3x  2 y  9
3(0)  2 y  9
2y  9
y  4.5
Graph each equation
y  x  4
(4,0)
(0,4)
Plot the ordered pairs and
connect them with a line.
An equation can also be graphed using the xand y-intercepts of the equation.
To find the x-intercept let y = 0.
y  x  4
0  x  4
x  4
To find the y-intercept let x = 0.
y  x  4
y  1(0)  4
y  4
Rate of change is a ratio that describes, on average, how
much one quantity changes with respect to a change in another
quantity. If x is the independent variable and y is the dependent
variable, then
The table below shows the distance a person has walked for different
amounts of time.
What would the rate of change
be?
4
1
This means the person
walked four feet per
second.
Use the table to find the rate of change.
Explain the meaning of the rate of change.
The rate of change
means that it costs
$39 per game.
Find the rate of change.
change in y 156  78 78 39



change in x
42
2
1
The slope of a line is the ratio of the change in the y-coordinates (rise) to the
change in the x-coordinates (run) as you move in the positive direction.
The graph shows a line that passes through
(1, 3) and (4, 5).
2
slope 
3
Any two points on the line can be
used to determine the slope.
KEY CONCEPT
The slope of a line is the ratio of the rise to the run.
The slope m of a nonvertical
y2  y1
line through any two points,
m
(x1, y1) and (x2, y2), can be
x2  x1
found as follows:
Slope
Find the slope of the line that passes through
the given points.
(-1, 2) and (3, 4)
Define the variables.
Use the slope formula.
y2  y1 4  2 2 1

 
m

x2  x1 3  1 4 2
Substitute and simplify.
(3, 6) and (4, 8)
Define the variables.
Use the slope formula.
y2  y1 8  6 2

 2
m
x2  x1 4  3 1
Substitute and simplify.
The slope of a line may be positive!
Find the slope of the line that passes through
the given points.
(-1, -2) & (-4, 1)
Define the variables.
Use the slope formula.

y2  y1 1  2 3 
m
      1
x2  x1
4 1 3
Substitute and simplify.
(-2, 2) & (-6, 4)
Define the variables.
Use the slope formula.
y2  y1 4  2
2 1
    
m
2
x2  x1 6  2 4
Substitute and simplify.
The slope of a line may be negative!
Find the slope of the line that passes through
the given points.
(6, 7) and (-2, 7)
Define the variables.
Use the slope formula.
0
y2  y1 7  7
 
  0
m
26
8
x2  x1
Substitute and simplify.
(1, 2) and (-1, 2)
Define the variables.
Use the slope formula.
y2  y1 2  2 0
 

m
0
11  2
x2  x1
Substitute and simplify.
The slope of a line may be zero!
Find the slope of the line that passes through
the given points.
(1, 2) and (1, 4)
Define the variables.
Use the slope formula.
Substitute and simplify.
(3, -2) and (3, 2)
Define the variables.
Use the slope formula.
Substitute and simplify.
y2  y1 4  2
2


m
11
0
x2  x1
Since division by zero is undefined, the slope is
undefined.
y2  y1 2  2 4

m

0
x2  x1 3  3

Since division by zero is undefined, the slope is
undefined.
Given the slope of a line and one point on the line,
you can find other points on the line using the slope
formula and your expertise in solving equations!