Download Linear Functions — 4.1

Math 2 Precalculus Algebra
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Linear Functions — 4.1
Slope or Rate of Change of a Line
All linear equations in two variables have a constant rate of change or slope. The slope of a line is a ratio of the vertical
vertical change
or
change to the horizontal change between two selected points on a line. In other words, slope is
horizontal change
rise
run
or
the change in the y-values
the change in the x-values
or
∆y
∆x
or
the change up or down
.
the change left or right
If two points on the line are labeled the first ordered pair: ( x1 , y1 ) and the second ordered pair: ( x2 , y2 ) , then the slope of
the line can be found by the following formula:
m =
For a linear function f =
( x)
y2 − y1
x2 − x1
m x + b , this slope formula looks like: m =
f ( x2 ) − f ( x1 )
.
x2 − x1
It does not matter which ordered pair is labeled the first ordered pair and which is labeled the second ordered pair.
Sketch the line through A and B, and find its slope, m.
1.
A ( 4 , − 1) B ( − 6 , − 3 )
8
7
6
5
4
3
2
1
0
-1 0 1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
2.
By using two points on the line and the slope formula:
A ( 4 , 3) B ( − 2 , 3)
8
7
6
5
4
3
2
1
0
-1 0 1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
3.
By counting the units of rise—vertical (y) change—and the units of
run—horizontal (x) change—between any
two points:
By counting the units of rise—vertical (y) change—and the units of
run—horizontal (x) change—between any
two points:
By using two points on the line and the slope formula:
A ( − 3 , 5 ) B ( − 3 , − 1)
8
7
6
5
4
3
2
1
0
-1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
-8
By counting the units of rise—vertical (y) change—and the units of
run—horizontal (x) change—between any
two points:
By using two points on the line and the slope formula:
•
A horizontal line has a slope of 0. The equation of a horizontal line is y = " t h e y - v a l u e o f t h e y - i n t e r c e p t " .
For example y = – 2 or y =
•
.
A vertical line has an undefined slope. The equation of a vertical line is x = " t h e x - v a l u e o f t h e x - i n t e r c e p t " .
For example x = – 2 or x =
•
•
3
5
3
5
. Note: Vertical lines are not functions of x.
The slopes of parallel lines are the same.
The slopes of perpendicular lines are opposite reciprocals.
Slope-Intercept Form=
y
mx + b
If an equation of a line is solved for y, and therefore in the form
=
y
m x + b , the slope of the line is the coefficient of x
or m, and the y-value of the y-intercept is b. Not all equations of lines are written in this form; however you can solve any
equation of a non-vertical line for y and get it in this form.
=
y
m x + b is called the slope-intercept form of an equation of a line because the slope and the y-value of the y-intercept
are readily seen.
Point Slope Form y − =
y1
m ( x − x1 )
If an equation of a line is in the form y − =
y1
m ( x − x1 ) , the slope of the line is m, and ( x1 , y1 ) is a point on the line.
Any point on the line may be used for ( x1 , y1 ) . Usually equations are not written in this form; however, this form is very
useful for writing an equation of a line.
General Form a x + b y =
c
Any equation in the form a x + b y =
c , is a line, and conversely, every line is the graph of a linear equation.
Linear Functions
f=
( x)
mx + b
The graph of a linear function is a line with slope m, and y-intercept ( 0 , b ) . Its domain is the set of all real numbers.
a)
b)
c)
d)
Determine the slope and the y-intercept of each function.
Use the slope and the y-intercept to graph the linear function.
Determine the average rate of change of each function.
Determine whether the linear function is increasing, decreasing or constant.
4.
g ( x) =
− 2x + 5
5.
f ( x) =
−3
8
7
6
5
4
3
2
1
0
-1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
-8
8
7
6
5
4
3
2
1
0
-1 0 1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
Determine whether the given function if linear or nonlinear. If it is linear, determine the slope.
6.
x
y = f ( x)
-2
-1
0
1
2
-4
-3.5
-3
-2.5
-2
x
y = f ( x)
-2
-1
0
1
2
1/4
1/2
1
2
4
7.
8.
Suppose that f (=
x)
a)
3 x + 5 and g ( x ) =
− 2x + 1 5 .
Solve f ( x ) = 0 .
b) Solve f ( x ) < 0 .
c)
Solve f ( x ) = g ( x ) .
d) Solve f ( x ) ≥ g ( x )
e)
Graph y = f ( x ) and y = g ( x ) and label the point that represents the solution to the equation
f ( x) = g ( x) .
9.
A phone company offers a domestic long distance package by charging $5 plus $0.05 per minute.
a) Write a linear function that relates the cost, C, in dollars, of talking x minutes.
b) What is the cost of talking 105 minutes?
c)
Suppose that your monthly bill is $45.20. How many minutes did you use the phone?