Transcript Section 1.3 – Linear Functions, Slope and Applications

Section 2.2 – Linear Equations in
One Variable
Linear Equations
An equation that can be written in the form mx + b = 0, where m and b are real
numbers is a linear equation.
Solutions to a linear equation are values of x that make the statement true.
Solve.
Ex 1:
Ex 2:
Linear Equations
Ex 3 :
2
6 1 5
x   x
3
5 2 6
6 1 5 
2
30   x      x   30
5 2 6 
3
20 x  36  15  25 x
 5 x  36  15
 5 x  51
51
x
5
Rational Equations
We will solve rational equations by multiplying both
sides of the equation by the LCD.
The resulting equation will be a linear equation,
which we know how to solve.
The solutions that we obtain must be checked to
determine if they are in the domain.
Rational Equations
Ex 3 :
1 1 3
 
x 5 2x
LCD : 10x
 1  1 3 
10 x        10 x
 x  5 2x 
10  2x  15
 5  2x
x
5
2
 5
 
 2
Slope
The slope m of a line containing the points (x1, y1) and (x2, y2) is given by
m
rise
run
m
m
change in y
change in x
m
y
x
y2  y1 y1  y2

x2  x1 x1  x2
***The slope is a measure of the slant of a line.
***You are often asked to find the rate of change in/of y over x  find the slope.
Slope & Lines
The slope of a line is constant. That is, the slope between every pair of points on a
line is equal.
y
f(x)=2x-4
Series 1
5
rise
m
run
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
6
7
8
9
Slope from Graph
Find the slope of the line.
Ex 5:
Ex 6:
m
rise  3

run
5
x  2
m
rise 3

Undefin ed
run 0
The slope of a vertical line is undefined.
Slope from Graph
Find the slope of the line.
y3
Ex 7:
m
rise 0
 0
run 3
The slope of a h
orizontal line is 0.
Ex 8:
m
rise 5

run 8
Slope & Lines
Decreasing
Slope is negative
Increasing
Slope is positive
Vertical
Slope is undefined
Horizontal
Slope is 0
y 1
x4
Slope from Points
Find the slope of the line containing the given points.
Ex 9 : (9,8)
(7,6)
m
8  (6) 8  6 14
7



97
 16  16
8
Ex 10 : ( 2 ,4)
(0.56,4)
m
 4  (4)
44
0


0
2  0.56
2  0.56
2  0.56
Slope-Intercept Form
The linear equation y = mx + b is written in slopeintercept form.
y
f(x) = mx + b
The graph of an equation in this
form is a straight line with a
slope equal to m and a yintercept at the point (0, b).
(0, b)
x
Graph Using Slope-Intercept
Graph the equation using the slope and the y-intercept.
Ex 11 :
y
3
x 1
2
3
m
2
b 1
Slope from Graph
Graph the linear equation and state its slope and y-intercept.
Ex 12 :
2y  x  8
2y  x 8
1
y  x4
2
1
f ( x)  x  4
2
rise 1

run 2
b  (0,4)
m
9
8
7
6
y
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
x
1 2 3 4 5 6 7 8 9
Find an Equation for a Line
Point-Slope Equation
y  y1  m( x  x1 )
In order to find an equation of a line, you MUST have
• slope  m
• point ( x1 , y1 )
Find an Equation for a Line
Write a slope-intercept equation for a line with the
given characteristics.
3
8
y - intercept (0,5)
Ex 13 : m  
3
y  (5)   ( x  (0))
8
3
y  5   ( x  0)
8
3
y 5   x 0
8
3
y   x5
8
This line crosses the y axis at 5 and it decreases.
3
y   x5
8
Find an Equation for a Line
Ex 14 :
m  2, passes through (5,1)
m  2
(5,1)
y  (1)  2( x  (5))
y  1  2( x  5)
y  1  2 x  10
y  2 x  9
y  2 x  9
This line crosses the y axis at -9 and it decreases.
Horizontal/Vertical Lines
Write equations of the horizontal and vertical lines that pass through the given
point.
 1 
Ex 15 :   ,7 
 4 
Horizontal:
y7
Vertical:
x
Ex16 :
0.03,0
Horizontal:
1
4
Vertical:
y0
x  0.03
Parallel/ Perpendicular Lines
Parallel lines have slopes that are equal.
m1  m2
Perpendicular lines have slopes that are
negative reciprocals.
m2  
m  3
1
m1
m||  3
m1  m2  1
1
m 
3
Parallel/ Perpendicular
Write a slope-intercept equation for a line passing through the given point that is parallel to
the given line.
Then write a second equation for a line passing through the point that is perpendicular to
the given line.
Ex 17 : (4,5), 2 x  y  4
Parallel Line:
m||  2
y  (5)  2( x  (4))
y  5  2( x  4)
y  5  2 x  8
y  2 x  13
y  2 x  13
m 2
(4,5)
2 x  y  4
y  2 x  4
Perpendicular Line: m 
y  (5) 
1
( x  (4))
2
1
( x  4)
2
1
y5  x2
2
1
y  x 3
2
1
2
y5 
1
y  x3
2
Parallel/ Perpendicular
1
y  x3
2
Graph all lines on the same set of axes.
y  2 x  4
y  2 x  13
y
f(x)=-2x - 4
f(x)=-2x - 13
f(x)=(1/2)x - 3
1
x
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
-6
-7
-8
2
3
4
5
6
7
8
Parallel/ Perpendicular
Ex 18 : (4,5), y  1
m 0
(4,  5)
y
f(x)=-1
 undefined
f(x)=-5
m||  0
Parallel Line:
y  5
Perpendicular Line: m
9
8
x=4
x4
7
6
5
4
3
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
-6
-7
-8
-9
2
3
4
5
6
7
8
9