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Linear Functions and Slope
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by
TweetJuan Jorrin
on 2 July 2018Transcript of Linear Functions and Slope
Linear Functions and Slope
Write and Graph the Slopeintercept Form of the Equation of a Line.
Slope  Intercept Form of a Line
Graph Horizontal and Vertical Lines
Example:
Graph
y = 2
.
Calculate a line’s slope.
Write the pointslope form
of the equation of a line.
Point  Slope Form
Slope of a Line
describes both the direction and the steepness of the line
m
is the common symbol for the slope
m=
∆y
∆y
∆x
∆x
y  y
x  x
2
2
1
1
=
Example: Find the slope of the line passing through each pair of points:
a. (3,6) and (1,2)
b. (1,1) and (7,5)
m =
x
x
y
y
1
1
2
2
x
x
y
y
1
2
2
1


2
1
4
2
= 2
=
=
x
1
y
1
x
2
y
2
m =
x
x
y
y
1
2
2
1


5
7
4
8
=
=
=
1
2
Possible for a Line's Slope
can be used if the slope
m
of the line and a point
(x , y )
on the line are given
1
1
y  y = m (x  x )
1
1
Example:
Write an equation in pointslope form for the line with slope 6 that passes through the point (2,5). Then solve the equation for y.
m = 6
x = 2
y = 5
1
1
Use the formula:
y  y = m (x  x )
1
1
Substitute the given values:
y  (
5
) =
6
(x 
2
)
Identify the given values:
y + 5 = 6x  12
Subtract both sides by 5:
y = 6x  17
Example: Write an equation in pointslope form for the line passing through the points
(2,4)
and
(1,2)
. Then solve the equation for y.
Remember that we need a
point
and a
slope
to use the point  slope form. A point is already given, (2,4) or (1,2). Then, the first step is to find the slope.
Substitute the given points (2,4) and (1,2) into the slope formula:
y  y
x  x
2
2
1
1
m =
2  4
1 (2)
=
=

2
3
Next, take either point on the line to be . Taking :
(x , y )
1
1
(x , y ) = (1,2)
1
1
y  y = m (x  x )
1
1
y  2 = (x  1)

2
3

2
3
3(y  2) = 2(x  1)
3y  6 = 2x + 2
3y = 2x + 8
y =
2x + 8
3

2
3
x +
8
3
or
y =
This is the point  slope form:
Substitute slope, and the point
(1,2)
:
Multiply both sides by 3:
Apply Distributive Property:
Divide both sides by 3:
y = mx + b
slope
y  intercept
Example:
y =
2
x +
6
slope:
2
y  intercept:
6
Example:
Graph the linear function
y =  x + 3.
3
4
Step 1.
Plot the point containing the yintercept on the yaxis. This is 3. Plot (0, 3).
Step 2.
Obtain a second point using the slope m. Write m as a fraction, and use rise over run, starting at the point containing the yintercept to plot this point. The slope, 3/4 is already written as a fraction.
Step 3.
Use a straightedge to draw a line through the two points.
y  intercept:
3
2nd Point
down: 3
right: 4
3
4
y =  x + 3
x
y = 2 (x,y)
1
0
1
2
2
2
(1,2)
(0,2)
(1,2)
Example:
Graph
x = 3
.
x = 3
3
3
3
y (x,y)
1
0
1
(3,1)
(3,0)
(3,1)
y =  2
x = 3
It's a vertical line.
It's a horizontal line.
Recognize and use the general form of a line’s equation
General Form of the Equation of a Line
Ax + By + C = 0
where
A, B,
and
C
are real numbers, and
A
and
B
are not both zero.
Example:
Find the slope and the y  intercept of the line whose equation is
4x − 2y − 5 = 0
.
Solution:
The equation is given in general form.We begin by rewriting it in the form y = mx + b.
Given equation:
Move all except y to the right:
Divide all by (−2):
Then, the slope
m = 2
and y  intercept
b =  5 / 2
.
4x − 2y − 5 = 0
−2y = −4x + 5
y = 2x − 5/2
Use intercepts to graph the general form of a line’s equation
y  intercept
x  intercept
Example:
Graph using intercepts
:
3x  y  6 = 0.
Step 1.
Find the x  intercept. Let y = 0 and solve for x.
3x 
0
 6 = 0
3x  6 = 0
3x = 6
x = 2
The x  intercept is 2, so the line passes through (2,0).
Step 2
.
Find the y  intercept. Let x = 0 and solve for y.
3(
0
)  y  6 = 0
0  y  6 = 0
 y = 6
y = 6
Step 3.
Graph the equation by drawing a line through the two points containing the intercepts.
The y  intercept is 6, so the line passes through (0,6).
(2,0)
(0,6)
3x  y  6 = 0
Summary of Equations of Lines
General Form:
Vertical Line:
Horizontal Line:
Slope  Intercept:
Point  Slope Form:
Ax + By + C = 0
x = a
y = b
y = mx + b
y  y = m(x  x )
1
1

6

3

(1)

1
(0,3)
(4,0)
Full transcriptWrite and Graph the Slopeintercept Form of the Equation of a Line.
Slope  Intercept Form of a Line
Graph Horizontal and Vertical Lines
Example:
Graph
y = 2
.
Calculate a line’s slope.
Write the pointslope form
of the equation of a line.
Point  Slope Form
Slope of a Line
describes both the direction and the steepness of the line
m
is the common symbol for the slope
m=
∆y
∆y
∆x
∆x
y  y
x  x
2
2
1
1
=
Example: Find the slope of the line passing through each pair of points:
a. (3,6) and (1,2)
b. (1,1) and (7,5)
m =
x
x
y
y
1
1
2
2
x
x
y
y
1
2
2
1


2
1
4
2
= 2
=
=
x
1
y
1
x
2
y
2
m =
x
x
y
y
1
2
2
1


5
7
4
8
=
=
=
1
2
Possible for a Line's Slope
can be used if the slope
m
of the line and a point
(x , y )
on the line are given
1
1
y  y = m (x  x )
1
1
Example:
Write an equation in pointslope form for the line with slope 6 that passes through the point (2,5). Then solve the equation for y.
m = 6
x = 2
y = 5
1
1
Use the formula:
y  y = m (x  x )
1
1
Substitute the given values:
y  (
5
) =
6
(x 
2
)
Identify the given values:
y + 5 = 6x  12
Subtract both sides by 5:
y = 6x  17
Example: Write an equation in pointslope form for the line passing through the points
(2,4)
and
(1,2)
. Then solve the equation for y.
Remember that we need a
point
and a
slope
to use the point  slope form. A point is already given, (2,4) or (1,2). Then, the first step is to find the slope.
Substitute the given points (2,4) and (1,2) into the slope formula:
y  y
x  x
2
2
1
1
m =
2  4
1 (2)
=
=

2
3
Next, take either point on the line to be . Taking :
(x , y )
1
1
(x , y ) = (1,2)
1
1
y  y = m (x  x )
1
1
y  2 = (x  1)

2
3

2
3
3(y  2) = 2(x  1)
3y  6 = 2x + 2
3y = 2x + 8
y =
2x + 8
3

2
3
x +
8
3
or
y =
This is the point  slope form:
Substitute slope, and the point
(1,2)
:
Multiply both sides by 3:
Apply Distributive Property:
Divide both sides by 3:
y = mx + b
slope
y  intercept
Example:
y =
2
x +
6
slope:
2
y  intercept:
6
Example:
Graph the linear function
y =  x + 3.
3
4
Step 1.
Plot the point containing the yintercept on the yaxis. This is 3. Plot (0, 3).
Step 2.
Obtain a second point using the slope m. Write m as a fraction, and use rise over run, starting at the point containing the yintercept to plot this point. The slope, 3/4 is already written as a fraction.
Step 3.
Use a straightedge to draw a line through the two points.
y  intercept:
3
2nd Point
down: 3
right: 4
3
4
y =  x + 3
x
y = 2 (x,y)
1
0
1
2
2
2
(1,2)
(0,2)
(1,2)
Example:
Graph
x = 3
.
x = 3
3
3
3
y (x,y)
1
0
1
(3,1)
(3,0)
(3,1)
y =  2
x = 3
It's a vertical line.
It's a horizontal line.
Recognize and use the general form of a line’s equation
General Form of the Equation of a Line
Ax + By + C = 0
where
A, B,
and
C
are real numbers, and
A
and
B
are not both zero.
Example:
Find the slope and the y  intercept of the line whose equation is
4x − 2y − 5 = 0
.
Solution:
The equation is given in general form.We begin by rewriting it in the form y = mx + b.
Given equation:
Move all except y to the right:
Divide all by (−2):
Then, the slope
m = 2
and y  intercept
b =  5 / 2
.
4x − 2y − 5 = 0
−2y = −4x + 5
y = 2x − 5/2
Use intercepts to graph the general form of a line’s equation
y  intercept
x  intercept
Example:
Graph using intercepts
:
3x  y  6 = 0.
Step 1.
Find the x  intercept. Let y = 0 and solve for x.
3x 
0
 6 = 0
3x  6 = 0
3x = 6
x = 2
The x  intercept is 2, so the line passes through (2,0).
Step 2
.
Find the y  intercept. Let x = 0 and solve for y.
3(
0
)  y  6 = 0
0  y  6 = 0
 y = 6
y = 6
Step 3.
Graph the equation by drawing a line through the two points containing the intercepts.
The y  intercept is 6, so the line passes through (0,6).
(2,0)
(0,6)
3x  y  6 = 0
Summary of Equations of Lines
General Form:
Vertical Line:
Horizontal Line:
Slope  Intercept:
Point  Slope Form:
Ax + By + C = 0
x = a
y = b
y = mx + b
y  y = m(x  x )
1
1

6

3

(1)

1
(0,3)
(4,0)