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More on Slope
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by
TweetJuan Jorrin
on 1 November 2017Transcript of More on Slope
Find Slopes and Equations of Parallel and Perpendicular Lines
Interpret Slope as Rate of Change
Slope
Find a Function’s Average Rate
of Change.
Average Rate of Change of a Function
Finding Average Velocity
Example:
The distance traveled by a ball rolling down a ramp is given by
s(t) = 3.5 t
, where t is the time in seconds after the ball is released, and s(t) is measured in feet. Find the average velocity of the ball in the interval from
1 second to 1.5 seconds
.
Writing Equations of a Line Parallel to a Given Line
Writing Equations of a Line Perpendicular to a Given Line
Example:
Find the slope of any line that is perpendicular to the line whose equation is
y = 4  3x
.
More on Slope
Parallel Lines
lines that will never intersect
Line 1
Line 2
have the SAME SLOPE
rise = 2
run = 1
rise = 2
run = 1
Slope =
2
1
=2
Slope =
2
1
= 2
Line 1 and Line 2 are parallel.
Perpendicular Lines
lines that intersect at 90°
their slopes are negative reciprocals of each other
Line 1
Line 2
rise
= 2
run = 1
slope =
2
1
= 2
run = 2
rise 1
slope =
1
2
90°
Line 1 and Line 2 are
perpendicular lines.
Example:
Write an equation of the line passing through
(1,7)
and parallel to the line whose equation is
y = 3x +5
. Express the equation in pointslope form and slopeintercept form.
Solution:
In order to write an equation in point  slope form, we need a point and the slope. We already have a point which is (1,7). We need to find the slope.
Step 1
.
Find the slope. We are given a line which is parallel to the equation that we are looking for. They are
PARALLEL
, so it only means that they have the
SAME SLOPE
!
Using the given equation y =
3
x + 5
slope
we can say that it's slope is 3.
Step 2.
Use the given point
(1,7)
and slope
m=3
to write an equation in point  slope form.
x = 1
y = 7
m = 3
y  y = m(x  x )
1
1
y  7 = 3(x  1)
y  7 = 3x  3
y = 3x + 4
Identify the given:
Use the point  slope form:
Plug  in the values:
Apply distributive property:
Solve for y by adding 7 on
both sides:
Solution:
Write y = 4  3x in slope  intercept form y=mx + b, where m is the slope.
y = 3 x + 4
slope
The given line has slope m = 3. Any line perpendicular to this line has a slope that is the negative reciprocal of 3.
Get the reciprocal of 3 which is 1/3. Multiply it by negative 1,
Then, the slope of any perpendicular line is 1/3.
m = (1/3)=1/3.
perpendicular
Writing Equations of a Line Perpendicular to a Given Line
Example:
Write the equation of the line passing through
(0, 5)
and perpendicular to the line whose equation is
y = 4  3x.
Express the equation in general form.
Identify the given:
The line passes through (0, 5), then
x = 0 and y = 5.
1
1
m =
1
3
*from the previous example
Use the point  slope form:
y  y = m(x  x )
1
1
Plug in the given values:
y  (5) = (x  0)
1
3
Simplify and express in general form:
y + 5 = x
1
3
3y + 15 = x
3y  x + 15 = 0
Multiply both sides by 3:
Subtract both sides by x:
used to describe the measurement of the steepness of a straight line
The higher the slope, the steeper the line.
The slope of a line is a
rate of change.
Slope =
Vertical Change
Horizontal Change
=
Rise
Run
Rise
Run
y  y
2
1
x  x
2
1
m =
y  y
2
1
x  x
2
1
Example:
Use the slope formula to calculate the slope of a line, using just the coordinates of two points. The line passes through the points
(1, 4)
and
(1, 8)
.
Solution:
Substitute the coordinates into the slope formula.
m =
y  y
x  x
1
1
2
2
=
8  4
1  1
=
4
2
= 2
The slope of the line is 2.
a process that calculates the amount of change in one item divided by the corresponding amount of change in another
Using function notation, we can define the Average rate of Change of a function f from a to x as
A(x) =
f(x)  f(a)
x  a
A is the name of the average rate of change function
x  a represents the change in the input of the function f
f(x)  f(a) represents the change in the function f as the input changes from a to x
Example: Find the average rate of change of as x changes to 3 from 0.
f(x) =
1
3
x  4
2
Step 1:
Step 2:
Step 3:
Step 4:
f(3) = 1 and f(0) = 4
Use the slope formula to create the ratio.
f(0)  f(3)
0  3
Simplify.
f(0)  f(3)
0  3
=
4  (1)
0  3
= 1
So the slope of the line going through the curve as x changes from 3 to 0 is 1.
f(x) =
1
3
x  4
2
2
Solution:
Use the formula for the average
velocity of an object:
∆s
∆t
s(t )  s(t )
t  t
2
2
1
1
=
Plug in the given values:
t
1
t
2
∆s
∆t
s(
1.5
)  s(
1
)
1.5 sec

1 sec
=
3.5(1.5)  3.5(1)
1.5 sec  1 sec
=
2
2
7.875 ft  3.5 ft
0.5 sec
=
4.375 ft
0.5 sec
=
= 8.75 ft/sec
∆s
∆t
1
1
Full transcriptInterpret Slope as Rate of Change
Slope
Find a Function’s Average Rate
of Change.
Average Rate of Change of a Function
Finding Average Velocity
Example:
The distance traveled by a ball rolling down a ramp is given by
s(t) = 3.5 t
, where t is the time in seconds after the ball is released, and s(t) is measured in feet. Find the average velocity of the ball in the interval from
1 second to 1.5 seconds
.
Writing Equations of a Line Parallel to a Given Line
Writing Equations of a Line Perpendicular to a Given Line
Example:
Find the slope of any line that is perpendicular to the line whose equation is
y = 4  3x
.
More on Slope
Parallel Lines
lines that will never intersect
Line 1
Line 2
have the SAME SLOPE
rise = 2
run = 1
rise = 2
run = 1
Slope =
2
1
=2
Slope =
2
1
= 2
Line 1 and Line 2 are parallel.
Perpendicular Lines
lines that intersect at 90°
their slopes are negative reciprocals of each other
Line 1
Line 2
rise
= 2
run = 1
slope =
2
1
= 2
run = 2
rise 1
slope =
1
2
90°
Line 1 and Line 2 are
perpendicular lines.
Example:
Write an equation of the line passing through
(1,7)
and parallel to the line whose equation is
y = 3x +5
. Express the equation in pointslope form and slopeintercept form.
Solution:
In order to write an equation in point  slope form, we need a point and the slope. We already have a point which is (1,7). We need to find the slope.
Step 1
.
Find the slope. We are given a line which is parallel to the equation that we are looking for. They are
PARALLEL
, so it only means that they have the
SAME SLOPE
!
Using the given equation y =
3
x + 5
slope
we can say that it's slope is 3.
Step 2.
Use the given point
(1,7)
and slope
m=3
to write an equation in point  slope form.
x = 1
y = 7
m = 3
y  y = m(x  x )
1
1
y  7 = 3(x  1)
y  7 = 3x  3
y = 3x + 4
Identify the given:
Use the point  slope form:
Plug  in the values:
Apply distributive property:
Solve for y by adding 7 on
both sides:
Solution:
Write y = 4  3x in slope  intercept form y=mx + b, where m is the slope.
y = 3 x + 4
slope
The given line has slope m = 3. Any line perpendicular to this line has a slope that is the negative reciprocal of 3.
Get the reciprocal of 3 which is 1/3. Multiply it by negative 1,
Then, the slope of any perpendicular line is 1/3.
m = (1/3)=1/3.
perpendicular
Writing Equations of a Line Perpendicular to a Given Line
Example:
Write the equation of the line passing through
(0, 5)
and perpendicular to the line whose equation is
y = 4  3x.
Express the equation in general form.
Identify the given:
The line passes through (0, 5), then
x = 0 and y = 5.
1
1
m =
1
3
*from the previous example
Use the point  slope form:
y  y = m(x  x )
1
1
Plug in the given values:
y  (5) = (x  0)
1
3
Simplify and express in general form:
y + 5 = x
1
3
3y + 15 = x
3y  x + 15 = 0
Multiply both sides by 3:
Subtract both sides by x:
used to describe the measurement of the steepness of a straight line
The higher the slope, the steeper the line.
The slope of a line is a
rate of change.
Slope =
Vertical Change
Horizontal Change
=
Rise
Run
Rise
Run
y  y
2
1
x  x
2
1
m =
y  y
2
1
x  x
2
1
Example:
Use the slope formula to calculate the slope of a line, using just the coordinates of two points. The line passes through the points
(1, 4)
and
(1, 8)
.
Solution:
Substitute the coordinates into the slope formula.
m =
y  y
x  x
1
1
2
2
=
8  4
1  1
=
4
2
= 2
The slope of the line is 2.
a process that calculates the amount of change in one item divided by the corresponding amount of change in another
Using function notation, we can define the Average rate of Change of a function f from a to x as
A(x) =
f(x)  f(a)
x  a
A is the name of the average rate of change function
x  a represents the change in the input of the function f
f(x)  f(a) represents the change in the function f as the input changes from a to x
Example: Find the average rate of change of as x changes to 3 from 0.
f(x) =
1
3
x  4
2
Step 1:
Step 2:
Step 3:
Step 4:
f(3) = 1 and f(0) = 4
Use the slope formula to create the ratio.
f(0)  f(3)
0  3
Simplify.
f(0)  f(3)
0  3
=
4  (1)
0  3
= 1
So the slope of the line going through the curve as x changes from 3 to 0 is 1.
f(x) =
1
3
x  4
2
2
Solution:
Use the formula for the average
velocity of an object:
∆s
∆t
s(t )  s(t )
t  t
2
2
1
1
=
Plug in the given values:
t
1
t
2
∆s
∆t
s(
1.5
)  s(
1
)
1.5 sec

1 sec
=
3.5(1.5)  3.5(1)
1.5 sec  1 sec
=
2
2
7.875 ft  3.5 ft
0.5 sec
=
4.375 ft
0.5 sec
=
= 8.75 ft/sec
∆s
∆t
1
1