Transcript EESlopedirectvariationlinearbutnotdirectvariation / Microsoft

8. EE. 6
Similar Triangles & Slope
y=mx
y=mx+b
Similar Triangles
If triangles are similar, their sides have the same ratios….
2
6
So these are…
similar triangles
6
1
=
12
2
2
1
=
4
2
3
=
6
1
2
12
4
3
All 3 sides
have the
same ratio…
6
Similar Triangles
If triangles are similar, their sides have the same ratios….
1
4
So these are…
12
similar triangles
4
1
=
12
3
1
1
=
3
3
1
=
3
1
3
1
All 3 sides
have the
same ratio…
3
3
Similar Triangles
If triangles are similar, their sides have the same ratios….
10
6
12
15
So these are…
similar triangles
6
2
=
9
8
3
9
10
2
=
15
3
8
=
12
2
3
All 3 sides
have the
same ratio…
E
L
I
Rise = 6
M
Rise = 4
S
Run = 3
Run = 2
S to I
4
2
=
M to E
2
6
1
3
=
2
1
Think about the rise and run between
2 points like similar triangles. Then it
will make sense that no matter which
points you pick on a line, the slope will
always be the same ratio.
H
T
A
M
Rise = 1
Run = 2
M to A
1
2
=
M to H
1
3
2
6
=
Rise = 3
1
2
Run = 6
Remember that the rise and run
between 2 points are like similar
triangles. So it makes sense that no
matter which points you pick on a line,
the slope will always be the same
ratio.
G
R
E
Rise =Rise
12 = 6
A
Run = 2
T
Run = 4
R to A
6
2
=
G to T
3
12
1
4
=
3
1
Rise and run between 2 points is like
similar triangles. So, no matter which
points you pick on a line, the slope will
always be the same.
To be a Direct Variation a line must…
• Go through the origin
• Be linear
x
Now let’s look at the
equation for this line…
How do you get
from the x-value to
the y-value?
y
2
Multiply by 2 =
4
1
Multiply by 2 =
2
0
Multiply by 2 =
0
-1
Multiply by 2 =
-2
-2
Multiply by 2 =
-4
y = 2x
This line
is a
direct
variation!
Goes through
the origin
To be a Direct Variation a line must…
• Go through the origin
• Be linear
Now let’s look at the
equation for this line…
x
How do you get
from the x-value to
the y-value?
y
4
Multiply by ½ =
2
2
Multiply by ½ =
1
0
Multiply by ½ =
0
-2
Multiply by ½ =
-1
-4
Multiply by ½ =
-2
y = ½x
This line
is a
direct
variation!
Goes through
the origin
To be a Direct Variation a line must…
• Go through the origin
• Be linear
x
Now let’s look at the
equation for this line…
How do you get
from the x-value to
the y-value?
y
2
Multiply by 3 =
6
1
Multiply by 3 =
3
0
Multiply by 3 =
0
-1
Multiply by 3 =
-3
-2
Multiply by 3 =
-6
y = 3x
This line
is a
direct
variation!
Goes through
the origin
Let’s look at the direct variation
equations….
y = 2x
y = ½x
y = mx
y = 3x
Direct variation can
always be in this format
This line
is linear but
not a
direct
variation!
Linear but not a Direct Variation
•Linear
•Does not go through the origin
x
Now let’s look at the
equation for this line…
How do you get from the
x-value to the y-value?
y
-2
Multiply by -2, subtract 1= 3
-1
Multiply by -2, subtract 1=
1
0
Multiply by -2, subtract 1= -1
1
Multiply by -2, subtract 1= -3
y = -2x -1
Does not go
through
the origin
This line
is linear but
not a
direct
variation!
Linear but not a Direct Variation
•Linear
•Does not go through the origin
x
y
1
Multiply by 3, add 2 =
5
0
Multiply by 3, add 2 =
2
Multiply by 3, add 2 =
-2
Multiply by 3, add 2 =
-5
-1
-2
Now let’s look at the
equation for this line…
How do you get
from the x-value to
the y-value?
y = 3x + 2
Does not go
through
the origin
This line
is linear but
not a
direct
variation!
Linear but not a Direct Variation
•Linear
•Does not go through the origin
How do you get
from the x-value to
the y-value?
y
2
Multiply by ½, add 4 =
5
0
Multiply by ½, add 4 =
x
-2
-4
Now let’s look at the
equation for this line…
4
Multiply by ½, add 4 = 3
Multiply by ½, add 4 = 2
y = ½x + 4
Does not go
through
the origin
Let’s look at the direct variation
equations….
y = -2x -1
y = 3x + 2
y = mx + b
y = ½x + 4
Linear but not direct
variation can always be
Written in this format.
• Explain…
• How are similar triangles and slope
related?
– What is the format of an equation that is a
direct variation?
– What is the format of an equation that is
linear, but not a direct variation?
Then click here to
check your answers!
Check your answers…
– How are similar triangles and slope related?
• Think about the rise and run between 2 points like
similar triangles. Then it will make sense that no
matter which points you pick on a line, the slope
will always be the same ratio.
– What is the format of an equation that is a
direct variation?
• y = mx
– What is the format of an equation that is
linear, but not a direct variation?
• y = mx + b