Download “point-slope” equation of a line - TangHua2012-2013

Linear
Functions
Amber & Skyfal l& Winnie & Bruce
Have you still worried about …
We are coming !!!
❤How to find the slope of a line?❤
The thing you should know is that…
The slope of a line
characterizes the general
direction in which a line
points.
❤Formula to find the slope of a line❤
❤Example 1：
The slope of a line going through the
point (1,2) and the point (4,3) is 1/3.
❤Example 2
The slope of a line through the points (3, 4) and (5, 1) is
-3/2 because every time that the line goes down by 3(the
change in y or the rise) the line moves to the right (the
run) by 2.
❤Question
Does it matter which
point you start with?
There is only one way to know! ♣
<树哥无敌！>
Let's try to find the
slope of a line through the
points (4,3) and (1,2) .First
we'll start with one point
and then we'll start with
the other.
First, let's start with the point (4,3)
Now, let's start with the point (1,2)
And the Answer is...
It does not matter which point you put
first. You can start with (4,3) or with (1,2)
and, either way, you end with the exact same
number!
❤Slope of vertical and horizontal lines❤
♣ The slope of a vertical line is undefined.
♣ This is because any
vertical line has a Δx or
“run” of zero.
Whenever zero is the
denominator of the
fraction in this case of
the fraction representing
the slope of a line, the
fraction is undefined.
The picture below shows
a vertical line （x=1）
♣ The slope of a horizontal line is zero
♣ This is because any
horizontal line has a Δy or
"rise" of zero. Therefore,
regardless of what the run
is (provided its' not also
zero!), the fraction
representing slope has a
zero in its numerator.
Therefore, the slope must
evaluate to zero. Below is a
picture of a horizontal
line--you can see that it
does not have any 'rise' to
it.
Parallel Lines
How do you know if two lines are parallel?
Their slopes are the same!
❤Example:
Find the equation of the line
that is:
parallel to y = 2x + 1
passes though the point (5,4)
The slope of y=2x+1 is:
2
The parallel line must have the same slope!
Let us put that in the
“point-slope” equation of a line
y - y1 = 2(x - x1)
And now put in the point (5,4):
y - 4 = 2(x - 5)
And that is a good answer!
But let's also put it in
“slope-intercept(y=mx+b)” form
y - 4 = 2x - 10
y = 2x - 6
Perpendicular Lines
Two lines are Perpendicular if they meet at a right angle (90°).
How do you know if two lines are perpendicular?
When you multiply their slopes, you get -1
This will show you what I mean:
If we multiply the two slopes we get:
2 × (-0.5) = -1
These two lines are perpendicular:
Line
Slope
y = 2x + 1
2
y = -0.5x + 4
-0.5
❤Example:
Find the equation of the line
that is:
perpendicular to y = -4x + 10
passes though the point (7,2)
The slope of y=-4x+10 is: -4
The negative reciprocal of that slope is:
So the perpendicular line will have a
slope of 1/4:
y - y1 = (1/4)(x - x1)
And now put in the point (7,2):
y - 2 = (1/4)(x - 7)
And that is a good answer!
But let's also put it in "y=mx+b" form:
y - 2 = x/4 - 7/4
y = x/4 + 1/4
❤General Form of Equation of a Line❤
The "General Form" of the
equation of a straight line is:
Ax + By + C = 0
A or B can be zero, but not both at the
same time.
The General Form is not always the
most useful form, and you may
prefer to use:
The Slope-Intercept Form of
the equation of a straight line:
y = mx + b
The Point-Slope Form of the
equation of a straight line:
y - y1 = m(x - x1)
❤Example:
We are heading for
y = mx + b
Start with
4x - 2y - 5 = 0
Move all except y to the left:
Divide all by (-2):
-2y = -4x + 5
y = 2x - 5/2
And we are done! (Note: m=2 and b=-5/2)
Amber & Skyfall
& Winnie & Bruce