Linear Functions

Amber & Skyfal l& Winnie & Bruce

Have you still worried about …

We are coming !!!

Part 1

Slope of a

Line

❤How to find the slope of a line?❤

The slope of a line characterizes the general direction in which a line points.

The thing you should know is that…

❤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 2The 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)

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!

And the Answer is...

❤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.

Part 2Slope of parallel and perpendicular lines

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:

2The 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 - 10y = 2x - 6

Perpendicular LinesTwo lines are Perpendicular if they meet at a right angle (90°).

When you multiply their slopes, you get -1

How do you know if two lines are perpendicular?

Line Slopey = 2x + 1 2

y = -0.5x + 4 -0.5

This will show you what I mean:

These two lines are perpendicular:

If we multiply the two slopes we get:

2 × (-0.5) = -1

❤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 - 7/4y = x/4 + 1/4

Part 3 Equation of

a Line in

Slope-

intercept

form and

General

form

❤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

Move all except y to the left:

Divide all by (-2):

And we are done! (Note: m=2 and b=-5/2)

4x - 2y - 5 = 0

-2y = -4x + 5

y = 2x - 5/2

Amber & Skyfall & Winnie & Bruce