Download 1.4 Linear Equations in Two Variables and Linear Functions

Section 1.4 Notes Page 1
1.4 Linear Equations in Two Variables and Linear Functions
Slope Formula
The slope formula is used to find the slope between two points ( x1 , y1 ) and ( x2 , y2 ) .
( x2 , y 2 )
( x1 , y1 )
•
•
•
•
The slope is the vertical change divided by the horizontal change.
From our picture, the horizontal change is y 2 − y1 and the horizontal
change is x 2 − x1 .
From this we get the formula for slope: m =
y 2 − y1
.
x 2 − x1
Positive slopes will rise as you move from left to right.
Negative slopes will fall as you move from left to right.
A slope of zero is a horizontal line.
An undefined or infinity slope is a vertical line.
EXAMPLE: Find the slope of a line passing through the following points. Indicate whether the line rises, falls,
is horizontal or vertical.
a.) (-1, 3) and (2, 4)
To do this problem we can label our point so we know what to put into the slope formula. It doesn’t matter
which point you call x1 or x 2 . I will label the point as the following: x1 = −1 , y1 = 3 , x 2 = 2 , y 2 = 4 . Now
1
4−3
we plug these into the slope formula: m =
= . Since the slope is positive we know this line rises.
2 − (−1) 3
b.) (4, -1) and (3, -1)
First we label the point as the following: x1 = 4 , y1 = −1 , x 2 = 3 , y 2 = −1 . Now we plug these into the slope
0
− 1 − (−1)
formula: m =
=
= 0 . Since the slope is zero we know this line is horizontal.
3−4
−1
c.) (3, -2) and (3, -5)
First we label the point as the following: x1 = 3 , y1 = −2 , x 2 = 3 , y 2 = −5 . Now we plug these into the slope
− 5 − (−2) − 3
formula: m =
=
= undefined . Since the slope is undefined we know this line is vertical.
3−3
0
Section 1.4 Notes Page 2
Now I will introduce two formulas you will need to know in this section:
Slope-Intercept Formula– this is the standard from of a line which allows you to easily identify the slope and
y-intercept.
y = mx + b
Here the slope is m and the y-intercept is (0, b).
Standard Form: Ax + By = C. The constants, A, B, and C should be written as integers if possible, and A
must be written as a positive number.
EXAMPLE: Given y + 9 = 7 x − 35 , write this in standard form.
In order to solve this, we must bring both variables over to one side of the equation, so I will subtract 7x from
both sides: y + 9 − 7 x + 35 = 0 . Now simplify and write the x and y terms first: − 7 x + y + 44 = 0 . Now
subtract 44 from both sides: − 7 x + y = −44 The A constant must be written as a positive number, so we will
multiply the whole equation by -1 to get: 7 x − y = 44 .
EXAMPLE: Write the following in slope-intercept form and identify the slope and y-intercept. Use this
information to graph the equation (parts a and b only).
a.) f ( x) =
3
x−3
4
This equation can be written in slope-intercept form by replacing f (x) with y. You will get: y =
3
and the y-intercept is (0, -3).
4
To graph this, first plot the y-intercept. Now we need another point on
our line. The definition of the slope is the change in the vertical distance
divided by the change in the horizontal distance. In our slope the top
number is 3. This is our vertical change. Because it is positive we will
move up 3 units from our y-intercept. The bottom number is 4, so we
will need to move 4 units to our right. So from our y-int we will move
up 3 units and 4 units to the right. This will give us our next point. Plot
this and connect our two points with a line.
From here we can identify that the slope is
More on next page…
3
x −3.
4
Section 1.4 Notes Page 3
b.) 4x + 6y = –12
We need to solve for y in order to put this into slope-intercept form. First isolate y:
6y = –4x – 12. Now
2
2
divide both sides by 6 to get: y = − x − 2 . Now we can identify that the slope is − and the y-intercept is
3
3
2
(0, -2). To graph this, first plot the y-intercept. Now the fraction − can be
3
2
−2
−2
written as either
or
. If we think of the slope as
then the
3
−3
3
our vertical change is -2. This means we move DOWN 2 units from our
y-intercept. The bottom number is 3, so we will need to move 3 units to our
right. So from our y-int we will move DOWN 2 units and 3 units to the right.
This will give us our next point. Plot this and connect our two points with a line.
2
If we used
then we would move UP two units and to the LEFT 3 units.
−3
Notice we will still get another point on the same line, so we can use either fraction.
c.) Passing through (-3, 6) and (3, -2)
This time we are not given a slope, so we first must use the slope formula. We label our points and put them
4
− 2−6 −8
into the slope formula: m =
=
= − . We will now use the slope-intercept formula. For this
3 − (−3)
6
3
formula we need to know m, x, and b. We already have m, so now we need x and y. For this, we can use
EITHER of our two given points. In this case I will use the first point. So x = −3 , and y = 6 . We can plug
4
these into our slope-intercept formula: y = mx + b: 6 = − (−3) + b . When we simplify we get: 6 = 4 + b .
3
Solve and we will get b = 2. Now we write the slope-intercept equation since we now know m and b:
4
y = − x+2.
3
d.) x-intercept = −
1
, y-intercept = 4
2
This time we are not given a slope, so again we must use the slope formula. We want to put these intercepts in
 1 
a point form, which would be:  − , 0  and 0, 4 . We label our points and put them into the slope formula:
 2 
4
4−0
= = 8 . When we use the slope-intercept we can use EITHER of our two given points for x
m=
 1 1
0 − − 
 2 2
and y. The reason for using the second point is that since this is the y-intercept, we already know that b = 4.
Since we also have our m, we are ready to write this in slope-intercept form: y = 8 x + 4 .
(
)
Section 1.4 Notes Page 4
EXAMPLE: Find the x and y intercepts and use them to graph the following equation: 6x + 9y = 18.
To find an x-intercept, put a zero in for y and solve: 6x + 9(0) = 18.
Simplifying will give us 6x = 18. Solve for x, so x = 3.
It is important to write our answer as a point. The x-intercept is (3, 0).
To find an y-intercept, put a zero in for x and solve: 6(0) + 9y = 18.
Simplifying will give us 9y = 18. Solve for y, so y = 2.
It is important to write our answer as a point. The y-intercept is (0, 2).
To graph, just plot each point and connect them with a line.
EXAMPLE: Find the x and y intercepts and use them to graph the following equation: 6x – 3y + 15 = 0.
To find an x-intercept, put a zero in for y and solve: 6x – 3(0) + 15 = 0.
5
Simplifying will give us 6x + 15 = 0. Solve for x: 6x = -15, so x = − .
2
 5 
It is important to write our answer as a point. The x-intercept is  − , 0  .
 2 
To find an y-intercept, put a zero in for x and solve: 6(0) – 3y + 15 = 0.
Simplifying will give us -3y + 15 = 0. Solve for y: -3y = -15, so y = 5.
It is important to write our answer as a point. The y-intercept is (0, 5).
To graph, just plot each point. For fractions, you can change them into
a decimal. Our x-intercept can be written as: (-2.5, 0).
Average Rate of Change (A.R.C.)
The A.R.C. is an estimate of the slope between x and c. Basically how much does something change between x
and c. The formula is as follows and is derived from the slope formula.
∆y f ( x 2 ) − f ( x1 )
=
x 2 − x1
∆x
EXAMPLE: Find the A.R.C. for the f ( x) = −3 x 2 + 3 from x1 = 0 to x 2 = 2 .
We can substitute these numbers into our general A.R.C. formula:
f (2) − f (0)
Now we will work this out. Find f (2) and f (0)
2−0
−9−3
= −6 So the slope is -6 between the x values of 0 and 2.
2
Section 1.4 Notes Page 5
EXAMPLE: Find the A.R.C. for the f ( x) = x − x + 2 from x1 = 1 to x 2 = 3 .
3
We can substitute these numbers into our general A.R.C. formula:
f (3) − f (1)
3 −1
26 − 2
= 12
2
Now we will work this out. Find f (3) and f (1)
So the slope is 12 between the x values of 1 and 3.
EXAMPLE: Find the A.R.C. for the f ( x) = x from x1 = 9 to x 2 = 16 .
We can substitute these numbers into our general A.R.C. formula:
f (16) − f (9)
Now we will work this out. Find f (16) and f (9)
16 − 9
4−3 1
1
So the slope is
between the x values of 9 and 16.
=
7
7
7