Fundamentals of Mathematics
§2.3 - Slope and Intercepts of Lines
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Lecture 18
October 7, 2013
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Fundamentals of Mathematics
Announcements
HW 8 is due next Tuesday!
Online Quiz 7 due Friday.
Test 2 starts at 10/11 to 10/14. Registration is available.
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Fundamentals of Mathematics
Correction!!!
I WAS WRONG on quadrants!!
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Fundamentals of Mathematics
Recall
Recall a line can be given by the linear function
y = ax + b,
where a, b are constants.
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§§2.3 - Slope and Intercepts of Lines
Outlines
There are two important notions of lines we shall study.
The slope of a line:
Geometric meaning
Algebraic meaning
The intercepts
Geometric meaning
Input-output meaning
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§§2.3 - Slope and Intercepts of Lines
Slope: Algebraic Meaning
The slope of a linear measures how steep a line is, i.e.
how much y changes per unit of change in x
Definition
Given two points (x1 , y1 ), (x2 , y2 ) of a line, the slope is given by
m=
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y2 − y1
x2 − x1
§§2.3 - Slope and Intercepts of Lines
Slope: Geometric Meaning
Pictorially, m is the ratio of differences in x and y.
Remark
Note that in the form y = ax + b, m is just a. Sometimes people
write it as y = mx + b.
Ricky Ng
§§2.3 - Slope and Intercepts of Lines
Here we point out some observations regarding to the slope m.
Basically m indicates the direction a line is heading to, from
left to right.
If m > 0, the line is going upward.
If m < 0, the line is going downward.
If m is more positive, the line is steeper in the upward
direction.
If m is more negative, the line is steeper in the downward
direction.
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§§2.3 - Slope and Intercepts of Lines
An illustration of positive slopes:
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§§2.3 - Slope and Intercepts of Lines
An illustration of negative slopes:
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§§2.3 - Slope and Intercepts of Lines
What about horizontal lines y = b and vertical lines x = a?
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§§2.3 - Slope and Intercepts of Lines
Horizontal lines y = b have slope 0.
Vertical lines x = a have slope undefined.
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§§2.3 - Slope and Intercepts of Lines
Geometric Examples
State whether the slope of each of the following lines is positive,
negative, zero, or undefined.
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§§2.3 - Slope and Intercepts of Lines
Algebraic Examples
Recall
m=
y2 − y1
x2 − x1
Find the slope of the line that passes through the following
points:
(3, 5) and (6, 7)
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§§2.3 - Slope and Intercepts of Lines
(−4, 5) and (5, −2)
(0, 3) and (0, 0)
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§§2.3 - Slope and Intercepts of Lines
( 12 , 1) and ( 23 , 41 )
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§§2.3 - Slope and Intercepts of Lines
Popper 11: Question 1
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§§2.3 - Slope and Intercepts of Lines
Popper 11: Question 2
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§§2.3 - Slope and Intercepts of Lines
Popper 11: Question 3
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§§2.3 - Slope and Intercepts of Lines
Another form of equation of a line
Sometimes we write
Ax + By = C,
where A, B, C are constants.
Remark
If A = 0, then we have a horizontal line.
If B = 0, then we have a vertical line.
If both A, B 6= 0, then it can be transformed into the
slope-point formula.
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§§2.3 - Slope and Intercepts of Lines
Intercepts: Geometric Meaning
Definition
x-intercept: intersection with x-axis.
y-intercept: intersection with y-axis.
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§§2.3 - Slope and Intercepts of Lines
Intercepts: Input-Output Meaning
Since the x-axis always has y-coordinate 0, the x-intercept is the
point (x, 0) of the line, or for short just the number x. Likewise,
the y-intercept is (0, y) of the line or just the number y.
Hence, finding x-intercept is to find the input x, when the
output = 0. And finding y-intercept is to find the output y
when the input = 0.
As long as we keep the input-output relation in mind with the
previous picture, finding intercepts are easy.
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§§2.3 - Slope and Intercepts of Lines
Examples
Find the intercepts of the following lines:
2x + 4y = 4
−3x + 2y = −3
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§§2.3 - Slope and Intercepts of Lines
y = 12 x − 2
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§§2.3 - Slope and Intercepts of Lines
2
5x
+ 34 y = 6
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§§2.3 - Slope and Intercepts of Lines
Popper 11: Question 4
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§§2.3 - Slope and Intercepts of Lines