lines chapter 1.1. increments 2 example 1: finding increments 3
TRANSCRIPT
Lines
Chapter 1.1
2
Increments
DEFINITIONIf a particle moves from a point to a point , the increments in its coordinates are
• Note that increments can be positive, negative, or zero.
• Note also that the order of subtraction does not matter, but the indices (subscript values) must be in the same relative position. In other words, if , then we cannot have
• The symbol is the Greek letter delta. The above are pronounced “delta-x” and “delta-y”.
• The text refers to them as increments, but it will be more helpful later to think of them as the change in x and the change in y.
3
Example 1: Finding Increments
Find the coordinate increments from:
a)
b)
4
Example 1: Finding Increments
Find the coordinate increments from:
a)
b)
5
Slope of a Line
DEFINITION:Let be points on a non-vertical line . The slope of is
6
Slope of a Line
7
Slope of a Line
• Along any given non-vertical line, if a point on the line rises as x goes from left to right, then the slope of the line is positive
• If a point on the line falls as x goes from left to right, then the slope of the line is negative
• If the y-coordinate remains unchanged (i.e., ) as x goes from left to right, the slope of the line is zero
• A vertical line has , so the slope of a vertical line is undefined (division by zero)
8
Positive Slope of a Line
9
Negative Slope of a Line
10
Theorems and Definitions
• As we continue in the course, it is important that you read definitions and theorems carefully
• A definition is a statement taken to be true and from which conclusions can be drawn
• A theorem is a conditional statement that must be proven using logical reasoning from definitions and/or other proven theorems
• Both definitions and theorems will state the conditions under which they can be applied
• Theorems can be stated in the form “if (hypotheses or conditions), then (conclusion)”
• In general, theorems are not reversible; that is, it is not always correct to say “if (conclusion as hypothesis), then (hypotheses as conclusion)
• Theorems that are “reversible” are known as bi-conditional or equivalence theorems
• The next two theorems are equivalence theorems
11
Parallel Lines
THEOREM:The slopes of two lines are equal if and only if the lines are parallel
• Note that this is really two theorems in one:• If the slopes of two lines are equal, then the lines are parallel• If two lines are parallel, then the slopes of the lines are equal
• What this means is that we could replace the phrase “lines with equal slope” with the phrase “parallel lines” and our meaning would not change
• You will now see a geometric proof of this theorem
Parallel Lines
We prove that, if two lines are parallel, then their slopes are equal
• Lines are parallel and the x-axis is a transversal line• (corresponding angles) so and
, i.e., the triangles are similar• Corresponding sides of similar triangles are proportional, so
• By cross multiplication
• The slope of is and of is , therefore
12
13
Parallel Lines
We prove that, if , then the slopes of the lines are equal• Lines are parallel and the x-axis is a
transversal line• (corresponding angles) so
and , i.e., the triangles are similar• Corresponding sides of similar triangles are
proportional, so
• By cross multiplication
• The slope of is and of is , therefore
Assume that is parallel to
14
Parallel Lines
• We must also prove that, if two lines have equal slopes, then the lines are parallel
• In this case, we need merely reverse our steps of the proof to arrive at this conclusion
• So we have proven that the slopes of two lines are equal if and only if (iff) the lines are parallel
15
Perpendicular Lines
THEOREM:Two lines with slopes , respectively, are perpendicular iff
Or equivalently, if
• This is also an equivalence; what are the two parts of the theorem?• If lines with slopes , respectively, are perpendicular, then • If lines with slopes , respectively, are such that , then the lines are perpendicular
Perpendicular Lines
• We assume that
• Since h, a, and b represent segment lengths, then
• The slope of is
• The slope of is
• By a theorem from geometry, so that
• By substitution,
• Now,
16
Parallel Lines
• As in the previous proof, we can prove the converse statement by reversing our steps
• So we have proven that, given lines with slopes , respectively, the lines are parallel iff
• Next, you will see several definitions associated with equations of lines (all of which should be familiar by now)
17
Equations of Lines
DEFINITION:The equation
is the point-slope equation of the non-vertical line through point with slope m.
18
Equations of Lines
• This definition is derived from the definition of slope
• If we take to be a point on a line , and allow to be any other point on the line, then
• This comes about because we assume that we know one of the points on the line and also the slope of the line
19
Equations of Lines
DEFINITION:
If a non-vertical line passes through a point on the y-axis, then we say that is the y-intercept of the line.
If a non-horizontal line passes through a point on the x-axis, then we say that a is the x-intercept of the line.
20
Equations of Lines
DEFINITION:The equation
is the slope-intercept equation of the line with slope m and y-intercept b
21
Equations of Lines
• We derive the slope-intercept form from the point slope form
• Let be the -intercept of a line
• Then
• This comes about because we assume that we know the -intercept and the slope of the line
22
Equations of Lines
DEFINITION:The equation
with A and B not both equal to zero, is the general linear equation in x and y
23
Equations of Lines
• If we take , and , then using the slope-intercept equation we get
• Multiplying by B gives which leads to
• For a vertical line, the slope is undefined and there is no y-intercept, but in the form above if , then is the equation of the line.
24
Equations of Lines
DEFINITION:The equation of a horizontal line passing through a point is
The equation of a vertical line passing through a point is
(It is often necessary to define special cases separately from general definitions. In this case, vertical lines cannot be represented by either the point-slope form or the slope-intercept form)
25
Example 2: Finding Equations of Vertical & Horizontal LinesFind the equation of
a) The vertical line through the point
b) The horizontal line through the point
26
Example 2: Finding Equations of Vertical & Horizontal LinesFind the equation of
a) The vertical line through the point :
b) The horizontal line through the point :
27
Example 3: Using the Point-Slope Equation
Write the point-slope equation for the line through the point with slope .
28
Example 3: Using the Point-Slope Equation
Write the point-slope equation for the line through the point with slope .We take and . The equation is
29
Example 4: Writing the Slope-Intercept EquationWrite the slope-intercept equation for the line through and .
30
Example 4: Writing the Slope-Intercept EquationWrite the slope-intercept equation for the line through and .First find the slope, taking
Now use either point and the slope to find the equation
31
Example 5: Analyzing & Graphing a General Linear EquationFind the slope and y-intercept of the line , then graph the line.
32
Example 5: Analyzing & Graphing a General Linear EquationFind the slope and y-intercept of the line , then graph the line.Recall that we defined and , so the slope is and the y-intercept is . You can now graph the line by using the intercept as a starting point and count rise and run.
33
Example 6: Writing Equations for Lines
Write an equation for the line through the point that is (a) parallel, and (b) perpendicular to the line L: .
34
Example 6: Writing Equations for Lines
Write an equation for the line through the point that is (a) parallel, and (b) perpendicular to the line L: .Since you know that “if two lines are parallel, then the slopes are equal”, we take the slope of the new line to be . The equation (in point-slope form) is
35
Example 6: Writing Equations for Lines
Write an equation for the line through the point that is (a) parallel, and (b) perpendicular to the line L: .You know that “if two lines are perpendicular, then the product of their slopes is ”, so
and the slope of the new line will be . The equation (in point-slope form) is
36
Example 7: Determining a Function
The following table gives values for the linear function . Determine m and b.
37
x
Example 7: Determining a Function
The following table gives values for the linear function . Determine m and b.You can determine the slope by taking using any two points:
Find the intercept by substituting any given point and the slope in the point-slope form and solving for
38
Example 8: Temperature Conversion
Find the relationship between Fahrenheit and Celsius temperature. Then find the Celsius equivalent of and the Fahrenheit equivalent of .
39
Example 8: Temperature Conversion
Find the relationship between Fahrenheit and Celsius temperature. Then find the Celsius equivalent of and the Fahrenheit equivalent of .Since the relationship between the two units is a linear one, the equation will take the form . We need two points for the slope. Water freezes at and at , so an ordered pair is . Water boils at and at , so another ordered pair is . Thus we have
40
Example 8: Temperature Conversion
Find the relationship between Fahrenheit and Celsius temperature. Then find the Celsius equivalent of and the Fahrenheit equivalent of .So far we have .
The formula is therefore . The temperature in degrees Celsius of is , . The temperature in degrees Fahrenheit of is .
41
Chapter 1.1 Exercises
• Finney page 9, #1-37 odds, #40, 41, 45, 46, 57
42