Lines

Chapter 1.1

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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.

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Example 1: Finding Increments

Find the coordinate increments from:

a)

b)

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Example 1: Finding Increments

Find the coordinate increments from:

a)

b)

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Slope of a Line

DEFINITION:Let be points on a non-vertical line . The slope of is

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Slope of a Line

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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)

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Positive Slope of a Line

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Negative Slope of a Line

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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

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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

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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

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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

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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,

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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)

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Equations of Lines

DEFINITION:The equation

is the point-slope equation of the non-vertical line through point with slope m.

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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

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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.

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Equations of Lines

DEFINITION:The equation

is the slope-intercept equation of the line with slope m and y-intercept b

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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

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Equations of Lines

DEFINITION:The equation

with A and B not both equal to zero, is the general linear equation in x and y

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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.

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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)

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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

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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 :

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Example 3: Using the Point-Slope Equation

Write the point-slope equation for the line through the point with slope .

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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

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Example 4: Writing the Slope-Intercept EquationWrite the slope-intercept equation for the line through and .

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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

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Example 5: Analyzing & Graphing a General Linear EquationFind the slope and y-intercept of the line , then graph the line.

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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.

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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: .

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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

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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

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Example 7: Determining a Function

The following table gives values for the linear function . Determine m and b.

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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

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Example 8: Temperature Conversion

Find the relationship between Fahrenheit and Celsius temperature. Then find the Celsius equivalent of and the Fahrenheit equivalent of .

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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

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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 .

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Chapter 1.1 Exercises

• Finney page 9, #1-37 odds, #40, 41, 45, 46, 57

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