Alternative University

Math Preliminaries

Graphs and Lines

Rectangle

A rectangle is four lines that are parallel and perpendicular and meet at four corners:

Figure 1: Rectangle

Since the sides are parallel and perpendicular, each of the four corners of a rectangle is a right angle:

Figure 2: Rectangle corners are right angles.

Consequently, opposing sides of a rectangle are equal in length. For example, in the following rectangle, the sides marked A are equal in length to each other, and sides marked B are equal in length to each other:

Figure 3: Opposite sides of a rectangle are equal in length.

A square is a rectangle with all 4 sides equal in length. In the following example, the length of each side of a square is labelled B:

Figure 4: Square

A diagonal of a rectangle is a line connecting opposite corners of the rectangle. Each rectangle has two diagonals, shown as dashed lines in this drawing:

Figure 5: Rectangle diagonals

The two diagonals of a rectangle are equal in length, because they can be shown to be a common side of equal reflected triangles:

Figure 6: Equal reflected triangles.

This property of a rectangle, that the two diagonals are equal length, makes it easier to lay out a rectangle. For example, in construction, when building a rectangle, to make sure it is a rectangle, measure both diagonals and tweak the structure until the diagonals are equal length before bracing:

Figure 7: Measuring a diagonal of a rectangular structure. [Navy]

Graph

For introductory mathematics, we can define a graph to be a plane with rectangular markings to indicate regularly spaced positions on the plane. The positions could be for points, lines, circles, etc.

We refer to such a graph as having a rectangular coordinate system, also referred to as a Cartesian coordinate system (after 17th Century mathematician Rene Descartes).

In this sytem, horizontal distances may be a variable denoted 𝒙, and vertical distances a variable denoted 𝒚.

Figure 8: Rectangular coordinate system. Red lines indicate horizontal distance from Origin (O), and green lines indicate vertical distances from Origin. Point is at (x,y) position (3,2).

Figure 8 shows a square grid coordinate graph with a point at position x = 3 and y = 2. We usually abbreviate such a position as (x,y), in this example (3,2). That point is 3 red lines over from the Origin (x = 3), and 2 green lines up from the Origin (y = 2).

Figure 9: A point positioned at (3,2) is 3 units in the x direction from the Origin, and 2 units in the y direction from the Origin.

Line

A line segment is the set of points that connects two points along the shortest (straightest) distance between the two points:

Figure 10: Line segment from (3,2) to (6,3).

The two points that a line segment connects are called “end points” of the line segment. In this example, the end points are (3,2) and (6,3).

A line includes a line segment, and also includes all points past the end points of the line segment that are “in-line” with the line segment (in the same direction as the line segment), so that any two points on the line are end points for a line segment that is on the line.

Figure 11:  A line through (3,2) and (6,3).

Figure 11 shows a line that contains the points (3,2) and (6,3). A line extends infinitely in both directions, past end points of any line segment on the line.

A snap line (chalk line) is actually a line segment, not a full line, because it has end points (does not extend infinitely). However, snap lines are often split into multiple line segments to make snapping long lines more accurate — in that sense it is appropriate to call it a line since often it includes multiple line segments within the overall “line”.

Figure 12:  Construction worker about to snap a segment of a chalk line (with blue chalk). The worker to his right is holding down endpoint for that line segment with a finger of his left hand. A worker at each end looks down the overall snap line to make sure it stays straight from segment to segment. (Note: The chalk line in this example is not for cutting, but rather will be used to determine how far to push in and fasten this wooden concrete form after it is tilted and lifted into place to the left of this photograph). [Navy]

Slope

Consider a line that contains the points (1,2) and (5,3).

Figure 13:  Line passing through (1,2) and (5,3).

In this graph, the horizontal direction arrow, denoted 𝒙 and pointing right, is called the 𝒙 axis, and the vertical direction arrow, denoted 𝒚 and pointing up, is called the 𝒚 axis.

The 𝒙 axis is said to be at 𝒚 = 0, and the 𝒚 axis is at 𝒙 = 0. The point (1,2) is at 𝒙 = 1 and 𝒚 = 2, and the point (5,3) is at 𝒙 = 5 and 𝒚 = 3.

Each line has a “slope” that specifies its orientation relative to the horizontal and vertical directions. The slope of a line is referred to as “rise over run”, and is the ratio of vertical change relative to horizontal change of the line.

The slope is usually denoted m, and is measured as the vertical difference of two points on the line, divided by the horizontal difference of those two points. This produces a slope that is the same regardless of which two points on the line are used to calculate the slope.

For the line that contains the points (1,2) and (5,3), the slope is (3−2)╱(5−1) which is 1/4 or 0.25. More generally, for a line that contains the points (𝑥₁ , 𝑦₁) and (𝑥₂ , 𝑦₂) the slope (m) is:

m = (𝑦₂ − 𝑦₁) ╱ (𝑥₂ − 𝑥₁)

Lines that are steeper have a higher slope since rise is the numerator of the slope ratio fraction. Just remember that slope is “rise over run”.

Y-Intercept

The y-intercept of a line is the point where it intercepts the y-axis. For this example, the line intercepts the y-axis at y = 1.75, which is the point (0,1.75) because the y-axis is at x = 0.

Figure 14:  Line intercepts the y-axis at y = 1.75

The y-intercept is usually denoted b. In this example, b = 1.75.

Knowing the slope and y-intercept of a line produces the following equation of a line:

y = mx + b

where m is the slope, and b is the y-intercept.

That formula lets you plug in a value of 𝒙 to find out a value of 𝒚.

Plugging x = 5 into that equation produces y = 3.  For x = 4, that equation produces y = 2.75.  For x = 3, y = 2.5, etc.

Other equations of a line can be converted to this slope-intercept form by plugging in x = 0 into those equations to find the y-intercept b, because the y-intercept equals y when x = 0.

For example, here is the two-point form of the equation of a line, for a line that contains the points (𝑥₁ , 𝑦₁) and (𝑥₂ , 𝑦₂) :

𝑦 − 𝑦₁  =  ❲(𝑦₂−𝑦₁)╱(𝑥₂−𝑥₁)❳ · (𝑥−𝑥₁)

For slope m = (𝑦₂−𝑦₁)╱(𝑥₂−𝑥₁) the equation becomes:

𝑦 − 𝑦₁  =  m (𝑥−𝑥₁)

where (𝑥₁ , 𝑦₁) are one of the points used to calculate the slope.

Setting x = 0:

𝑦 − 𝑦₁  =  m (−𝑥₁)

And adding 𝑦₁ to both sides:

𝑦  =  𝑦₁ − m𝑥₁

provides the y-intercept b (the value of y for x = 0).

Using that equation for the line containing points (1,2) and (5,3):

𝑦  
𝑦₁ − m𝑥₁
=  2 − (1/4) · 1
=  2 − (1/4)
=  2 − 0.25
=  1.75

which is the y-intercept of that line.

Exercise

Problem: Try using the other point, (5,3) instead of (1,2), in that equation to find the y-intercept. Is the result the same?

Solution: Yes, the result is the same:

𝑦  
𝑦₂ − m𝑥₂
=  3 − (1/4) · 5
=  3 − (5/4)
=  3 − 1.25
=  1.75

The solution is the same because the slope m is the same for any set of two points on the line, including reversing the order of the points.

Let's see how that works, by calculating the slope first with the points (1,2) and (5,3), then with the points (5,3) and (1,2) — the same points but in reverse order.

For the points (1,2) and (5,3):

m  
=  (𝑦₂ − 𝑦₁) ╱ (𝑥₂ − 𝑥₁)
=  (3 − 2) ╱ (5 − 1)
=  (1) ╱ (4)
=  1 ╱ 4

For the points (5,3) and (1,2):

m  
=  (𝑦₂ − 𝑦₁) ╱ (𝑥₂ − 𝑥₁)
=  (2 − 3) ╱ (1 − 5)
=  (−1) ╱ (−4)
=  (1╱4) · (−1╱−1)
=  (1╱4) · 1
=  1 ╱ 4

Thus, the formula for calculating the y-intercept b of a line is:

𝑏  =  𝑦₁ − m𝑥₁

where m is the slope of the line, and ( 𝑥₁ , 𝑦₁ ) is any point on the line.


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2020–Nov–25  19:20  UTC