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PRACTICE WITH SLOPE

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Consider the equation  y=m⁢ x+b .

As shown below, this equation defines a very special relationship between  x  and  y :
equal changes in  x  (the input) give rise to equal changes in  y  (the output).

Indeed, if  x  changes by an amount  Δx  (read aloud as "delta ex" or "change in ex"),
then  y  changes by  mΔx .
That is,  y  changes  m  times as fast as  x .

For example, if  m=2 , then  y  changes twice as fast as  x :
if (say)  x  changes by  3 , then  y  will change by  2(3) = 6 .

For this reason, the graph of  y=m⁢ x+b  is always a line (see sketch below),
and the number  m  gives information about the "steepness" of the line.
This number  m  is called the slope of the line.


PROOF THAT EQUAL CHANGES IN  x  GIVE RISE TO EQUAL CHANGES IN  y 

Let  (x 1,y 1)  be a point on the graph of   y=m⁢ x+b , so that  y1 =m⁢ x1+ b .
Let  x1   change by an amount  Δx , to get a new x-value:  x2 =x1 +Δx  .
Then, the new y-value is:
y2 =mx2+b
    =m(x 1+ Δx)+b
    =mx1+ mΔx+b
    =(mx1+b)+ mΔx
    =y1 +mΔx
Thus,  y  has changed by  mΔx .
That is,  Δy=mΔx , or, equivalently,   m= ΔyΔx  .
This leads us to:

SLOPE OF A LINE
The slope of a line is a number that measures the "steepness" or "slant" of the line.
Given any two different points  (x 1,y 1)  and  (x 2,y 2)  on a line,

slope = y 2-y 1x 2-x 1= change in ychange in  x= Δy Δx  .

Since two points on a horizontal line have the same y-value,
the slope of a horizontal line is  0 Δx=0 .
Thus, horizontal lines have ZERO slope.

Since two points on a vertical line have the same x-value,
an attempt to compute the slope results in   Δy0 , which is not defined, since division by zero is not allowed.
Thus, you cannot compute the slope of a vertical line, and we say that vertical lines have NO slope.

Note that "zero slope" and "no slope" mean two entirely different things!
Horizontal lines have zero slope; vertical lines have no slope.

EXAMPLES:

Question:  Find the slope of the line through  (-1,3)  and  (2,-5) :
Answer:  slope = (-5-3)/(2-(-1)) = (-8)/3 = -8/3
Write your answers as fractions, in simplest form.

Question:  Find the slope of the line through  (2,5)  and  (-7,5) :
Answer:  slope = 0 (horizontal line)

Question:  Find the slope of the line through  (2,5)  and  (2,-7) :
Answer:  no slope (vertical line)

Question:  Horizontal lines have (circle one):    no slope        zero slope
Answer:   zero slope

Question:  Vertical lines have (circle one):    no slope        zero slope
Answer:   no slope

Question:  Suppose a fraction has a zero in the numerator, and a nonzero denominator.
What is the value of the fraction?
Answer:   zero

Question:  Suppose a fraction has a zero in the denominator, and a nonzero numerator.
What is the value of the fraction?
Answer:   it is not defined

Question:  Start at a point  (x,y)  on a line.
To get to a new point, move up 3 and to the right 4. What is the slope of the line?
Answer:  slope = 3/4

Question:  Start at a point  (x,y)  on a line.
To get to a new point, move down 2 and to the left 6. What is the slope of the line?
Answer:  slope = -2/(-6) = 1/3
Write your answers as fractions, in simplest form.

Question:  Start at a point  (x,y)  on a line.
To get to a new point, move straight up 5 units. What is the slope of the line?
Answer:  no slope (vertical line)

Question:  Start at a point  (x,y)  on a line.
To get to a new point, move directly to the left 5 units. What is the slope of the line?
Answer:  slope = 0 (horizontal line)

Question:  Suppose you are walking along a line, moving from left to right. You are going uphill.
Then, the slope of the line is (circle one):    positive        negative
Answer:  positive