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The following images are the graphs that we generated for the functions
Let’s take a moment to analyze the slopes of these functions and the effect it has on their respective graphs.
Special Cases
Special Cases
When the slopes of linear equations are either positive or negative values we get slanted lines. Positive slopes slant upward from left to right and negative slopes slant downward. What if the slope is neither positive nor negative? There are two special cases where slope is not positive or negative, these are zero slope and undefined slope. Zero slope means the line is not increasing or decreasing, it stays flat. When slope is equal to zero the line is horizontal. Consider the equation y = 0x + 2 , this can be simplified to y = 2. Whenever an equation has the format of y equals a number, it is a horizontal line. Another way to think about the equation is the collection of all points with a y–coordinate of two.
{ . . . (–3 , 2) ; (–2 , 2) ; (–1 , 2) ; (0 , 2) ; (1 , 2) ; (5 , 2) . . . }
Notice that all y–coordinates equal two. If you plug any two points into the slope formula the result is a fraction with a zero in the numerator, which simplifies to zero.
Undefined slope means that there is no number that can represent the slope. It is the result of the slope formula such that the denominator equals zero. Zero in the denominator is undefined because division by zero is undefined. You cannot divide by zero because it would unravel all the laws and properties of our number system. For example, consider the mathematical statement 8 ÷ 2 = 4, we can apply the inverse operation of multiplication and say that 4 ⨉ 2 = 8. However, as soon as you define division by zero this inverse property fails. How would you define 8 ÷ 0 = ?
If 8 ÷ 0 = 0, then the multiplicative inverse would yield 0 ⨉ 0 = 8 which is false!
Hence, zero in the denominator is undefined. The graph of a line with undefined slope is a vertical line. A vertical line consists of all points with the same x–coordinate. Consider the equation x = –1. It is the collection of points where many different y–values are paired with x = –1. The following is a list of points with x = –1.
Notice that all x–coordinates equal .
If you plug any two points into the slope formula the result is a fraction with a zero in the denominator, which is undefined.
Undefined slope is often called “No Slope”. A common mistake that students make when considering phrases like “no slope” and “zero slope” is thinking that they are the same. They are not! No slope means there is no number that can represent the slope, where zero is a valid number.
Finally, we can classify horizontal lines as linear functions and vertical lines as linear relations. This is because horizontal lines will pass the vertical line test with exactly one intersection for each input value of the domain, but a vertical line will fail the vertical line test. A vertical line has only one input value in its domain and it gets paired up with infinitely many y–values.
y = a constant is a linear function
x = a constant is a linear relation
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