What do the sign and the value of a slope tell us regarding the nature of the straight line?

Question

What could be the minimum and maximum possible value for a slope of a straight line?

Is it -1 to +1 ?

What do the sign and the value tell us regarding the nature of the straight line?

Answer 1

Slopes can be more than one and less than negative one.

A slope determines how steep a line is and the sign indicates if it's going "uphill" or "downhill".

The lowest absolute slope (the absolute value of a slope) is $0$ which means the line is perfectly horizontal. A very small slope, so $\frac 1{10}$ means it's slightly up hill. (And a slope of $-\frac 1{10}$ means it's slightly downhill.)

A slope of $1$ means it rises just as fast as it goes forward. The slope is at a $45^{\circ}$ angle. (And a slope of $-1$ means it sinks just as fast as it goes forward). This is not a limit. You can have lines that are steeper.

A slope of more than $1$ means it raises faster than it goes forward. A huge slope such ans $1000$ would mean if you go forward $1$ foot you will go up $1000$ feet. (Same for negatives but downhill.)

Now there is a problem. Do you see what it is?

What if the line is perfectly vertical? Then we say the slope is infinite. The equation of such a line can not be written as $y = mx + b$. It must be written as $x = c$. ($y$ can be any point and $x$ will always be $c$.)

For a vertical line with infinite slope, it doesn't make sense to talk of negative or positive or going "uphill" or "downhill". It's simple a straight drop with no forward or backward motion.

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Also. You can write a line if $y=mx +b$ form for any $m$ or $b$. As $m$ can be any number the slope can have any value and if you think of it that way, of course it isn't limited to being between $-1$ and $1$.

The line $y=7x -3$ will be a line that has a steepness that for every one unit you go forward, you will go up $7$ units. That's a slope of $7$.