Question 24641
There are lots of ways to remember the meaning of slope.
Say you're climbing up a ramp that goes to the top of a vertical wall.
You get to the top of the wall, and you say "Boy are my legs tired.
My legs are twice as tired as yesterday when I climbed that other ramp,
but I know the wall was the same height as this one."
The slope of this ramp must be greater.
What do I mean by slope? Steepness. If it's really steep, I'm going to
struggle and my legs will get very tired.
OK, mathematically what do I mean by steep?
It's how much the ramp goes up divided by the distance to the wall that
I paced off on the ground. That's steepness.
If you draw it on a graph with x and y coordinates, it's the same thing.
The height of the ramp is:
y at the top of the ramp MINUS y at the bottom of the ramp
The distance to the wall on the ground is:
x at the wall MINUS x where the ramp began
The slope formula is:
{{{(y(top) - y(bottom)) / (x (at_wall) - x(start_of_ramp))}}}
or, the same thing:
{{{(y1 - y0) / (x1 - x0)}}}
Look at the equation you were given. What must the numbers represent?
y1 = 5
y0 = 4
x1 = -1
x0 = -3
m = 1/2
Now you can write the formula this way:
{{{(y - y0) / (x - x0)}}} That's where (x,y) is any old point on the line
(ramp), but everything still holds true.
{{{1/2 = (y - 4) / (x - (-3))}}}
reorder this so it looks like y = mx + b
{{{(y - 4) = (1/2) * ( x - (-3))}}}
{{{y = (1/2) * x + 3/2 + 4}}}
{{{y = (1/2) * x + 11/2}}}
lets check this. The slope is 1/2 That's OK
What about the 11/2?
That's what y is if you make x = 0
It's the point (0, 11/2). If I plug it into the slope formula, I should get 1/2.
(0, 11/2) is my new (x1, y1)
{{{(11/2 - 4) / (0 - (-3))}}}
{{{(3/2) / 3}}}
this equals 1/2, so all is well
I hope you're OK with all of this