Question

I need help understanding rate of change and slope.

Answer 1

"Rate of change" is a measure of how much some dependent variable changes with respect to a change in the independent variable. Given a function
`y(x)`, the average rate of change over some interval
`[a,b]` is given by what's called the difference quotient,

`(y(b)-y(a))/(b-a)`

If
`y(x)` is a linear function, then the average rate of change is constant regardless of the interval chosen.

A line represents a linear function. The slope of the line represents the linear function's rate of change. We pick any two points on the line,
`(a,y(a))` and
`(b,y(b))` where
`a<b`, and the slope of the line through them is exactly the value of the expression above. Then there are 3 possible scenarios:

(1) If
`a\\eq b`, then the slope can be any real number. If
`y(a)=y(b)` happens to be true, then the slope is 0 and the line is horizontal.

(2) If
`a=b` and
`y(a)\\eq y(b)`, the slope is undefined (some might say infinite) and the line is vertical.

(3) If
`a=b` and
`y(a)=y(b)`, then we're talking about just one point. But there are infinitely many possible lines through a single point, so the slope is undefined.

Some examples in practice:

23. The slope of the line through (-5, 0) and (-5, 5) is

`(5-0)/(-5-(-5))`

The
`x`-coordinates match but the
`y`-coordinates don't, so this line is vertical and the slope is undefined (or infinite).

27. The slope is

`(\frac37-\frac47)/(\frac25-\frac15)=(-\frac17)/(\frac15)=-\frac57`

Notice the order in which we plug in the given points' coordinates. Always take
`a` to be the lesser of the two points'
`x`-coordinates! The convention is to always take points left to right.

We use the same principles to work backwards:

31. Given a slope of 1/4 through two points (7, 4) and (3, y), we have

`(4-y)/(7-3)=\frac{4-y}4=\frac14\implies4-y=1\implies y=3`

Answer 2

Rate of Change and Slope are both the same thing. They show the amount of change on the “y-axis” as the “x-axis” moves. I hope it helps... we learned this in class about a month ago.