Question

I need help understanding rate of change and slope.

Answer 1

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.

Answer 2

"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,

$\dfrac{y(b)-y(a)}{b-a}$

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 $a$, and the slope of the line through them is exactly the value of the expression above. Then there are 3 possible scenarios:

(1) If $a\neq b$, 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 $y(a)\neq y(b)$, 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

$\dfrac{5-0}{-5-(-5)}$

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

$\dfrac{\frac37-\frac47}{\frac25-\frac15}=\dfrac{-\frac17}{\frac15}=-\dfrac57$

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

$\dfrac{4-y}{7-3}=\dfrac{4-y}4=\dfrac14\implies4-y=1\implies y=3$