• When the supply of pineapples goes up does the price go down?
• When medicine dosage goes up does the pain go down?
• When the speed limit goes down does the number of accidents go down?

Our words for this comparison are increase and decrease.

• When the supply of pineapples increases does the price decrease?
• When medicine dosage increases does the pain decrease?
• When the speed limit decreases does the number of accidents decrease?

Graphically, increasing appears as dots to the right being higher than dots to the left. Decreasing appears as dots on the curve being lower to the right.

Increasing and Decreasing

Let $g(k)$ be a function. The graph of $y= g(k)$ is displayed below.

• $g$ is increasing decreasing neither on the interval $(-8,-3)$.
• $g$ is increasing decreasing neither on the interval $[-1,3)$.
• $g$ is increasing decreasing neither on the interval $[4, 8)$.

Decreasing

Let $m(t)$ be a function. The graph of $y = m(t)$ is displayed below.

• $m$ is increasing decreasing neither on the interval $(-8,-3)$.
• $m$ is increasing decreasing neither on the interval $[-3, 8)$.

However, $m$ is not decreasing on the interval $(-8, 8)$. To show that a function is not decreasing on an interval, we merely have to give ONE counterexample. That is we need to come up with ONE pair of numbers $a < b$ in the interval where $f(a) \ngtr f(b)$. Let’s select $a = -4$ and $b = -3$.

$$-4 < -3 \text { but, } f(-4) = 4 \ngtr f(-3) = 6$$

Decreasing

Let $N(z)$ be a function. The graph of $y = N(z)$ is displayed below.

Select all of the correct statements

$N$ is decreasing on the interval $(-3,2)$ $N$ is decreasing on the interval $(2,8)$ $N$ is decreasing on the interval $(-3,8)$ $N$ is decreasing on the interval $[-3,8)$ $N$ is decreasing on the interval $(-3,8]$ $N$ is decreasing on the interval $[-3,8]$

Measuring Rates

Comparing how the connected values in the domain and range change is called a rate. Our symbol for a change in an amount is a Greek Delta, $\Delta$. And, we use fractions to quantify the comparison of changes.

$$rate = \frac {\Delta \, pineapples}{\Delta \, price}$$

• If the changes in pineapples and price are both positive, then this rate is positive.
• If the changes in pineapples and price are both negative, then this rate is again positive.
• If the changes in pineapples and price are of different sign, one increases while the other decreases, then this rate is negative.

Using rates, we can compare and measure changes in function values over a domain interval.

The rate-of-change measures how the function values change relative to changes in the domain.

Rates

Let $V(x)$ be a function. The graph of $y = V(x)$ is displayed below.

From the graph, we can estimate that over the interval $[-7, -5]$, $V$ had a rate-of-change of $\frac {-6 - 0}{-5 - (-7)} = \frac {-6}{2} = -3$.

From the graph, we can estimate that over the interval $[1, 8]$, $V$ had a rate-of-change of $\frac {-8 - (-1)}{8 - 1} = \frac {-7}{7} = -1$.

All of this might sound familiar. You have probably seen this when working with slopes of lines, $\frac {rise}{run}$. The rise is the vertical change (change in the value of the function). The run is the horizontal change (change in the domain).

$\blacktriangleright$ Rate-of-change is a function idea that we will connect up to the geometric idea of slope.

The rate of change over an interval is an overall measurement. It compares the changes occuring over the whole interval. It compares final values to initial values.

Let $M(t)$ be a function. The graph of $y = M(t)$ is displayed below.

The rate of change of $M$ over the interval $[-5,5]$ is

positive negative

The rate of change of $M$ over the interval $[-2,1]$ is

positive negative