We use functions to compare information(measurements). One aspect we are interest in is how one measurement changes compared to changes in the other.

• 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 up does the number of accidents go up?

Our words for this comparison are increase and decrease.

• If the price of pineapples goes down when the supply of pineapples goes up, then we say the price is decreasing with respect to the supply.
  
• If the pain goes down when the dosage goes up, then we say that pain decreases with respect to dosage.
  
• If the number of accidents goes up when the speed limit goes up, then we say that accidents increases with respect to speed limit.

Note: You cannot test the endpoints of an interval to show a function is increasing. It has to increase for EVERY pair of numbers in the interval.

Note: You cannot test the endpoints of an interval to show a function is decreasing. It has to decrease for EVERY pair of numbers in the interval.

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.

Or, the reverse....

Increasing also appears as dots to the left being lower than dots to the right.

Decreasing also appears as dots on the curve being higher to the left.

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 uppercase delta, \(\Delta \). And, we use fractions to quantify the comparison of changes.

• 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

Constant Rate of Change

Linear functions are functions that can be represented with formulas of the form \(L(x) = A \, x + B\).

Given an interval, we can measure the change of al inear funciton over this interval.

Let \(L(x)\) be the linear function \(L(x) = \frac {1}{2} x - 4\). The graph of \(y = L(x)\) is displayed below.

The rate of change of \(L\) over the interval \([-6, 0]\) is

\[ \frac {L(0) - L(-6)}{0 - (-6)} = \frac {-4 - (-7)}{6} = \frac {3}{6} = \frac {1}{2} \]

The rate of change of \(L\) over the interval \([-4, 6]\) is

\[ \frac {L(6) - L(-4)}{6 - (-4)} = \frac {-1 - (-6)}{10} = \frac {5}{10} = \frac {1}{2} \]

The rate of change of \(L\) over the interval \([-2, 2]\) is

\[ \frac {L(2) - L(-2)}{2 - (-2)} = \frac {-3 - (-5)}{4} = \frac {2}{4} = \frac {1}{2} \]

The rate of change of \(L\) over the interval \([8, 10]\) is

\[ \frac {L(10) - L(8)}{10 - 8} = \frac {1 - 0}{2} = \frac {1}{2} \]

Linear functions have a special property that other function do not have. No matter what interval you select, the rate of change over that interval is always the same.

Let \(L\) be the linear function in the example above, \(L(x) = \frac {1}{2} x - 4\).

Let \(a\) and \(b\) be any distinct real numbers.

The rate of change of \(L\) over the interval \([a, b]\) is

\(\blacktriangleright \) Linear functions have a constant rate of change, which measures the constant slope of the corresponding line.

We will use this fact to help us understand how function values change.