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MAT091 :: Lecture Note :: Week 13
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### Increasing and Decreasing Functions

Functions are sometimes categorized as either increasing or decreasing.

An increasing function is a function whose output values increase (i.e. get larger) when the inputs increase.

```   f(0) = 5
f(1) = 10
f(2) = 15
f(3) = 20

f(x) appears to be an increasing function
```

A decreasing function is a function whose output values decrease (i.e. get smaller) when the inputs increase.

```   g(0) = 100
g(1) = 90
g(2) = 80
g(3) = 70

g(x) appears to be a decreasing function
```

A constant function `f(x) = k` (where 'k' is a constant) is neither increasing nor decreasing.

The identity function `f(x) = x` is an increasing function.

Some functions can be both increasing and decreasing. For example, over an interval A, a function might be increasing, while it decreases over an interval B. Review the absolute value function.

```   abs(x) is decreasing from (-INF, 0)
abs(x) is zero at 0
abs(x) is increasing from (0, INF)
```

Technical definitions for increasing and decreasing functions.

```   increasing function:
====================
Function f(x) increases on an interval I if f(b) > f(a)
for all b > a, where a,b are in I.

decreasing function:
====================
Function f(x) decreases on an interval I if f(b) < f(a)
for all b > a, where a,b are in I.
```

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### Linear Functions

M-W.com defines linear as follows.

```   "of, relating to, resembling, or having a graph that
is a line and especially a straight line"

[...and...]

"having or being a response or output that is directly
proportional to the input"
```

Observe that the word linear contains the word line.

A linear function is a function is a "first degree polynomial function of one variable. These functions are called 'linear' because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line."

The following are equations for a line.

```   slope-intercept form:  y = mx + b
... used when slope and y-intercept are known

point-slope form:  y - y1 = m(x - x1)
... where (x1, y1) is a point on the line having slope m

standard form:  Ax + By = C
```

Slope is another way of saying "rate of change" and linear functions have a constant rate of change.

Slope is often described as the ratio "rise over run."

Linear functions with a positive constant rate of change are increasing functions. Linear functions with a negative constant rate of change are decreasing functions.

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### Slope (Linear Functions)

The slope of a line is a measure of its average rate of change (steepness?). The slope also indicates if the line is increasing (uphill?) or decreasing (downhill?). In linear equations, slope is represented by the letter m.

The following is how to find the slope between two unique points on a line.

```   point 1:  (x1, y1)
point 2:  (x2, y2)

slope = m = (y2 - y1) / (x2 - x1)

note:  x1 ≠ x2
```

Slope represents the ratio of change-in-output over change-in-input.

```   m = Δy / Δx    (change in output / change in input)

Δ is the "delta" character
Δy = y2 - y1
```

Slope is often described as the ratio "rise over run."

```
rise
m = ------
run
```

Some slope notes.

```   horizonal lines have slope 0   (y = f(x) = k, where k is a consant)
vertical lines have undefined slope   (x = k, where k is a consant)
parallel lines have equal slopes     (m1 = m2)
perpendicular lines have negative inverse slopes  (m2 = -1/m1)
```

The following is a verbose form the slope-intercept equation.

```   output = slope * input + initial_value_if_any

...or...

output = average_rate_of_change * input + initial_value_if_any
```

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### Intercepts (Linear Functions)

We frequently need to know the points where lines (graphs in general) intersect (cross) the horizonal-axis and vertical-axis (often called the x-intercept and y-intercept, respectively).

The vertical-intercept for a function is found by evaluating the function when its input is zero. The orderd-pair for a vertical-intercept will be `(0, something)`.

The horizontal-intercept for a function is found by setting its output to zero and finding what input value results in that output. The orderd-pair for a horizonal-intercept will be `(something, 0)`.

```   f(x) = 5(x) + 3

vertical-intercept (y-intercept)...
set the input to 0 and evaluate:  f(0) = 5(0) + 3 = 3
vertical-intercept is (0, 3)

horizontal-intercept (x-intercept)...
set the output to 0 and solve:  0 = 5(x) + 3
-3 = 5(x)
-3/5 = x
horizontal-intercept is (-3/5, 0)
```

Generalizations.

```   given  f(x) = m(x) + b

vertical-intercept:  (0, b)        [notice input is 0]
horizontal-intercept:  (-b/m, 0)     [notice output is 0]
```

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