1. Graphs of
FUNCTIONs
3.2

2. A Function May be Defined by a Graph
• “x” usually the independent or input variable
• “y” usually the dependent or output variable
• A graph is essentially a plot of inputs and their
associated outputs
• A point (x, y) on the graph can also be labeled
as (x, f(x)) [remember y is the same as f(x)]
• An open circle indicates the point is not part
of the graph
• A solid circle indicates the point is part of the
graph

3. Function Features
• Vertical Line Test: If a vertical line intersects
the graph no more than once, it’s a function!
• Increasing function: the graph always rises as
you move from left to right
• Decreasing function: the graph always falls as
you move from left to right
• Constant function: the graph is horizontal

4. Function Features
• Local Maximum/Minimum: Peaks are local
maximums, valleys are local minimums
– TI 83/4: Play w/zoom and/or window size if
necessary
– CALC, 3: minimum or 4: maximum
– Use arrows to select left bound, then right bound
– Find min/max for f(x) = x3 – 1.8x2 + x +1

5. Function Features
• Concave Up: up = cup; if you connect two
points the line segment is above the graph
• Concave Down: down = frown; if you connect
two points the line segment is below the
graph
• Inflection Points: a point where the graph
changes concavity

6. Graphs of Piecewise Functions
• Combine the graphs of the formulas
• Graph first formula as Y1, second part as Y2
• Inequalities found in TEST menu
• Must use proper syntax
Y1 = X2/(X≤1)
Y2 = X + 2/((X>1)(X≤4))
• Calculator display will not show which
endpoints are included or excluded
f(x) = x+2 if 1< x ≤ 4
x2 if x ≤ 1

7. Graph of Absolute Value Function
• f(x) = |x| is a special case piecewise function
• TI 83/4: MATH, NUM, 1:abs(
- x if x < 0
f(x) = |x|= x if x ≥ 0

8. Graph of Greatest Integer Function
• For any number x, round down to the nearest
integer less than or equal to x
• Remember negative numbers round
down, which is left on the number line!
• TI-83/4: MATH, NUM, 5: int(
• Graphs better in DOT mode vs CONNECTED
• Easy to see why it is called a step function
• Open circles are on the right side of each step
f(x) = [x]

9. Parametric Graphing
• x and y are each a function of a third
variable, t, which is called the “parameter”
• The functions for x and y are called parametric
equations
– Note: x and y are functions of t, but y may or may
not be a function of x
• Parametric graph can be thought of as
representing the function
f(t) = (x,y)
where
x = x(t) and y = y(t)
• TI-83/4: Select PAR mode (instead of FUNC)