Graphs of FUNCTIONs

3.2

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

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

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

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

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 ≤ 4x2 if x ≤ 1

Graph of Absolute Value Function

• f(x) = |x| is a special case piecewise function

• TI 83/4: MATH, NUM, 1:abs(

- x if x < 0f(x) = |x|= x if x ≥ 0

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]

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 functionf(t) = (x,y)

wherex = x(t) and y = y(t)

• TI-83/4: Select PAR mode (instead of FUNC)

Parametric Graphing• To graph y = f(x) in parametric mode, let

x = ty = f(t)

• To graph x = f(y) in parametric mode, letx = f(t)

y = t• Graph the curve given by

x = 2t + 1y = t2 + 3

• Graph the following equations in parametric modea. y = ()2 - 3 b. x = y2 -3y +1