2.6 Graphs of Basic Functions

Continuity
Graphs of basic functions.

1. 2.6 Graphs of Basic Functions
Chapter 2 Graphs and Functions

2. Concepts and Objectives
⚫Graphs of Basic Functions
⚫Continuity
⚫Identifying the Identity, Squaring, Cubing, Square
Root, Cube Root, and Absolute Value function graphs
⚫Graphing Piecewise functions, including the Greatest
Integer function

3. Continuity
⚫Roughly speaking, a function is
continuousover an interval of its
domain if its hand-drawn graph
over the interval can be sketched
without lifing the pencil from the
paper.
⚫If a function is not continuous at at
point, then it has a discontinuity
there.
Discontinuity
at (3, 1)

4. Graphs of Basic Functions
⚫Graphs of the basic functions we study can be sketched
by plotting points or by using a program such as
Desmos.
⚫Once you understand the shape of the basic graph, it is
much easier to understand what transformations do to
it.
⚫You should already be familiar with most, if not all, of
these from either Algebra I or Algebra II.

5. Identity Function f(x)= x
⚫f(x)= xis increasing on its entire domain, (–∞∞).
⚫Itiscontinuousonitsentiredomain,(–∞∞)
Domain: (–∞∞) Range: (–∞∞)
x y
–2 –2
–1 –1
0 0
1 1
2 2

6. Squaring Function f(x)= x
2
⚫f(x)= x
2
decreases on the interval (–∞0] and increases
on the interval [0, ∞).
⚫Itiscontinuousonitsentiredomain,(–∞∞)
Domain: (–∞∞) Range: [0∞)
x y
–2 4
–1 1
0 0
1 1
2 4
vertex

7. Cubing Function f(x)= x
3
⚫f(x)= x
3
increases on its entire domain, (–∞∞).
⚫Itiscontinuousonitsentiredomain,(–∞∞)
Domain: (–∞∞) Range: (–∞∞)
x y
–2 –8
–1 –1
0 0
1 1
2 8

8. SquareRootFunction
⚫ increases on its entire domain, [0∞).
⚫Itiscontinuousonitsentiredomain, [0∞)
Domain: [0∞) Range: [0∞)
x y
0 0
1 1
4 2
9 3
16 4()fxx= ()fxx=

9. CubeRootFunction
⚫ increases on its entire domain, (–∞∞).
⚫Itiscontinuousonitsentiredomain, (–∞∞)
Domain: (–∞∞) Range: (–∞∞)
x y
-8 -2
-1 -1
0 0
1 1
8 2()
3
fxx= ()
3
fxx=

10. Absolute Value Function
⚫ decreases on the interval (–∞0] and
increases on the interval [0, ∞).
⚫Itiscontinuousonitsentiredomain, (–∞∞)
Domain: (–∞∞) Range: (–∞∞)
x y
-2 2
-1 1
0 0
1 1
2 2()fxx= ()fxx=

11. Piecewise-Defined Functions
⚫The absolute value function is defined by different rules
over different intervals of its domain. Such functions are
called piecewise-defined functions.
⚫Ifyouaregraphingapiecewisefunctionbyhand,graph
eachpieceoveritsdefinedinterval.Ifnecessary, use
open and closed circles to mark discontinuities.
⚫IfyouareusingDesmostographa piecewise function,
you can control the interval graphed by putting braces
after the function.
⚫Youcanmakeopencirclesbyplottingthepointand
changingthetypeofpointused.

12. Piecewise-Defined Functions
⚫Example: Graph the function.()
25 if 2
1 if 2
xx
fx
xx
−+
=
+

13. Greatest Integer Function
⚫The greatest integer function, , pairs every
real number xwith the greatest integer less than or
equal to x.
⚫Forexample,8.4= 8, –5= –5, = 3, and –6.4= –7.
⚫Ingeneral, if , then()fxx= ()fxx= ()fxx= ()
2 if 21
1 if 10
0 if 01, etc.
1 if 12
2 if 23
x
x
fxx
x
x
−−−

−−


=





14. Greatest Integer Function
⚫ is constant on the intervals …, [–2, –1), [–1, 0),
[0, 1), [1, 2), [2, 3), ….
⚫Itisdiscontinuous at all integers values in itsentire
domain, (–∞∞).
Domain: (–∞∞) Range: {y | y∊ ℤ}
x y
-2 -2
-0.5-1
0 0
1 1
2.5 2()fxx= ()fxx=

15. Greatest Integer Function
⚫TographthisinDesmos,usethe“floor”function.Make
onetableofpointswithclosedcirclesandonetable with
open circles.
⚫Example: Graph ()fxx= ()
1
1
2
fxx=+

16. The Relation x= y
2
⚫This is nota function, but you should see the relation
between it and the graphs of y= x
2
and .
⚫Itiscontinuousonitsentiredomain, [0∞)
Domain: [0∞) Range: (–∞∞)
x y
0 0
1 –1
1 1
4 –2
4 2yx=

17. Classwork
⚫2.6 Assignment (College Algebra)
⚫Page255:2-20;page242:22-26,36,38;
page 227:54-68
⚫2.6ClassworkCheck
⚫Quiz 2.5