This document discusses graphing functions. It begins by introducing common functions like linear, quadratic, cubic, rational, absolute value, square root, and cube root functions. Examples of their graphs are shown. It then discusses using the vertical line test to determine if something is a function. Piecewise functions are introduced next along with steps for graphing them which include graphing each piece and indicating open and closed points. Examples of piecewise functions are given and discussed. The document concludes by noting T-tables can be used to graph but plotting each piece of piecewise functions is more accurate.

1. Math 1000
Stuart Jones
Section2.2
Graphs of Functions

2. Math 1000
Stuart Jones
This section begins our focus over the next several sections of
graphing. We are going to skip the basic graphing technique
using T-tables, as it is assumed everyone knows how to graph
using a T-table. We will then begin our study of graphs with
the common graphs you need to know.

3. Math 1000
Stuart Jones
Linear Function: y = x
−5 −4 −3 −2 −1 1 2 3 4 5
−5
−4
−3
−2
−1
1
2
3
4
5

4. Math 1000
Stuart Jones
Quadratic Function: y = x2 −5 −4 −3 −2 −1 1 2 3 4 5
5
10
15
20
25

5. Math 1000
Stuart Jones
Cubic Function: y = x3
−3 −2 −1 1 2 3
−27
−18
−9
9
18
27

6. Math 1000
Stuart Jones
Rational Function: y = 1
x
−5 −4 −3 −2 −1 1 2 3 4 5
−2
−1
1
2

7. Math 1000
Stuart Jones
Absolute Value Function: y=—x—
−5 −4 −3 −2 −1 1 2 3 4 5
1
2
3
4
5

8. Math 1000
Stuart Jones
Square Root Function: y =
√
x
4 8 12 16
1
2
3
4

9. Math 1000
Stuart Jones
Cube Root Function: y = 3
√
x
−10 −5 5 10
−2
−1
1
2

10. Math 1000
Stuart Jones
We can test to see if a function is a function using the vertical
line test. You can check that all of the graphs shown previously
in this section are functions. Can you draw some graphs that
are not functions?

11. Math 1000
Stuart Jones
To Be Continued...Really! The rest of these slides will be done
AFTER we complete 2.6, which we will be doing next. (Yes,
we’re skipping ahead.) Then, we will come back and graph
piecewise functions.

12. Math 1000
Stuart Jones
Steps for graphing a Piecewise Function:
1 Graph the first piece.
2 Erase where the first piece doesn’t apply.
3 Graph the second piece. Etc.
4 Place your open holes/closed dots.

13. Math 1000
Stuart Jones
Graph the following:
f (x) =
2x + 1 x ≤ −1
−x − 4 x > −1

14. Math 1000
Stuart Jones
Graph the following:
f (x) =



−x − 1 x ≤ 0
x2 + 1 0 < x ≤ 2
4 x > 2

15. Math 1000
Stuart Jones
Graph the following:
f (x) =
−x3 x < 0
√
x x ≥ 0

16. Math 1000
Stuart Jones
The Bottom Line
You can use a T-table to primitively graph functions. This
is somewhat inaccurate and slow, though.
Piecewise functions can be plotted by plotting each graph
separately, erasing areas where its rules don’t apply, then
plotting the open/closed points.
Open/closed circles will always come in pairs. But
sometimes, the piecewise function will join together and
not have jumps.