(Section 8)
A student preparing to take calculus should easily recognize, recall, and sketch many basic equations. This could be called having a toolkit of functions. The functions that should be in the toolkit are listed below.
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Click on any of the functions to view a graph of that function. In addition students should also remember some of the special characteristics about the function's graph. Some of these characteristics are listed below each graph. Study what the graphs are showing and what the values in the table are saying about the graph.
I.
| Function is positive (y>0) | when x >0 | y is positive when x is positive |
| Function is negative (y<0) | when x<0 | y is negative when x is negative |
| Function equal to zero | when x = 0 | y is zero when x is zero |
| Slope of function | constant (always equal to 1) | slope is constant--always 1 |
II.
or
| Function is positive (y>0) | when x >0 or x <0 |
y values are positive for any value of x no equal to zero. |
| Function equal to zero | when x = 0 | y equals 0 only when x equals zero |
| Slope of function | constant when x <0 (always equal to -1) and constant when x>0 (always equal to +1) | two different slopes |
III.
| Function is positive (y>0) | when x >0 or x<0 |
y is positive when x is negative |
| Function equal to zero | when x = 0 | y equals zero when x equals zero |
| Slope is negative and changing | when x<0 | slope is negative and changing when x is negative-- As x values increase, the slope increases |
| Slope is zero | when x = 0 | slope is zero when x is zero |
| Slope is positive and changing | when x>0 | slope is positive and changing when x is positive-- As x increases the slope increases |
IV.
| Function is positive (y>0) | when x >0 | y is positive when x is positive |
| Function is negative (y<0) | when x<0 | y is negative when x is negative |
| Function equal to zero | when x = 0 | y is zero when x is zero |
| Slope is always positive | when x is not equal to zero | slope is positive and
changing when x is negative--As x increases the slope
increases
slope is positive and changing when x is positive--As x increases the slope increases |
| Slope is zero | when x = 0 | slope is zero when x is zero |
V.
| Function is positive (y>0) | when x >0 | y is positive when x is
positive
y values are increasing as x values increase |
| Function is negative (y<0) | when x<0 | y is negative when x is
negative
y values are increasing as x values increase |
| Function equal to zero | when x = 0 | y is zero when x is zero |
| Slope is always positive | when x is not equal to zero | slope is positive and
increasing as x values are negative and increasing
slope is positive and decreasing when x is positive and increasing |
| Slope is undefined | when x = 0 | slope is undefined at (0,0) or you can draw a tangent line with no slope at the point (0,0) |
VI.
| Function is positive (y>0) | when x >0 | y is positive when x is positive |
| Function is negative (y<0) | when x<0 | y is negative when x is negative |
| Function is not defined | when x = 0 | y is undefined when x is zero |
| Slope is negative and changing | when x does not equal zero | slope is negative and
decreasing as x is negative and increasing
slope is negative and increasing when x is positive and increasing |
VII.
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This equation is not a
function of x or y but can be written as two functions
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is positive |
for all values of x between -2 and 2 inclusive | y is positive for all x between -2 and 2 inclusive |
 is
negative |
for all values of x between -2 and 2 inclusive | y is negative for all x between -2 and 2 inclusive |
|
has
positive slope |
when x>-2 and x<0 | slope is positive and decreasing as x goes from -2 to 0 |
has
negative slope |
when x>-2 and x<0 | slope is negative and increasing as x goes from -2 to 0 |
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has
zero slope |
when x = 0 | slope is zero when x is zero |
has
zero slope |
when x = 0 | slope is zero when x is zero |
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has
negative slope |
when x>0 and x<2 | slope is negative and decreasing as x goes from 0 to 2 |
has
positive slope |
when x>0 and x<2 | slope is positive and increasing as x goes from 0 to 2 |
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has
undefined slope |
when x = -2 or x = 2 |
Slope is undefined when x is -2 or 2 |
has
undefined slope |
when x = -2 or x = 2 | slope is undefined when x is -2 or 2 |
VIII.
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is positive |
for all values of x between -2 and 2 inclusive | y is positive when x is between -2 and 2 |
|
is
zero |
for x = -2 and x = +2 | y is zero when x equals -2 and -2 |
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has
positive slope |
when x>-2 and x<0 | slope is positive and decreasing as x changes from -2 to 0 |
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has
zero slope |
when x = 0 | slope is zero when x = 0 |
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has
negative slope |
when x>0 and x<2 | slope is negative and decreasing as x changes from 0 to 2 |
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has
undefined slope |
when x = -2 or x = 2 |
slope is undefined at x equals -2 and 2 |
IX.
| This equation is not a
function of x or y but can be written as two functions
or
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is positive |
for all values of x between -2 and 2 inclusive | y is positive when x is between -2 and 2 |
is
negative |
for all values of x between -2 and 2 inclusive | y is negative when x is between -2 and 2 |
|
has
positive slope |
when x>-2 and x<0 | slope is positive and decreasing as x increases from -2 to 0 |
has
negative slope |
when x>-2 and x<0 | slope is negative and increasing as x increases from -2 to 0 |
|
has
zero slope |
when x = 0 | slope is zero when x = 0 |
has
zero slope |
when x = 0 | slope is zero when x is zero |
|
has
negative slope |
when x>0 and x<2 | slope is negative and decreasing as x increases from 0 to 2 |
has
positive slope |
when x>0 and x<2 | slope is positive and increasing as x increases from 0 to 2 |
|
has
undefined slope |
when x = -2 or x = 2 |
slope is undefined at x equals -2 and +2 |
has
undefined slope |
when x = -2 or x = 2 | slope is undefined at x equals -2 and +2 |
X.
This equation is not a
function of x or y but can be written as two functions
or
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| is
positive |
for all values of x less than -2 or all x values greater than 2 | y is positive when x is greater than 2 or x is less than -2. |
is
negative |
for all values of x between -2 and 2 inclusive | y is negative when x is greater than 2 or x is less than -2 |
|
has
negative slope |
when x<-2 | slope is negative and decreasing as x increases from negative infinity to -2 |
has
positive slope |
when x<-2 | slope is positive and increasing as x increase from negative infinity to -2 |
|
has positive slope |
when x>2 | slope is positive and decreasing as x increases from 2 to infinity |
has
negative slope |
when x>2 | slope is native and increases as x increases from 2 to infinity |
|
has
undefined slope |
when x = -2 or x = 2 | slope is undefined at x equal -2 or 2 |
has
undefined slope |
when x = -2 or x = 2 | slope is undefined at x equal -2 or 2 |
XI.
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is positive |
for all values of x | y is positive for all x values |
|
has
positive slope |
for all values of x | slope is positive and
increasing as x increases--
slope approaches zero as x approaches negative infinity |
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never equals zero |
for any values of x | y is never zero or negative |
XII.
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is positive |
for all values of x>1 | y is positive when x is greater than 1 |
|
is
negative |
for all values of x>0 and X<1 | y is negative when x is less than 1 |
|
is
equal to zero |
for x=1 | y is zero when x is 1 |
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is positive slope |
for any values of x>0 | slope is positive for
all x
slope is approaching zero as x approaches positive infinity |
Note samples VII, IX, and X are not functions.
Recall that a function is a rule that assigns each number x to exactly one number y.
Samples VII, IX, and X are examples of equations which are just relations.
A set of points which does not satisfy the conditions of a function is called a relation.
Some of the functions above intersect the x-axis. The value of x where this occurs is called a zero of the function.
The zero for function I is x = 0.
The zero for function XII is x = 1.
Students entering a study of calculus should be familiar with each of the above graphs. They should know the general shape of each function and the x and y intercepts.
Check Your Understanding: (Remember to write down the answers to each question.)
1. Use your calculator to create each of the graphs I-XII. You should be able to obtain each of the graphs using one equation except for VII, IX, and X . These three graphs will require two or more equations. Write down the equation(s) which create each graph.
This page was modified on 05/06/09
© Rahn, 2000