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(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. 
       
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.  
graph
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

 

graph
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. 
graph
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. 
graph
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. 
graph
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. 
graph
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. 
graph

This equation is not a function of x or y but can be written as two functions or

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
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
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 
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. 
graph
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
has positive slope when x>-2 and x<0 slope is positive and decreasing as x changes from -2 to 0
has zero slope when x = 0 slope is zero when x = 0
has negative slope  when x>0 and x<2 slope is negative and decreasing as x changes from 0 to 2
has undefined slope when x = -2 or x = 2 slope is undefined at x equals -2 and 2
IX. 
graph
 
This equation is not a function of x or y but can be written as two functions or equation
equation is positive  for all values of x between -2 and 2 inclusive y is positive when x is between -2 and 2 equationis negative for all values of x between -2 and 2 inclusive y is negative when x is between -2 and 2
equationhas positive slope when x>-2 and x<0 slope is positive and decreasing as x increases from -2 to 0 equationhas negative slope when x>-2 and x<0 slope is negative and increasing as x increases from -2 to 0
equationhas zero slope when x = 0 slope is zero when x = 0 equationhas zero slope when x = 0 slope is zero when x is zero
equationhas negative slope  when x>0 and x<2 slope is negative and decreasing as x increases from 0 to 2 equationhas positive slope  when x>0 and x<2 slope is positive and increasing as x increases from 0 to 2
equationhas undefined slope when x = -2 or x = 2 slope is undefined at x equals -2 and +2 equationhas undefined slope when x = -2 or x = 2 slope is undefined at x equals -2 and +2

 

X. 
graph
This equation is not a function of x or y but can be written as two functions or
 equationis 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. equationis 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
equationhas negative slope when x<-2 slope is negative and decreasing as x increases from negative infinity to -2 equationhas positive slope   when x<-2 slope is positive and increasing as x increase from negative infinity to -2
equation has positive slope when x>2 slope is positive and decreasing as x increases from 2 to infinity equationhas negative slope when x>2 slope is native and increases as x increases from 2 to infinity
equationhas undefined slope when x = -2 or x = 2 slope is undefined at x equal -2 or 2 equationhas undefined slope when x = -2 or x = 2 slope is undefined at x equal -2 or 2

 

XI. 
graph
 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

never equals zero  for any values of x y is never zero or negative

 

XII. 
graph
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
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.

Table of Contents

Section 7

Answer Section

Section 9

 

This page was modified on 03/21/16

Rahn, 2000