Functions and Graphs
STUDY TIPS
Domain : values of x for which the function is defined.
What to do?
Look for x values!
We don’t want x values that makes the function not defined.

Pay attention to square roots – no square root of negative numbers!

Denominators - if f ( x) 

No log of negative numbers!
g ( x)
, must exclude values of x for which h(x) = 0.
h( x )
Range : values that the function can take.
What to do?
Look for y values!
Remember the standard graphs & their main features such as x &
y intercepts, maximum or minimum values, asymptotes.
x-intercepts
Sketching graphs using transformation
First identify the standard graph f(x) involved.
Reflection about the y-axis: y = f(-x)
3
( x 2)
1
2
2
1
(  x 2)
2
1
10
5
0
5
10
x
Graph of
x  2 on the right and
 x  2 on the left.

Any initial x intercept, x = a is reflected to the point x = –a.

Any vertical asymptote at the line x = b is reflected to the line x = -b.
Reflection about the x-axis: y = -f(x)
4
2
1
( x 2)
2
1
 ( x 2)
4
2
0
2
4
2
2
4
x
Graph of
x  2 above the x-axis and  x  2 below the x-axis.

Any initial y intercept, y= c is reflected to the point y = –c.

Any horizontal asymptote at the line y = b is reflected to the line y = -b.
X shift :
y = f(x-a)
5
ln( x)
ln( x 2)
0
1
2
3
4
5
x
Graph of ln x on the left and ln( x  2) on the right.

The whole graph slides to the right or left.

In the example above, it is a shift to the right because a is positive.

If a is negative, it is shift to the left.

Don’t forget to move everything with f, a units to the right or right: including
any x-intercepts, asymptotes etc.
Y Stretch :
y = k * f(x)
Graph is stretched along the y-axis.
The whole function
is multiplied by k.
5
ln( x)
5 ln( x)
0
0.5 ln( x)
1
2
3
4
5
x
FOR MULTIPLE TRANSFORMATIONS, DO ONE AT A TIME!.
Make sure you determine the correct initial standard function involved.
For this, try to write the final function in terms of the initial function.
For example, if
f ( x)  log( 1  2 x)  1
Start with
1. g ( x)  log( 2 x)

The shape of this graph is like that of log(x) with different x intercept. Find it
first! When log( 2 x)  0 , x 
1
.
2

Vertical asymptote at x = 0.
5
log( 2x)
0
1
2
3
4
5
x
2. g ( x)  log( 2 x) means reflection
about the y-axis.
1
2

x-intercept of this graph is at x  

Remember the y-axis is still a vertical asymptote.
5
log(  2x)
4
2
0
5
1x
1
3. But log( 1  2 x)  log[ 2( x  )] means that moving the graph above
unit to the
2
2
right.
1 1
  0 now!
2 2

The x-intercept is at 

The vertical asymptote is also moved
x
1
.
2
1
unit to the right as well. Now at
2
5
log( 12x)
2
0
2
5
x
4. Finally,
f ( x)  log( 1  2 x)  1
shifts the graph 1 unit upwards cutting the y-axis at y = 1.
5
log( 12x)  1
2
0
5
x
2