Radical Functions and
Equations
radical
f x  x
index
or
3
x or
4
x , etc.
radicand
L. Waihman
2002
A radical function is a function that has
a variable in the radicand.
y x
y3x
y4 x
You can apply the same transformations
to the graphs of radical functions as
you can to polynomial functions.
y  x 1
y  x 3
Horizontal shift left one
Vertical shift down three
y2 x
Vertical stretch two.
Radical Parent functions
y x
 0, 0 
1,1
 4, 2 
 9,3
As x, f(x)
As x0+, f(x)0
y3x
 8, 2 
 1, 1
 0,0 
1,1
8, 2 
As x, f(x)
As x–, f(x)–
Click on each graph above to link to interactive page!
y4x
 0,0 
1,1
16, 2 
 81,3
As x, f(x)
As x0+, f(x)0
Transformations
We apply the transformations on these functions in the same
manner as we did with the polynomial functions.
y 3 x
y  x 2
Vertical stretch of 3
Vertical shift up 2
Domain: 0, 
Domain:
Range 0, 
Range:
0, 
2, 
y  x 3
Horizontal shift right 3
3, 
Range: 0, 
Domain:
Applying the transformations
f  x  3 x 1 1
Vertical stretch 3
Horizontal shift left 1
Vertical shift down 1
Domain:  1,   Range:  1,  
As x , f(x) ;
As x -1+, f(x) -1
f ( x )   12 x  2  4
Reflect @ x-axis
Vertical shrink of ½
Horizontal shift right 2
Vertical shift up 4
Domain: 2, 
Range:  ,4
As x , f(x) - ;
As x 2+, f(x) 4
Applying the transformations
f  x   33 x  2  4
Vertical stretch of 3
Horizontal shift left 2
Vertical shift down 4
Domain:  ,   Range:  ,  
As x , f(x)  ;
As x - , f(x) - 
g  x    12 3 x  1  2
Reflect @ x-axis
Vertical shrink of ½
Horizontal shift right 1
Vertical shift up 2
Domain:  ,   Range:  ,  
As x , f(x) -  ;
As x - , f(x) 
Radical Equations
•To solve a radical equation that has only one variable
in the radicand, isolate that term on one side of the
equation. If the index is 2, then square both sides of the
equation.
Given: x  6  x  14
x  6  14  x
Isolate the radical.
Square both sides.  x  6   14  x   x  6  196  28x  x
Simplify and set equal to zero. x  29x  190  0
Factor.
 x  19 x  10  0
x  19 or x  10
Solve.
2
2
2
2
Be careful! The new equation you created when you
Squared both sides might have extraneous solutions!
Solving Radical Equations
These solutions may not be solutions to the
original equation.
Check your solutions!
x  6  x  14
?
19  6  19  14
24  14
x  6  x  14
?
10  6  10  14
14  14 √
10
The graph below illustrates that
the derived equation may be
different from the original
equation.
Try:
x  10  x  12
x  10  x  12
x  10  x2  24 x  144
Check:
?
14  10  14  12
0  ( x  11)( x  14)
12  12
x  11 or x  14
x  10  x  2
11  10  11  12
10  12
0  x2  25x  154
Try:
?
14
Check
?
1  10  (1)  2
3  (1)  2
x  10  x  2
x  10  x 2  4 x  4
0  x2  5x  6
?
6  10  6  2
0   x  6  x  1
x  6 or x  1
4  6  2
6
More Solving Radical Equations
A radical equation may contain two radical
expressions with an index of 2.
To solve these, rewrite the equation with one
of the radicals isolated on one side of the
equals sign.
Then, square both sides.
If a variable remains in a radicand, you must
repeat the squaring process.
More Solving Radical Equations
Try:
x  8  x  2
Isolate one radical on each side of equals sign.
x  8  2  x
Square both sides.

x 8
 
2
 2  x

2
 x 8  4  4 x  x
Collect like terms on each side of equals sign.
12  4 x
Simplify. 3  x
Square both sides again.
3
2

 x
9x
2
Check:
x  8  x  2
?
98  9   2
2   2
9
More Solving Radical Equations
Try:
Check:
x  4  5x  8
x  4   5x  8
x  4  5x  16 5x  64
4 x  60  16 5 x
x  15  4 5 x
x2  30 x  225  80 x
x2  50 x  225  0
 x  5 x  45  0
x  5 or x  45
?
45  4  5(45) 8
5  4  5(5) 8
7  15  8
35  8
?
5
√
Radical equations with indexes greater
than 2 can be solved using similar
techniques.
After isolating the term containing the
radical, raise each side of the equation to the
power equal to the index of the radical.
3

3x  2  2  0
3
3x  2  2
3
3
3 x  2  23
Check:

22  0
3x  2  8
3x  6
x2
3 2   2  20
?
3
8  2 0
?
3
2
√
Bonus Questions!
One type of transformation was not covered in
this PowerPoint. Identify it and give and
example of an equation with this type of
transformation and then sketch its graph to
illustrate the transformation. Identify the domain
and range as well. click
Solve the equation:
1 x 
3
 x  1 x  13
1 x 
1  x 
3
3
 x  1 x  13


3
 x  1 x  13 
3
1  3 x  3 x  x   x  1 x  13
2
3
1  3 x  3 x 2  x 3  x 2  14 x  13
 x  2 x  11x  12  0
3
2
x  2 x  11x  12  0
3
2
 x  3 x  1 x  4   0
3,1, 4