Graphs of Radical Functions
1
The Graph of a Square Root Function
On your calculator, graph y 
x
The graph should look like half
of a sideways parabola with the
vertex at the origin.
In fact, it is a sideways parabola.
If you start with y  x
and square both sides of the
2
equation, you get y  x
The graph of y2 = x is the same as y = x2. They are both parabolas.
2
The only difference is the y2 = x opens right instead of up.
The Graph of a Square Root Function
The graph of y  x does
not include the bottom half of
the parabola. This part is
excluded, so that the graph
will be a function.
What is the domain of this
function?
x  0
What is the range of this
function?
y  0
3
Shifting the Graph of a Square Root Function
The graph of y  x has
a vertex at the origin and
opens right.
What does the graph of y  x  1
look like? Where’s the vertex?
What is the domain of this
function?
x  0
What is the range of this
function?
y 1
The + 1 shifted the graph
vertically up one.
4
Shifting the Graph of a Square Root Function
What does the graph of y  x  5
look like? Where’s the vertex?
What is the domain of this
function?
x  0
What is the range of this
function?
y  5
The - 5 shifted the graph
vertically down five.
5
Shifting the Graph of a Square Root Function
What does the graph of y  x  2
look like? Where’s the vertex?
What is the domain of this
function?
x  2
What is the range of this
function?
y  0
The + 2 shifted the graph
horizontally two to the left.
6
Shifting the Graph of a Square Root Function
What does the graph of y  x  4
look like? Where’s the vertex?
What is the domain of this
function?
x  4
What is the range of this
function?
y  0
The -4 shifted the graph
horizontally four to the right.
7
Shifting the Graph of a Square Root Function
What does the graph of y   x
look like? Where’s the vertex?
What is the domain of this
function?
x  0
What is the range of this
function?
y  0
The minus sign in front
made the graph go down.
8
Shifting the Graph of a Square Root Function
What does the graph of y  2 x
look like? Where’s the vertex?
What is the domain of this
function?
x  0
What is the range of this
function?
y  0
The 2 in front made the
graph go up twice as fast
(made it steeper).
9
Shifting the Graph of a Square Root Function
What does the graph of
y 
1
x
2
look like? Where’s the vertex?
What is the domain of this
function?
x  0
What is the range of this
function?
y  0
The 1/2 in front made the
graph go up half as fast
10
(made it less steep).
Shifting the Graph of a Square Root Function
What does the graph of y 
3 x
look like? Where’s the vertex?
What is the domain of this
function?
x  3
What is the range of this
function?
y  0
The horizontal shift is 3 and
the negative in front of the x
11
makes it open left.
Shifting the Graph of a Square Root Function
What does the graph of y 
2 x
look like? Where’s the vertex?
What is the domain of this
function?
x  2
What is the range of this
function?
y  0
The horizontal shift is 2 to the
left and the negative in front
of the x makes it open left. 12
Vertex Form of a Radical Function
The vertex form of a radical function is:
y  a  x  h  k
The sign of
the
coefficient
makes it go
up or down.
The
coefficient
determines
the steepness
of the graph
(like a slope)
Whatever
makes the
inside of the
radical equal
zero is the
horizontal
shift.
The sign
of the x
makes it
open left
or right.
The k is the
vertical
shift.
13
Vertex Form of all the Functions that you
have learned.
The vertex form of a absolute value function is:
y  a x  h  k
The vertex form of a quadratic function is:
y   ax  h   k
2
The vertex form of a radical function is:
y  a  x  h  k
14
Graph the equation. Then state the
domain and range.
y  2 x7 3
What is the domain of this
function?
x  7
What is the range of this
function?
y 3
15
Graph the equation. Then state the
domain and range.
y  
2
x2 4
3
What is the domain of this
function?
x  2
What is the range of this
function?
y  4
16
Graph the equation. Then state the
domain and range.
y   2 x 3
What is the domain of this
function?
x  2
What is the range of this
function?
y  3
17
Graph the equation. Then state the
domain and range.
y 3  x35
What is the domain of this
function?
x  3
What is the range of this
function?
y  5
18
Graphs of Cube Roots Functions
What does the graph of
y 
3
x look like?
Why does the cube root
function have points on
both sides of the y axis
but the square root
function does not?
Domain:
x
(all real numbers)
Range:
y
19
(all real numbers)
Graph & state the domain and range.
y  2
3
x35
vertex at (3,5)
the coefficient is positive so the
graph goes up
the 2 makes it twice as steep as a
regular graph.
Domain:
x
(all real numbers)
Range:
y
20
(all real numbers)
Graphs of cube root functions
Vertex form of a cube root function is:
y  a
The sign of
the
coefficient
makes it go
up or down.
The
coefficient
determines
the steepness
of the graph
(like a slope)
3
 xh k
Whatever
makes the
inside of the
radical equal
zero is the
horizontal
shift.
The sign of
the
The k is the
coefficient vertical
of x makes shift.
it open left
or right.
Note: the signs of the 2 coefficients may cancel either out.
21
Graph & state the domain and range.
y  
1
3
x4 6
2
vertex at (-4,-6)
the coefficient is negative so the
graph goes down
the 1/2 makes it half as steep as a
regular graph.
Domain:
x
(all real numbers)
Range:
y
(all real numbers)
22
Graph & state the domain and range.
y  
3
5 x 2
vertex at (5,2)
the coefficient is negative so the
graph goes down
the negative in front of the x
makes the direction of the graph
flip left to right.
Domain:
x
(all real numbers)
Range:
y
(all real numbers)
23
y 
y 
x
5
x
Higher Degree Roots
3
y 
y  x
y 
6
x
4
y 
x
7
x
24
Sketch the graph
y 
1
6
x2 3
3
25
Sketch the graph
y  3
19
x5 7
26