Lesson 7-6: Graphing Square Root and Cube Root Functions
Oh yeah! Graphing baby! Our favorite right? <cough> Yeah, right Mr. T.
Okay, I know you’re not fond of graphing…but…please don’t let this throw you. You
already know the basics. All we’re going to do today is learn what the graphs of
f ( x)  x and f ( x)  3 x functions look like and how the basic translations work. You
already know how the translations work. It will be exactly the same as all the graphs
we’ve been working with. You will just apply what you know to some new graphs.
Graphs of the basic radical functions
All we’re going to worry about is graphs for square root f ( x)  x and cube root
f ( x)  3 x functions. The first step is being able to recognize graphs of these two
functions and tell them apart. Here are their graphs…square root on the left, cube
root on the right:
Square root graph f ( x)  x
Cube root graph f ( x)  3 x

It looks like ½ of a parabola on its side.

It looks like a squashed “S”.

It starts at the vertex.

The “twist” happens at the vertex.

Domain: x  0

Domain:

Range: y  0

Range:

Key plotting points:

Key plotting points:
Stepping: Up 1…right 1, 3, 5
… all real numbers
o (-8,-2), (-1,-1), (0, 0), (1,1), (8, 2)
o (0, 0), (1, 1), (4, 2), (9, 3)

… all real numbers

Stepping: Up 1…right 1, 7
Down 1…left 1, 7
So, S shape…cubed root. Half a parabola on its side…square root.
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Lesson 7-6: Graphing Square Root and Cube Root Functions
Translations
The basic equation for a square root function is f ( x)  x  a  b .
The basic equation for a cube root function is f ( x)  3 x  a  b .

What do you guess adding a number outside the radical (b) would do? It’ll
do exactly what it has done to the other functions we’ve played with…move
the vertex up (+b) or down (-b).

What do you guess adding a number inside the radical (a) will do? It’ll move
the vertex left (+a) or right (-a).

What will a negative in front of the radical do? It’ll flip the graph up-side-down.
Matching a function with its graph
Take a look at the following functions. For each, answer the following questions:
1. What is its shape? ½ parabola or squashed S?
2. Where is its vertex?
3. Is it right-side-up, or up-side-down?
Then, using that information, determine which graph goes with which function.
1) f ( x)   3 x  2  1
2) f ( x)  x  3
3) f ( x)  3 x  2  1
4) f ( x)   x  1  2
5) f ( x)   3 x  3
6) f ( x)  x  1  2
1) S, left 2, down 1 → vertex (-2, -1), up-side-down
2) ½ parabola, right 0, down 3 → vertex (0, -3), right-side-up
3) S, right 2, up 1 → vertex (2, 1), right-side-up
4) ½ parabola, left 1, down 2 → vertex (-1, -2), up-side-down
5) S, right 0, down 3 → vertex (0, -3), up-side-down
6) ½ parabola, right 1, up 2 → vertex (1, 2), right-side-up
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Lesson 7-6: Graphing Square Root and Cube Root Functions
a)
b)
c)
d)
e)
f)

Graph a is an S, right-side-up, shifted right and up.

Graph b is a ½ parabola, right-side-up, shifted a little right and more up.

Graph c is a ½ parabola, up-side-down, shifted left and down.

Graph d is a ½ parabola, right-side-up, just shifted down.

Graph e is an S, up-side-down, shifted left and down.

Graph f is an S, up-side-down, just shifted down.
So, using the information we extracted from the functions on the prior page, we can
pair things up as follows:
Function 1 → graph e
Function 2 → graph d
Function 3 → graph a
Function 4 → graph c
Function 5 → graph f
Function 6 → graph b
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Lesson 7-6: Graphing Square Root and Cube Root Functions
Sketching quick graphs
These translations work exactly the same as all the others we’ve done. All you
need to do is be able to quickly sketch out a reasonably accurate graph of either
function.
For our purposes, all I really care about is can you draw the correct basic shape for
the function. I am not looking for a perfect graph with every point correctly and
carefully plotted.
My criteria for grading your sketch of a square root or cube root function are:
1. Did you plot the vertex correctly?

This comes from the left/right, up/down translation shifts.
2. Did you sketch a rough picture of the graph with reasonable accuracy?

Square root function: just draw ½ a parabola going in the right direction.

Cube root function: draw a very squashed, flat S shaped curve.
3. Is your sketch correctly right-side-up or up-side-down?

This comes from the sign in front of the radical.
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