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Graphing Square and Cube Roots {4.7} Standard: F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Standard: F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Compare and contrast square root, cubed root, and step functions with all other functions.

Standard: F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

Lingo: Radical

Square root

√ Cube root

√ 3

The Parent Fucntions

Sq

ua

re

Ro

ots

𝑓(𝑥) = √𝑥

x Work y (x,y)

0 √𝟎 = 𝟎 0 (0, 0)

1 √𝟏 = 𝟏 1 (1, 1)

4 √𝟒 = 𝟐 2 (4, 2)

9 √𝟗 = 𝟑 3 (9, 3)

Domain: [0, ∞)

Range: [0, ∞)

Cu

be

Ro

ots

𝑓(𝑥) = √𝑥3

x Work y (x,y)

-1 √−𝟏𝟑

= −𝟏 -1 (-1, -1)

0 √𝟎𝟑

= 𝟎 0 (0, 0)

1 √𝟏𝟑

= 𝟏 1 (1, 1)

Domain: (−∞, ∞)

Range: (−∞, ∞)

radical

index

Starting Point

Center Point

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Graphing Square and Cube Roots Cont. {4.7} Transformation Review

𝑓(𝑥 − 𝑘) or √𝑥 − 𝑘

Moves the parent function

OR

*Remember! If it “lies” inside the parenthesis

or “lies” underneath the radical, it ____ lies

____to you.

𝑓(𝑥) + 𝑘 or √𝑥 + 𝑘

Moves the parent function

OR

−𝑓(𝑥) or −√𝑥

over the x-axis

𝑓(−𝑥) or √−𝑥

over the y-axis

Example 1) Given 𝑓(𝑥) = √𝑥 describe the transformation of 𝑓(𝑥 + 3).

Left 3

Example 2) How has 𝑓(𝑥) = √𝑥 − 2 + 10 transformed from the parent function?

Right 2 and up 10

Example 3) Given 𝑓(𝑥) = √𝑥3

describe the transformation of 𝑓(−𝑥) − 8. Reflection over the x-axis and down 8

Example 4) How has 𝑓(𝑥) = −√𝑥 + 53

+ 4 transformed from the parent function? Reflection over the y-axis, left 5, and up 4.

Example 5) Given 𝑓(𝑥) = √𝑥 describe the transformation of 𝑓(𝑥 − 5) − 1.

Right 5 and down 1

Example 6) How has 𝑓(𝑥) = √𝑥3

+ 8 transformed from the parent function? up 8

64

Graphing Square Root and Cube Root Functions

Example 7) Graph 𝑓(𝑥) = √𝑥 − 3 + 2

Steps to Graphing:

Step 1 Identify the transformation

Step 2 Move the x and y-axis up/down or right/left

based on step 1.

Step 3 Identify the points of the parent function

Step 4 If the transformation has a reflection, use the

chart below.

Reflection over x-axis Reflection over y-axis

Switch all the signs of the y-values

Switch all the signs of the x-values

Step 5 Graph the parent function using your “new” x

and y-axis

x y

0 0

1 1

4 2

9 3

What is the domain of the function above?

[3, ∞) What is the range of the function above?

[2, ∞)

Transformation:

3 Right and Up 2

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Graphing Square and Cube Roots Cont. {4.7} Example 8) Graph 𝑔(𝑥) = −√𝑥 + 1 − 2 x y

0 0

1 -1

4 -2

9 -3

Transformation: left 1 and down 2 and reflects over x-

axis

Domain: [−1, ∞)

Range: [−2, ∞)

Example 9) Graph 𝑓(𝑥) = √𝑥 − 13

+ 2 Transformation: right 1 and up 2

x y

-1 -1

0 0

1 1

Domain:(−∞, ∞) or all real

numbers

Range:(−∞, ∞) or all real numbers

Example 10) 𝑓(𝑥 + 1) − 3

a) Describe the transformation that will occur to the parent function 𝑓(𝑥).

Left 1 and down 3

b) Graph the transformed function on the graph to the right using the parent function given.

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Example 11) −𝑓(𝑥 + 3) − 2

c) Describe the transformation that will occur to the parent function 𝑓(𝑥).

Reflect over the x-axis, Left 3 and down 2

Example 12) Graph the transformed function on

the graph to the right using the parent function given.

Example 13) Write an equation for the function 𝑓(𝑥) = √𝑥 by transforming the equation as described below:

Translated to the right 7, up 4, and reflected over the y-axis

𝑔(𝑥) = √−(𝑥 − 7) + 4

Example 14) Write an equation to the graph to the right.

𝑓(𝑥) = √𝑥 − 1 + 1

Example 15) Write an equation to the graph to the right.

𝑓(𝑥) = √𝑥 + 23

+ 4