CONFIDENTIAL 1

Algebra1Algebra1

Square-RootSquare-RootFunctionsFunctions

CONFIDENTIAL 2

Warm UpWarm Up

1) A = 42,000(1.0125)4t ; $48751.69 2) 1.38g

1) Blake invested $42,000 at a rate of 5% compounded quarterly. Write a function to model this situation. Then find the value of Blake’s investment after 3 years.

2) Lead-209 has a half-life of about 3.25 hours. Find the

amount of lead-209 left from a 230-mg sample after 1 day. Round your answer to the nearest hundredth.

CONFIDENTIAL 3

Square-Root FunctionsSquare-Root Functions

A square-root function is a function whose rule contains a variable under a square-root sign.

EXAMPLES

y = x2

y = 2 . x + 1y = √ 3 x

y = √xy = 2x + 1y = 3 – x - 6 2

NONEXAMPLES

CONFIDENTIAL 4

After an object has fallen 4 feet, its speed is 16 ft/s.

Square-Root FunctionsSquare-Root Functions

A) Find the speed of an object in free fall after it has fallen 4 feet.

y = 8 √x

= 8 √4

= 8 (2)

= 16

Write the speed function.

Substitute 4 for x.

Simplify.

CONFIDENTIAL 5

After an object has fallen 50 feet, its speed is about 56.6 ft/s.

B) Find the speed of an object in free fall after it has fallen 50 feet. Round your answer to the nearest tenth.

y = 8 √x

= 8 √50

≈ 56.6

Write the speed function.

Substitute 50 for x.

Use a calculator.

CONFIDENTIAL 6

1a) Find the speed of an object in free fall after it has fallen 25 feet.

1b) Find the speed of an object in free fall after it has fallen 15 feet. Round your answer to the nearest hundredth.

Now you try!

1a) 40 ft/s

1b) 30.98 ft/s

CONFIDENTIAL 7

Recall that the square root of a negative number is not a real

number. Thedomain (x-values) of a square-

root function is restricted to numbers that

make the value under the radical sign greater than or equal to 0.

CONFIDENTIAL 8

The domain is the set of all real numbers greater than or equal to -4.

Square-Root FunctionsSquare-Root Functions

A) y = x + 4 - 3

x + 4 ≥ 0 The expression under the radical sign must be greater than or equal to 0.

Solve the inequality. Subtract 4 from both sides.

Find the domain of each square-root function.

- 4 - 4

x ≥ -4

CONFIDENTIAL 9

The domain is the set of all real numbers greater than or equal to 2.

3 ( x – 2) ≥ 0 The expression under the radical sign must be greater than or equal to 0.

Solve the inequality. Distribute 3 on the left side.

B) y = 3 ( x – 2)

+ 6 + 4

3x ≥ 6

3x – 6 ≥ 0

x ≥ 2

Add 6 to both sides.

Divide both sides by 3.

CONFIDENTIAL 10

Now you try!

2a) x ≥ 1/2

2b) x ≥ 5/3

Find the domain of each square-root function.

2a) y = 2 x - 1

2b) y = 3x - 5

CONFIDENTIAL 11

The parent function for square-root functions, f (x) = √x , is graphed at right.

Notice there are no x-values to the left of 0 because the domain is x ≥ 0.

CONFIDENTIAL 12

Translations of the Graph of f (x) = √x

If a square-root function is given in one of these forms, you can graph the parent function f(x) = √x

and translate it vertically or horizontally.

CONFIDENTIAL 13

Graphing Square-Root FunctionsGraphing Square-Root Functions

A) Graph f (x) = x – 4

Since this function is in the form f (x) = x - a , you can graph it as a horizontal translation of the graph of f (x) = √x .

Graph f (x) = √x and then shift the graph 4 units to the right.

CONFIDENTIAL 14

B) Graph f (x) = 2x + 3

This is not a horizontal or vertical translation of the graph of f (x) = √ x .

Step1: Find the domain of the function.

2x ≥ 0

x ≥ 0

The expression under the radical sign must be greater than or equal to 0.

Solve the inequality by dividing both sides by 2.

The domain is the set of all real numbers greater than or equal to 0.

CONFIDENTIAL 15

Step2: Choose x-values greater than or equal to 0 and generate ordered pairs.

Step3: Plot the points. Then connect them with a smooth curve.

CONFIDENTIAL 16

Now you try!

Graph each square-root function.

3a) f (x) = √x + 2

3b) f (x) = 2√x + 3

CONFIDENTIAL 17

Assessment

1) Explain why y = x + √3 is not a square-root function.

1 )There is no variable under the square root sign.

2 )16.12 cm

2) In a right triangle, c = a2 + b2 , where c is thelength of the hypotenuse (the longest side) and a and b are the lengths of the other two sides, called the legs.

What is the length of the hypotenuse of a right triangle if its legs measure 14 cm and 8 cm? Round your answer to

the nearest hundredth.

CONFIDENTIAL 18

Find the domain of each square-root function.

3 )x ≥ -6 4 )x ≤ 35 )x ≥ 06 )x ≥ -27 )x ≥ -3

8 )x ≥ 5

3) y = x + 6

4) y = 4 - 3 - x

5) y = 2x - 5

6) y = x + 2

7) y = 3x + 9

8) y = x + x - 5

CONFIDENTIAL 19

Graph each square-root function.

9) y = x - 1

10) y = 2x

CONFIDENTIAL 20

Square-Root FunctionsSquare-Root Functions

A square-root function is a function whose rule contains a variable under a square-root sign.

EXAMPLES

y = x2

y = 2 . x + 1y = √ 3 x

y = √xy = 2x + 1y = 3 – x - 6 2

NONEXAMPLES

Let’s review

CONFIDENTIAL 21

After an object has fallen 4 feet, its speed is 16 ft/s.

Square-Root FunctionsSquare-Root Functions

A) Find the speed of an object in free fall after it has fallen 4 feet.

y = 8 √x

= 8 √4

= 8 (2)

= 16

Write the speed function.

Substitute 4 for x.

Simplify.

CONFIDENTIAL 22

The domain is the set of all real numbers greater than or equal to -4.

Square-Root FunctionsSquare-Root Functions

A) y = x + 4 - 3

x + 4 ≥ 0 The expression under the radical sign must be greater than or equal to 0.

Solve the inequality. Subtract 4 from both sides.

Find the domain of each square-root function.

- 4 - 4

x ≥ -4

CONFIDENTIAL 23

The parent function for square-root functions, f (x) = √x , is graphed at right.

Notice there are no x-values to the left of 0 because the domain is x ≥ 0.

CONFIDENTIAL 24

Translations of the Graph of f (x) = √x

If a square-root function is given in one of these forms, you can graph the parent function f(x) = √x

and translate it vertically or horizontally.

CONFIDENTIAL 25

B) Graph f (x) = 2x + 3

This is not a horizontal or vertical translation of the graph of f (x) = √ x .

Step1: Find the domain of the function.

2x ≥ 0

x ≥ 0

The expression under the radical sign must be greater than or equal to 0.

Solve the inequality by dividing both sides by 2.

The domain is the set of all real numbers greater than or equal to 0.

CONFIDENTIAL 26

Step2: Choose x-values greater than or equal to 0 and generate ordered pairs.

Step3: Plot the points. Then connect them with a smooth curve.

CONFIDENTIAL 27

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