11-1 Square-Root Functions

Warm UpWarm Up

Lesson Presentation

California Standards

PreviewPreview

11-1 Square-Root FunctionsWarm Up

Find each square root.

1. 3.

Solve each inequality.

5. x + 5 ≥ 0

7. 0 ≤ 3x

Compare. Write <, >, or =.

9. 7 10. 3

62. 4.

6. 0 ≤ 4x – 8

8. 10 – 3x ≥ 0

12 –20 undefined

x ≥ –5 x ≥ 2

x ≥ 0

< >

11-1 Square-Root Functions

California Standards

Extension of 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.Also covered: 17.0

11-1 Square-Root Functions

square-root function

Vocabulary

11-1 Square-Root Functions

The function gives the speed in feet per second of an object in free fall after falling x feet. This function is different from others you have seen so far. It contains a variable under the square-root sign, .

11-1 Square-Root Functions

11-1 Square-Root FunctionsAdditional Example 1A: Evaluating Square-Root Functions

Write the speed function.

Substitute 16 for x.

Simplify.= 8(4)

= 32

After an object has fallen 16 feet, its speed is 32 ft/s.

The function gives the speed in feet per second of an object in free fall after falling x feet. Find the speed of an object in free fall after it has fallen 16 feet.

11-1 Square-Root Functions

Write the speed function.Substitute 20 for x. Use a

calculator to find the square root.

Simplify.≈ 35.8

Additional Example 1B: Evaluating Square-Root Functions

After an object has fallen 20 feet, its speed is about 35.8 ft/s.

The function gives the speed in feet per second of an object in free fall after falling x feet. Find the speed of an object in free fall after it has fallen 20 feet. Round your answer to the nearest tenth.

11-1 Square-Root Functions

Write the speed function.

Substitute 25 for x.

Simplify.

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

= 8(5)

= 40

After an object has fallen 25 feet, its speed is 40 ft/s.

Check It Out! Example 1a

11-1 Square-Root FunctionsCheck It Out! Example 1b

Write the speed function.

Substitute 15 for x. Use a calculator to find the square root.

Simplify.

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

≈ 30.98

After an object has fallen 15 feet, its speed is about 30.98 ft/s.

11-1 Square-Root Functions

Recall that the square root of a negative number is not a real number. The domain (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.

11-1 Square-Root FunctionsAdditional Example 2: Finding Domain of Square-Root

Functions Find the domain of each square-root function.

x– 4 ≥ 0

x ≥ 4

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

Solve the inequality. Add 4 to both sides.

A.

x + 3 ≥ 0

x ≥ –3

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

Solve the inequality. Subtract 3 from both sides.

B.

11-1 Square-Root Functions

Check It Out! Example 2

Find the domain of .

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

Solve the inequality. Add 1 to both sides.

2x – 1 ≥ 0

2x ≥ 1

Divide both sides by 2.

11-1 Square-Root Functions

The function for is graphed at right. Notice there are no x-values to the left of 0 because the domain is x ≥ 0.

is y ≥ 0.

Remember that the symbol indicates the positive square root only. For this reason the range of

11-1 Square-Root Functions

Additional Example 3: Graphing Square-Root Functions

Graph .

Step 1 Find the domain of the function.

x ≥ 3 The expression under the radical sign must be greater than or equal to 3.

11-1 Square-Root FunctionsAdditional Example 3 Continued

Step 2 Choose x-values greater than or equal to 3 and generate ordered pairs.

Step 3 Plot the points. Then connect them with a smooth curve.

27

1.415

1 4

0 3

x

Graph .

11-1 Square-Root Functions

In Example 3, when generating ordered pairs, choose x-values that make the expression under the radical sign a perfect square.

Helpful Hint

11-1 Square-Root Functions

Check It Out! Example 3

Graph square root function.

Step 1 Find the domain of the function.

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

11-1 Square-Root FunctionsCheck It Out! Example 3 Continued

Graph

Step 2 Choose x-values greater than or equal to 4 and generate ordered pairs.

Step 3 Plot the points. Then connect them with a smooth curve.

28

1.416

1 5

0 4

x

11-1 Square-Root Functions

Lesson Quiz: Part I

1. Use the formula to find the radius of a circle

whose area is 28 in2. Use 3.14 for . Round your

answer to the nearest tenth of an inch.

Find the domain of each square-root function.

3.0 in.

x ≥ 0

x ≥ 5

2.

3.

4.

11-1 Square-Root Functions

Lesson Quiz: Part II

5.

Graph each square-root function.

6.