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Understanding Square-Root Functions: Graphing and Domain

Learn about square-root functions, evaluate speeds of falling objects, find domains, and graph functions. Engage with interactive examples and practice solving inequalities.

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Understanding Square-Root Functions: Graphing and Domain

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  1. 11-5 Square-Root Functions Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1

  2. Warm Up Find each square root. 1. 3. Solve each inequality. 5. x + 5 ≥ 0 7. 0 ≤ 3x 2. 6 12 4. –20 undefined x ≥ 2 6. 0 ≤ 4x – 8 x ≥ –5 8. 10 – 3x ≥ 0 x ≥ 0

  3. Warm Up Continued Compare. Write <, >, or =. 9. 7 10. 3 < >

  4. Objectives Identify square-root functions and their domains and ranges. Graph square-root functions.

  5. Vocabulary square-root function

  6. 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, .

  7. The function gives the speed in feet per second of an object in free fall after falling x feet. Example 1A: Evaluating Square-Root Functions Find the speed of an object in free fall after it has fallen 16 feet. Write the speed function. Substitute 16 for x. = 8(4) Simplify. = 32 After an object has fallen 16 feet, its speed is 32 ft/s.

  8. The function gives the speed in feet per second of an object in free fall after falling x feet. Example 1B: Evaluating Square-Root Functions Find the speed of an object in free fall after it has fallen 20 feet. Round your answer to the nearest tenth. Write the speed function. Substitute 20 for x. Use a calculator to find the square root. = 8(4.47) Simplify. ≈ 35.8 After an object has fallen 20 feet, its speed is about 35.8 ft/s.

  9. The function gives the speed in feet per second of an object in free fall after falling x feet. Check It Out! Example 1a Find the speed of an object in free fall after it has fallen 25 feet. Write the speed function. Substitute 25 for x. = 8(5) Simplify. = 40 After an object has fallen 25 feet, its speed is 40 ft/s.

  10. The function gives the speed in feet per second of an object in free fall after falling x feet. Check It Out! Example 1b Find the speed of an object in free fall after it has fallen 15 feet. Round your answer to the nearest hundredth. Write the speed function. Substitute 15 for x. Use a calculator to find the square root. = 8(3.87) ≈ 30.98 Simplify. After an object has fallen 15 feet, its speed is about 30.98 ft/s.

  11. 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.

  12. + 4 + 4 x ≥ 4 Example 2A: Finding the Domain of Square-root Functions Find the domain of the square-root function. The expression under the radical sign must be greater than or equal to 0. x– 4 ≥ 0 Solve the inequality. Add 4 to both sides. The domain is the set of all real numbers greater than or equal to 4.

  13. –3 –3 x ≥ –3 Example 2B: Finding the Domain of Square-root Functions Find the domain of the square-root function. The expression under the radical sign must be greater than or equal to 0. x +3 ≥ 0 Solve the inequality. Subtract 3 from both sides. The domain is the set of all real numbers greater than or equal to –3.

  14. +1 +1 2x ≥ 1 The domain is the set of all real numbers greater than or equal to . Check It Out! Example 2a Find the domain of the square-root function. The expression under the radical sign must be greater than or equal to 0. 2x – 1 ≥ 0 Solve the inequality. Add 1 to both sides. Divide both sides by 2.

  15. + 5 +5 3x ≥ 5 The domain is the set of all real numbers greater than or equal to . Check It Out! Example 2b Find the domain of the square-root function. The expression under the radical sign must be greater than or equal to 0. 3x – 5 ≥ 0 Solve the inequality. Add 5 to both sides. Divide both sides by 3.

  16. The parent function for square-root functions, is graphed at right. Notice there are no x-values to the left of 0 because the domain is x ≥ 0.

  17. If a square-root function is given in one of these forms, you can graph the parent function and translate it vertically or horizontally.

  18. Since this function is in the form f(x) = , you can graph it as a horizontal translation of the graph of f(x) = Graph f(x) = and then shift the graph 3 units to the right. Example 3A: Graphing Square-Root Functions Graph .

  19. Graph . This is not a horizontal or vertical translation of . Example 3B: Graphing Square-Root Functions Step 1 Find the domain of the function. x ≥ 0 The expression under the radical sign must be greater than or equal to 0. The domain is the set of all real numbers greater than or equal to 0.

  20. Graph . x 0 4 1 7 4 10 6 11.35 Example 3B Continued Step 3 Plot the points. Then connect them with a smooth curve. Step 2 Choose x-values greater than or equal to 0 and generate ordered pairs.

  21. Since this function is in the form f(x) = , you can graph it as a vertical translation of the graph of f(x) = Graph f(x) = and then shift the graph 2 units up. Check It Out! Example 3a Graph each square root function.

  22. This is not a horizontal or vertical translation of . Check It Out! Example 3b Graph each square root function. Step 1 Find the domain of the function. The expression under the radical sign must be greater than or equal to 0. x ≥ 0 The domain is the set of all real numbers greater than or equal to 0.

  23. Graph . x 0 3 1 5 4 7 6 7.89 Check It Out! Example 3b Continued Step 3 Plot the points. Then connect them with a smooth curve. Step 2 Choose x-values greater than or equal to 0 and generate ordered pairs.

  24. 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. 3.0 in. Find the domain of each square-root function. x ≥ 0 2. 3. x ≥ 5 4.

  25. Lesson Quiz: Part II Graph each square-root function. 5. 6.

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