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Understanding Square Root Functions and Inverses

This lesson covers the parent function of square root functions, identifying domain and range, graphing square root functions, analyzing real-world examples, finding inverses, and graphing square root inequalities.

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Understanding Square Root Functions and Inverses

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 6–2) CCSS Then/Now New Vocabulary Key Concept: Parent Function of Square Root Functions Example 1: Identify Domain and Range Key Concept: Transformations of Square Root Functions Example 2: Graph Square Root Functions Example 3: Real-World Example: Use Graphs to Analyze Square Root Functions Example 4: Graph a Square Root Inequality Lesson Menu

  3. A.{(–4, –3), (8, 0), (5, –6)} B.{(1, 5, 2), (0, 4), (–3, 2.5)} C. D.{(3, 4), (0, 8), (6, –5)} Find the inverse of {(4, 3), (8, 0), (–5, 6)}. 5-Minute Check 1

  4. Find the inverse of f(x) = x + 3. A.f–1(x) = 4x – 12 B.f–1(x) = 4x + 3 C.f–1(x) = 4x + D.f–1(x) = x + 3 1 3 1 __ __ __ 4 9 3 5-Minute Check 2

  5. A. B. C. D. Find the inverse of f(x) = 2x – 1. 5-Minute Check 3

  6. Determine whether the pair of functions are inverse functions. Write yes or no.f(x) = x + 7g(x) = x – 7 A. yes B. no 5-Minute Check 4

  7. Determine whether the pair of functions are inverse functions. Write yes or no.f(x) = 3x + 4g(x) = x – 4 1 __ 3 A. yes B. no 5-Minute Check 5

  8. Which of the following functions has an inverse that is also a function? A.f(x) = x2 + 3x – 2 B.g(x) = x3 – 8 C.h(x) = x5 + x2 – 4x D.k(x) = 2x2 5-Minute Check 6

  9. Content Standards F.IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 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. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. CCSS

  10. You simplified expressions with square roots. • Graph and analyze square root functions. • Graph square root inequalities. Then/Now

  11. square root function • radical function • square root inequality Vocabulary

  12. Concept

  13. Identify Domain and Range The domain only includes values for which the radicand is nonnegative. x – 2 ≥ 0 Write an inequality. x ≥ 2 Add 2 to each side. Thus, the domain is {x | x ≥ 2}. Find f(2) to determine the lower limit of the range. So, the range is {f(x)| f(x) ≥ 0}. Answer: D: {x | x ≥ 2}; R: {f(x)| f(x)≥ 0} Example 1

  14. A.D: {x | x ≥ –4}; R: {f(x)| f(x) ≤ 0} B.D: {x | x ≥ 4}; R: {f(x)| f(x) ≥ 0} C.D: {x | x ≥ –4}; R: {f(x)| f(x) ≥ 0} D.D: {x | all real numbers}; R: {f(x)| f(x) ≥ 0} Example 1

  15. Concept

  16. Graph Square Root Functions Example 2

  17. Graph Square Root Functions Notice the end behavior; as x increases, y increases. Answer:The domain is {x | x≥ 4} and the range is {y | y≥ 2}. Example 2

  18. Graph Square Root Functions Example 2

  19. Graph Square Root Functions Notice the end behavior; as x increases, y decreases. Answer:The domain is {x | x≥ –5} and the range is {y | y≤ –6}. Example 2

  20. A.B. C.D. Example 2

  21. A. D: {x | x≥ –1}; R: {y | y≤ –4} B. D: {x | x≥ 1}; R: {y | y≥ –4} C. D: {x | x≥ –1}; R: {y | y≤ 4} D. D: {x | x≥ 1}; R: {y | y≤ 4} Example 2

  22. A. PHYSICSWhen an object is spinning in a circular path of radius 2 meters with velocity v, in meters per second, the centripetal acceleration a, in meters per second squared, is directed toward the center of the circle. The velocity v and acceleration a of the object are related by the function . Graph the function in the domain {a|a≥ 0}. Use Graphs to Analyze Square Root Functions Example 3

  23. The function is . Make a table of values for {a | a≥ 0} and graph. Use Graphs to Analyze Square Root Functions Answer: Example 3

  24. Use Graphs to Analyze Square Root Functions B. What would be the centripetal acceleration of an object spinning along the circular path with a velocity of 4 meters per second? It appears from the graph that the acceleration would be 8 meters per second squared. Check this estimate. Original equation Replace v with 4. Square each side. Divide each side by 2. Answer: The centripetal acceleration would be 8 meters per second squared. Example 3

  25. A. GEOMETRY The volume V and surface area A of a soap bubble are related by the function Which is the graph of this function? A. B. C. D. Example 3

  26. B. GEOMETRY The volume V and surface area A of a soap bubble are related by the function What would the surface area be if the volume was 3 cubic units? A. 10.1 units2 B. 31.6 units2 C. 100 units2 D. 1000 units2 Example 3

  27. Graph . Graph the boundary . Since the boundary should not be included, the graph should be dashed. Graph a Square Root Inequality Example 4

  28. The domain is . Because y is greater than, the shaded region should be above the boundary and within the domain. ? Graph a Square Root Inequality Select a point to see if it is in the shaded region. Test (0, 0). Answer: Shade the region that does not include (0, 0). Example 4

  29. Which is the graph of ? A. B. C. D. Example 4

  30. End of the Lesson

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