Splash Screen

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

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

Find the inverse of f(x) = x + 3. __ 1 4 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 __ 3 9 5-Minute Check 2

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

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

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

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

Mathematical Practices 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

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

square root inequality square root function radical function square root inequality Vocabulary

Concept

The domain only includes values for which the radicand is nonnegative. 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

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

Concept

Graph Square Root Functions Example 2

Notice the end behavior; as x increases, y increases. 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

Graph Square Root Functions Example 2

Notice the end behavior; as x increases, y decreases. 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

A. B. C. D. Example 2

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

Graph the function in the domain {a|a ≥ 0}. Use Graphs to Analyze Square Root Functions A. PHYSICS When 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}. Example 3

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

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

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

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

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

Select a point to see if it is in the shaded region. Graph a Square Root Inequality The domain is . Because y is greater than, the shaded region should be above the boundary and within the domain. 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

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

End of the Lesson