Page 1 of 12 Radical Functions Attendance Problems. Identify the domain and range of each function. 1. f ( x) = x 2 + 2 2. f ( x) = 3x 3 Use the description to write the quadratic function g based on the parent function f(x) = x 2. 3. f is translated 3 units up. 4. f is translated 2 units left. I can graph radical functions and inequalities. I can transform radical functions by changing parameters. Vocabulary radical function square-root function Common Core CCSS.MATH.CONTENT.HSF.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* CCSS.MATH.CONTENT.HSF.BF.B.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. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well. The graphs show the inverses of the quadratic parent function and cubic parent function.
Page 2 of 12 Notice that the inverses of f(x) = x 2 is not a function because it fails the vertical line test. However, if we limit the domain of f(x) = x 2 to x 0, its inverse is the function f 1 x. A radical function is a function whose rule is a radical expression. A square-root function is a radical function involving x. The square-root parent function is 3 f x. The cube-root parent function is. Video Example 1. Graph each function identifying its domain and range. A. f (x) = x ( ) = x ( ) = x f ( x) = x
Algebra II Page 3 of 12 B. 3 f ( x) = 2 x +1 1 Graphing Radical Functions Graph the function, and identify its domain and range. A f (x) = x Make a table of values. Plot enough ordered pairs to see the shape of the curve. Because the square root of a negative number is imaginary, choose only nonnegative values for x. x f (x) = x (x, f (x)) 0 f (0) = 0 = 0 (0, 0) 1 f (1) = 1 = 1 (1, 1) 4 f (4) = 4 = 2 (4, 2) 9 f (9) = 9 = 3 (9, 3) 4 (0, 0) 0-2 y (1, 1) 2 (9, 3) (4, 2) x 4 6 8 10 The domain is x x 0, and the range is y y 0.
Page 4 of 12 Example 1. Graph each function identifying its domain and range. A. f ( x) = x 3 B. 3 f ( x) = 2 x 2 Guided Practice. 3 5. f ( x) = x
Page 5 of 12 6. f ( x) = x +1 The graphs of radical functions can be transformed by using methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions.
Page 6 of 12
Page 7 of 12 Video Example 2. Using the graph of f ( x) = x as a guide, describe the transformation and graph the function. g( x) = x + 3 A. B. g( x) = 2 x 2 Transforming Square-Root Functions Using the graph of f (x) = x as a guide, describe the transformation and graph each function. A g (x) = x - 2 B g (x) = 3 x g (x) = f (x) - 2 g (x) = 3 f (x) Translate f 2 units down. 4 2 (0, 0) 0-2 y (4, 2) (4, 0) 4 (0, -2) f g x Stretch f vertically by a factor of 3. 6 4 2 (0, 0) 0 y g (4, 6) (4, 2) 2 4 6 f x Example 2. Using the graph of f x as a guide, describe the transformation and graph the function g( x) = x + 5. ( ) = x
Page 8 of 12 Guided Practice. Using the graph of f ( x) = x as a guide, describe the transformation and graph the function. g( x) = x +1 7. 8. g( x) = 1 2 x
Page 9 of 12 Video Example 3. Using the graph of f x as a guide, describe the transformation and graph the function. A. g(x) = 3 x 2 B. x +1 ( ) = x 3 Applying Multiple Transformations Using the graph of f (x) = x as a guide, describe the transformation and graph each function. A g (x) = 2 x + 3 B g (x) = -x - 2 Stretch f vertically by a factor Reflect f across the y-axis, of 2, and translate it 3 units left. and translate it 2 units down. 6 4 2 (-3, 0) (0, 0) -4-2 0 y (1, 4) g f (4, 2) x 2 4 2 g (0, 0) 0 (-4, 0) -2-4 y (4, 2) f x 2 4 (0, -2)
Page 10 of 12 Example 3. Using the graph of f x as a guide, describe the transformation and graph the function g(x) = x 4. Guided Practice. Using the graph of f ( x) = x as a guide, describe the transformation and graph the function. 9. g( x) = x + 3 10. g( x) = 3 x 1 ( ) = x Video Example 4. Use the description to write the square-root function g. the parent function f ( x) = x is stretched horizontally by a factor of 3, reflected across the y-axis, and translated 2 units right.
Page 11 of 12 4 Writing Transformed Square-Root Functions Use the description to write the square-root function g. The parent function f (x) = x is stretched horizontally by a factor of 2, reflected across the y-axis, and translated 3 units left. Step 1 Identify how each transformation affects the function. Horizontal stretch by a factor of 2: b = 2 b = -2 Reflection across the y-axis: b is negative Translation 3 units left: h = -3 Step 2 Write the transformed function. g (x) = _ 1 (x - h) b g (x) = _ 1-2 x- (-3) Substitute -2 for b and -3 for h. g (x) = -_ 1 (x + 3) Simplify. 2 Check Graph both functions on a graphing calculator. The graph of g indicates the given transformations of f. Example 4. Use the description to write the square-root function g. The parent function f ( x) = x is reflected across the x-axis, compressed vertically by a factor 1 of, and translated down 5 units. 5
Page 12 of 12 11. Guided Practice. Use the description to write the square-root function g. The parent function f ( x) = x is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up. 5-7 Assignment (p 372) 25-41 odd, 49, 51-54, 57, 68, 73-77.