Unit 3B Graph Radical Functions

Notice that the inverse of f(x) = x2 is not a function because it fails the vertical line test. However, if we limit the domain of f(x) = x2 to x ≥ 0, its inverse is the function . A radical function is a function whose rule is a radical expression. For example, we will study the square-root parent function and the cube-root parent function .

Graph of Square Root Function x y -1 i 1 4 2 Note: We cannot graph imaginary numbers on the coordinate plane. Therefore, the graph stops at x = 0.

Graph of the Cube Root x y -4 -1 1 4 1.59 -1.59 1 4 1.59 Note: Since the index number is odd, we can graph the function for all x values. Therefore, the domain is all reals.

Transformations of square root and cube root parent functions. The general form of the square root function is The cube root function is

Changing a a > 1 vertical stretch 0< a < 1 vertical shrink a is - (flip vertically)

• • • • x (x, f(x)) Check It Out! Example 1b Graph each function, and identify its domain and range. x (x, f(x)) –1 (–1, 0) 3 (3, 2) 8 (8, 3) 15 (15, 4) • • • • The domain is {x|x ≥ –1}, and the range is {y|y ≥0}.

Shift left h. Shift right h. Up k. Down k.

Shift left h. Shift right h. Up k. Down k.

Example 1 Solution Comparing Two Graphs h = -2 and k = -4 Describe how to create the graph of y = x + 2 – 4 from the graph of y = x . Solution h = -2 and k = -4 shift the graph to the left 2 units & down 4 units

v Example 2 Solution Graphing a Square Root Sketch the graph of y = -3 x (dashed). It begins at the origin and passes through point (1,-3). Graphing a Square Root Graph y = -3 x – 1 + 3 . (1, 3) (0,0) (2,0) 2) For y = -3 x – 1 + 3, h = 1 & k = 3. Shift both points 1 to the right and 3 up. (1,-3)

Graphing a Square Root Graph y = 2 x – 2 + 1 . (3,3) (1,2) (2, 1) (0,0)

v Example 3 Solution Graphing a Cube Root Sketch the graph of y = 2 3 x (dashed). It passed through the origin & the points (1, 2) & (-1, -2). Graphing a Cube Root Graph y = 2 3 x + 3 – 4 . (1, 2) (0,0) 2) For y = 2 x + 3 – 4, h = -3 & k = -4. Shift the three points Left 3 and Down 4. (-1,-2) (-2,-2) (-3,-4) (-4,-6)

v Example 4 Using the graph of as a guide, describe the transformation and graph the function. f(x)= x ● g is f reflected across the y-axis and translated 3 units up.