Unit 3 B Graph Radical Functions Notice that

Unit 3 B Graph Radical Functions

Notice that the inverse 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. 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 -1 0 1 4 y i 0 1 2 Note: We cannot graph imaginary numbers on the coordinate plane. Therefore, the graph stops at x = 0.

Graph of the Cube Root x -4 -1 0 1 4 y -1. 59 -1 0 1 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)

Check It Out! Example 1 b Graph each function, and identify its domain and range. x (x, f(x)) – 1 (– 1, 0) 3 8 15 (3, 2) (8, 3) (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.

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

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

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

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

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