Topic VIII: Radical Functions and Equations 8.2 Graphing Radical Functions

Domain: x   0, Range: y  0 Domain and range: all real numbers Graphing Radical Functions You have seen the graphs of y = x and y = x. These are examples of radical functions. 3

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 radical functions.

Transformations of square-root functions are summarized below.

Domain: x  0, Range: y  0 Domain and range: all real numbers Graphing Radical Functions You have seen the graphs of y = x and y = x. These are examples of radical functions. 3 In this lesson you will learn to graph functions of the form y = a x – h + k and y = a x – h + k. 3

SOLUTION

Graphing Radical Functions GRAPHS OF RADICAL FUNCTIONS 1 STEP Shift the graph h units horizontally and k units vertically. 2 STEP To graph y = a x – h + k or y = a x – h + k, follow these steps. 3 Sketch the graph of y = a x or y = a x. 3

Comparing Two Graphs SOLUTION Describe how to obtain the graph of y = x + 1 – 3 from the graph of y = x. To obtain the graph of y = x + 1 – 3, shift the graph of y = x left 1 unit and down 3 units. Note that y = x + 1 – 3 = x – ( – 1) + ( – 3), so h = –1 and k = –3. The y-intercept is

Graphing a Square Root Function SOLUTION 1 2 So, shift the graph right 2 units and up 1 unit. The result is a graph that starts at (2, 1) and passes through the point (3, –2). Graph y = –3 x – Note that for y = –3 x – 2 + 1, h = 2 and k = 1. Sketch the graph of y = –3 x (shown dashed). Notice that it begins at the origin and passes through the point (1, –3).

Finding Domain and Range State the domain and range of the functions in the previous examples. SOLUTION From the graph of y = –3 x – 2 + 1, you can see that the domain of the function is x  2 and the range of the function is y  1.

Graphing a Square Root Function SOLUTION Graph y = –2 x – 1 – 4. 3

Using Radical Functions in Real Life When you use radical functions in real life, the domain is understood to be restricted to the values that make sense in the real-life situation. The model that gives the speed s (in meters per second) necessary to keep a person pinned to the wall is where r is the radius (in meters) of the rotor. Use a graphing calculator to graph the model. Then use the graph to estimate the radius of a rotor that spins at a speed of 8 meters per second. s = 4.95 r AMUSEMENT PARKS At an amusement park a ride called the rotor is a cylindrical room that spins around. The riders stand against the circular wall. When the rotor reaches the necessary speed, the floor drops out and the centrifugal force keeps the riders pinned to the wall.

Modeling with a Square Root Function SOLUTION You get x  The radius is about 2.61 meters. Graph y = 4.95 x and y = 8. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the x -coordinate of that point.

Modeling with a Cube Root Function Use a graphing calculator to graph the model. Then use the graph to estimate the age of an elephant whose shoulder height is 200 centimeters. SOLUTION The elephant is about 8 years old. You get x  7.85 Biologists have discovered that the shoulder height h (in centimeters) of a male African elephant can be modeled by h = 62.5 t where t is the age (in years) of the elephant. Graph y = 62.5 x and y = 200 with your calculator. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the x -coordinate of that point. 3

Time 2 Work !!!