Unit 3 Chapter 2 Radical Functions (Square Root Functions)
Radical Functions Sketch graphs of radical functions by applying translations, stretches and reflections to the graph of y x Analyze transformations to identify the domain and range of radical functions. The graph of y f ( x ) is related to the graph of y = f (x) and the domain and range for each function are compared. Graphically finding approximate solutions to radical equations. (This was done algebraically in Math 2200)
Graphing square root functions What would be the base graph for the square root function? What is the table of values? y x y=f(x) x
A)Sketch the following using y x B) State the mapping rule and the transformational form of the function C) Determine the domain and range 1. Vertical translation of -3, Horizontal translation of -2 (x, y) x 2, y 3 y (0, 0) (1, 1) (4, 2) (9, 3) x
A)Sketch the following using y x B) State the mapping rule and the transformational form of the function C) Determine the domain and range (x, y) (0, 0) (1, 1) (4, 2) (9, 3) 2. Vertical stretch 2, Horizontal translation of 1 left y x
3.A) Sketch the following y1 3 2( x 2) B) State the mapping rule C) Determine the domain and range (x, y) (0, 0) (1, 1) (4, 2) (9, 3) y x
4. Explain how to transform the graph of y xto obtain y 2 4x 3 1. Sketch the graph of each function. Then, identify the domain and range of each function. Solution Begin by identifying the parameters and the effect each has on the base function. Parameter a =, resulting in a by a factor of. Since a is negative, the graph is reflected in the. Parameter b =, resulting in a by a factor of. Parameter h =, so the graph is translated by units. Parameter k =, so the graph is translated by units.
Apply the transformations to sketch the graph of transformed function. (x, y) y 2 4 x 3 1 The domain of the base function is {x x 0, x R} and its range is {y y 0, y R}. The domain and range of the transformed function are domain: range:
5. Explain how to transform the graph of y x to obtain 1 y 1 3 x 6. Sketch the graph of each function. Then, 2 identify the domain and range of each function. Solution Begin by identifying the parameters and the effect each has on the base function. Parameter a =, resulting in a by a factor of. Since a is negative, the graph is reflected in the. Parameter b =, resulting in a by a factor of. Parameter h =, so the graph is translated by units. Parameter k =, so the graph is translated by units.
Apply the transformations to sketch the graph of transformed function. y (x, y) x The domain of the base function is {x x 0, x R} and its range is {y y 0, y R}. The domain and range of the transformed function are domain: range:
6. Using the four points on the graph of the base function, y complete the table to determine the resulting coordinates on the graph of y 4 2x 3 5 x domain: range:
Now that we have worked through a number of examples, can you see a pattern that would allow us to determine the domain and range of a radical expression such without creating an accurate graph. This can be done by examining the reflections and translations. The stretches do not affect the domain and range of radical functions.
We have a reflection in the y-axis. Horizontal Translation is 5 left. Thus the domain is x -5 Vertical translation is 7 up Thus the range is y 7
Examples 1. Determine the domain and range of A) y 3 2x 4 5 First, rewrite the equation in a form to read the transformations. y 3 2( x 2) 5 Horizontal Translation is 2 left. Thus the domain is x -2 We have a reflection in the x-axis. Vertical translation is 5 down Thus the range is y -5
Examples 1. Determine the domain and range of B) 2y x 1 3 Rewrite the equation in a form to read the transformations. 1 3 y ( x 1) 2 2 We have a reflection in the y-axis. Horizontal Translation is 1right. Thus the domain is x 1 We have a reflection in the x-axis. Vertical translation is 1.5 down Thus the range is y -1.5
2. Write the equation of a radical function with each domain and range. (Don t use stretch factors) A) Domain: x x 7 B) Domain: (,2] Range: y y 2 Range: [-8, )
Page 72-75 2, 3, 4, 5. 11, 13
Section 2.2 Sketch the graph of the function y f ( x), given the equation or graph of the function y = f(x), and explain the strategies used. Compare the domain and range of the function y f ( x), to the domain and range of the function y = f(x), and explain why the domains and ranges may differ.
Working Example 1: Compare Graphs of a Linear Function and the Square Root of the Function a) Given f (x) = 4x 3, graph the functions y = f (x) and y f ( x) b) Compare the graphs.
Solution a) Determine the y-value in the second column of the table. Then, complete the third column by taking the square root of the second column. Use the table of values to sketch the graphs of y = f (x) and. y f ( x) (Hint: You could graph y = f (x) on your graphing calculator and then use the table function to complete the second column of the table.)
From your table of values, determine the points of intersection: (, 0); (, 1 ) How is the x-intercept of the graph of y = 4x - 3 related to the graph of the function? Same point Why are these points of intersection referred to as invariant points? These points are the same on both graphs because when f(x) = 0, or when f(x) = 1, f ( x ) f ( x ) For which values of x is the graph of above the graph of y =4x - 3? Between x = 0.75 and x = 1 How are these values related to the invariant points? They are between the invariant points Why is the graph of above y = 4x - 3 between these points? Because between these points the y values of y = 4x 3 are between 0 and 1. Taking the square root of these number result in a larger number. For example: 1 4 For which values of x is the graph of below the graph of y = 4x - 3? x > 1
To see a similar question, refer to Example 1 on pages 80 81 in Pre-Calculus 12.
Example 2. Compare Graphs of a Quadratic Function and the Square Root of the Function 2 Graph y x 1 Consider the graph of y = x 2 1 y x One method to produce 2 the graph of y x 1 is to first generate a table of values for y = x 2-1 from the given graph. 2 Then, to graph y x 1, take the square root of the y-values.
x f( x ) f( x) -2-1 0 1 2 For y = f(x) state: Domain: Range:
Solve: 2 x 1 0 Why is the graph undefined from x ( 1, 1)? y f ( x ) is undefined where f(x) < 0 You cannot take the square root of a negative number Are there any invariant points? If so, what are they? and The invariant points occur where f(x) = 0 or f(x) = 1. 2 x 1 1
y f ( x ) y f ( x ) y f ( x ) y f ( x )
Practice: Given the graph of y = f(x) sketch y f ( x ) A) y x
B) y x
2. What are all of the invariant points for the graphs of f x = x 2 + 6x 8 and y = f(x)? Remember: The invariant points occur where f(x) = 0 or f(x) = 1.
Another way to sketch the graph of y is to use equation y = f(x) to generate a table of values. From this, you could then graph y f ( x ). f ( x ) The graph and table of values of y = f(x) can also be generated with the use of technology. The graphs of y = f(x) are limited to linear and quadratic functions.
Example 1. Graph y f ( x ) A) f x = x 2 + 9 y x -4-2 0 2 4 f( x ) f( x) For f( x) state: Domain: Range: x
Example 1. Graph y f ( x ) B) f x = 4 x 2 y x -3 f( x ) f( x) -2 x -1 0 1 2 3 For f( x) state: Domain: Range:
Example 1. Graph y f ( x ) C) f x = (x 1) 2 2 y x -3-2 -1 0 1 2 3 f( x ) f( x) For f( x) state: Domain: Range: x
In General:
Text Page 86-7: # 1, 2, 3, 8 Page 89 # 16, 17
Finding Domain and Range of Square Root Functions Consider: y f ( x) To find the domain simply solve the inequality: f( x) 0 The range consists of the square roots of all of the values in the range of f(x) for which f(x) is defined.
Examples: Find the domain and range of the following: A) y = x 2 + 1 Hint: Find the x-intercepts to find the domain. Use the vertex to help find the range.
Examples: Find the domain and range of the following: B) y = x 2 5
Examples: Find the domain and range of the following: C) 2 y x 2x 8
Examples: Find the domain and range of the following: D) y = x 2 + 2x + 4
Page 87 5a) b) 6, 10, 11
2.3 Solving Radical Equations Graphically In Mathematics 2200, you solved radical equations algebraically. This year we will solve radical equations graphically.
Solving Radical Equations Algebraically Isolate one of the radicals Square both sides to remove a radical. Repeat if necessary Find the roots of the equation. Check the solution ensuring that it does not contain extraneous roots solutions that do not satisfy the original equation or restrictions when substituted in the original equation.
Examples : Solve A) x 5 3 0
Examples : Solve B) 3x 27 x 3
C) x 2x 15 6
D) 2 x 4 x 6
Solving Radical Equations Graphically Method 1: Graph a Single Equation Graph the corresponding function and find the zero(s) of the function. Example: 2 x 4 x 6 x 4 x 4 0 Graph y x 4 x 4 Method 2: Graph Two Equations Graph each side of the equation on the same grid, and find the point(s) of intersection. Example: Graph and 2 x 4 x 6 y 2 x 4 y x 6 This is difficult to graph without using technology. This is easy to graph.
Graph y x 4 x 4 y y x x
Graph and y 2 x 4 y x 6 y y x x
Examples : Solve A) x 5 3 0 y x
Examples : Solve B) 3x 27 x 3 y x
C) x 2x 15 6 y y x x
D) 2 x x 1 1 3 2 2 y x
E) 2 x 4 2 25 4x y x
Text Page 96-97 # 1, 2, 4 a) b), 6a) d)