Radical Functions and Equations

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Presentation transcript:

Radical Functions and Equations index radicand L. Waihman 2002

A radical function is a function that has a variable in the radicand. You can apply the same transformations to the graphs of radical functions as you can to polynomial functions. Horizontal shift left one Vertical shift down three Vertical stretch 2.

Radical Parent functions As x, f(x) As x0+, f(x)0 As x, f(x) As x0+, f(x)0 As x, f(x) As x–, f(x)–

Transformations We apply the transformations on these functions in the same manner as we did with the polynomial functions. Vertical stretch of 3 Vertical shift up 2 Horizontal shift right 3 Domain: Domain: Domain: Range Range: Range:

Applying the transformations Vertical stretch 3 Horizontal shift left 1 Vertical shift down 1 Domain: Range: As x , f(x) ; As x -1+, f(x) -1 Reflect @ x-axis Vertical shrink of ½ Horizontal shift right 2 Vertical shift up 4 As x , f(x) - ; As x 2+, f(x) 4 Domain: Range:

Applying the transformations Vertical stretch of 3 Horizontal shift left 2 Vertical shift down 4 Domain: Range: As x , f(x)  ; As x - , f(x) -  Reflect @ x-axis Vertical shrink of ½ Horizontal shift right 1 Vertical shift up 2 As x , f(x) -  ; As x - , f(x)  Domain: Range:

Radical Equations To solve a radical equation that has only one variable in the radicand, isolate that term on one side of the equation. If the index is 2, then square both sides of the equation. Given: Isolate the radical. Square both sides. Simplify and set equal to zero. Factor. Solve. Be careful! The new equation you created when you Squared both sides might have extraneous solutions!

Solving Radical Equations These solutions may not be solutions to the original equation. Check your solutions! The graph below illustrates that the derived equation may be different from the original equation. √

Try: Check: Check Try:

More Solving Radical Equations A radical equation may contain two radical expressions with an index of 2. To solve these, rewrite the equation with one of the radicals isolated on each side of the equals sign. Then, square both sides. If a variable remains in a radicand, you must repeat the squaring process.

More Solving Radical Equations Try: Isolate one radical on each side of equals sign. Square both sides. Collect like terms on each side of equals sign. Simplify. Check: Square both sides again.

More Solving Radical Equations Try: Check: √

Radical equations with indexes greater than 2 can be solved using similar techniques. After isolating the term containing the radical, raise each side of the equation to the power equal to the index of the radical. Check: √

Bonus Questions! One type of transformation was not covered in this PowerPoint. Identify it and give and example of an equation with this type of transformation and then sketch its graph to illustrate the transformation. Identify the domain and range as well. Solve the equation: