Algebra 2 Unit 4 Flashcards

Questions and Answers

What is the vertical asymptote(s) of the function y=(x+6)/(x+2)(x-5)?

x=-2, x=5

How do you graph the function y=(1/x-2)+5?

First graph the parent function 1/x and then translate the graph right 2 units and up 5 units.

How do you graph the function y=(3/x+7)?

First graph the parent function 3/x and then translate the graph left 7.

What are the vertical asymptotes of the function y=(x+2)/(x^2-5x+6)?

x=2, x=3 Signup and view all the answers

What is the horizontal asymptote of the function y=(x+2)/(x^2-5x+6)?

y=0 Signup and view all the answers

What is the horizontal asymptote of the function y=(4x+1)/(x+2)?

y=4 Signup and view all the answers

What is the domain of the function y=(4x+1)/(x+2)?

x is all real numbers except -2 Signup and view all the answers

What is the domain of the function y=(x+2)/(x^2-5x+6)?

x is all real numbers except 2 and 3 Signup and view all the answers

What is the range of the function y=(4x+1)/(x+2)?

y is all real numbers except 4 Signup and view all the answers

What is the range of the function y=(x+2)/(x^2-5x+6)?

y is all real numbers except 0 Signup and view all the answers

What is the oblique asymptote of the equation y=(x^2+7x+10)/(x+2)?

y=x+5 Signup and view all the answers

What is the oblique asymptote of the equation y=(x^2-8x+12)/(x+1)?

y=x-9 Signup and view all the answers

Using synthetic division, what is (x^2-5x-6)/(x-1)?

x-4-(10/x-1) Signup and view all the answers

When graphing rational functions, what is synthetic division used for?

Finding oblique asymptotes when the denominator is a linear binomial Signup and view all the answers

Using long division, what is (x^3-6x^2+3x-1)/(x^2-2x-2)?

y=x-4+{-3(x+9)/(x^2-2x-2)} Signup and view all the answers

When graphing rational functions, what is long division used for?

Finding oblique asymptotes when the denominator isn't a linear binomial Signup and view all the answers

What are the zeros in the function y=(x^2-5x+6)/(x+7)?

(2,0) & (3,0) Signup and view all the answers

What is the y-intercept of the function y=(x^2-5x+6)/(x+7)?

(0, 6/7) Signup and view all the answers

What is the y-intercept of the function y=(x+6)/(x-3)?

(0, -2) Signup and view all the answers

What are the zeros of the function y=(x+6)/(x-3)?

(-6, 0) Signup and view all the answers

Study Notes

Vertical Asymptotes

For the function ( y=\frac{x+6}{(x+2)(x-5)} ), vertical asymptotes occur at ( x=-2 ) and ( x=5 ).
For ( y=\frac{x+2}{x^2-5x+6} ), vertical asymptotes are found at ( x=2 ) and ( x=3 ).

Horizontal Asymptotes

The function ( y=\frac{x+2}{x^2-5x+6} ) has a horizontal asymptote at ( y=0 ).
For ( y=\frac{4x+1}{x+2} ), the horizontal asymptote is ( y=4 ).

Domain and Range

The domain of ( y=\frac{4x+1}{x+2} ) excludes ( x=-2 ), allowing all other real numbers.
The domain for ( y=\frac{x+2}{x^2-5x+6} ) excludes ( x=2 ) and ( x=3 ).
The range of ( y=\frac{4x+1}{x+2} ) excludes ( y=4 ).
The range for ( y=\frac{x+2}{x^2-5x+6} ) excludes ( y=0 ).

Oblique Asymptotes

The oblique asymptote for ( y=\frac{x^2+7x+10}{x+2} ) is given by ( y=x+5 ).
For ( y=\frac{x^2-8x+12}{x+1} ), the oblique asymptote is ( y=x-9 ).

Graphing Rational Functions

To graph ( y=\frac{1}{x-2}+5 ), start by graphing the parent function ( \frac{1}{x} ), then shift right by 2 units and up by 5 units.
For ( y=\frac{3}{x+7} ), graph ( \frac{3}{x} ) and translate left by 7 units.

Synthetic and Long Division

Synthetic division for ( \frac{x^2-5x-6}{x-1} ) yields ( x-4-\frac{10}{x-1} ).
Long division for ( \frac{x^3-6x^2+3x-1}{x^2-2x-2} ) results in ( y=x-4+\frac{-3(x+9)}{x^2-2x-2} ).
Synthetic division is used to find oblique asymptotes when the denominator is a linear binomial.
Long division serves this purpose when the denominator is not a linear binomial.

Zeros and Y-Intercepts

The zeros of ( y=\frac{x^2-5x+6}{x+7} ) occur at ( (3,0) ) and ( (2,0) ).
The y-intercept for ( y=\frac{x^2-5x+6}{x+7} ) is ( (0,\frac{6}{7}) ).
For ( y=\frac{x+6}{x-3} ), the y-intercept is ( (0,-2) ).
The zeros for ( y=\frac{x+6}{x-3} ) are found at ( (-6,0) ).

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Test your understanding of vertical asymptotes and graphing rational functions with these flashcards for Algebra 2 Unit 4. Each card provides a specific function and the steps needed for graphing and identifying characteristics. Perfect for review and practice before your upcoming exam.