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)?
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What is the horizontal asymptote of the function y=(x+2)/(x^2-5x+6)?
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What is the horizontal asymptote of the function y=(4x+1)/(x+2)?
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What is the domain of the function y=(4x+1)/(x+2)?
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What is the domain of the function y=(x+2)/(x^2-5x+6)?
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What is the range of the function y=(4x+1)/(x+2)?
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What is the range of the function y=(x+2)/(x^2-5x+6)?
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What is the oblique asymptote of the equation y=(x^2+7x+10)/(x+2)?
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What is the oblique asymptote of the equation y=(x^2-8x+12)/(x+1)?
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Using synthetic division, what is (x^2-5x-6)/(x-1)?
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When graphing rational functions, what is synthetic division used for?
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Using long division, what is (x^3-6x^2+3x-1)/(x^2-2x-2)?
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When graphing rational functions, what is long division used for?
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What are the zeros in the function y=(x^2-5x+6)/(x+7)?
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What is the y-intercept of the function y=(x^2-5x+6)/(x+7)?
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What is the y-intercept of the function y=(x+6)/(x-3)?
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What are the zeros of the function y=(x+6)/(x-3)?
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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.