Algebra 2 Unit 4 Flashcards

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) ).