HMA/AAT_Unit 3_Graph Rational F(x) 1

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
What is the domain of this function? 

Answer explanation

denominator factors to (x - 4)(x - 3); write the domain before cancelling any factors

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Give the extended function if there is one.  If there is not one, answer none.

Answer explanation

what remains of the function after you cancel factors

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Determine the x-intercept(s).  Write the answer(s) as coordinates. 

Answer explanation

use extended function; set numerator factor equal to 0 and solve for x

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Determine the y-intercept(s).  Write the answer as coordinates. 

Answer explanation

use extended function; plug in 0 for x and calculate

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Determine the location of the vertical asymptote(s).

Answer explanation

use extended function; set denominator factor equal to 0 and solve for x

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Determine the location of the hole in the graph if there is one.  Write the answer as coordinates.  If there is no hole, answer none. 

Answer explanation

set the cancelled factor equal to 0 and solve for x; plug in the x-value into the extended function and get the y-value

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Determine the location of the horizontal asymptote.  

Answer explanation

use the extended function; denominator is the divisor; numerator is the dividend; quotient is the HA

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Is there an intersection point with the horizontal asymptote? If yes, give the location of the intersection point.  If no, answer no.  

Answer explanation

the HA long division remainder does not contain an "x," so there is no intersection with the HA

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Does this function have symmetry?  Answer odd, even, or none.  

Answer explanation

use extended function; plug in (-x) for x; if all signs change or no signs change, the symmetry is even; if all signs change in the numerator or denominator (but not both), the symmetry is odd

 $f\left(x\right)\ =\ \frac{2x^2\ -\ 5x\ -\ 3}{x^2\ -\ 7x\ +\ 12}$  
Give the location of the infinite (non-removeable) discontinuity if there is one. If there is not one, answer none  

Answer explanation

vertical asymptotes are infinite (non-removeable) discontinuities