QUADRATIC FUNCTIONS

Question

QUADRATIC FUNCTIONS:

For each of the following,

a) Find the number of real roots -- find b^2-4ac: positive means 2 real roots, zero means 1 real root, negative means 0 real roots

b) Find all roots. You can use quadratic formula, completing the square, or a factoring method where applicable.

c) Find the vertex -- x = -b/2a, plug into the function to get y.

d) Plot the approximate graph -- plot the vertex exactly, then draw an upwards U shape for positive a, and downwards U shape for negative a.



1) y = x^2

2) y = x^2 - 2x + 1

3) y = -2x^2 - 3x - 4



RATIONAL FUNCTIONS:

For each of the following,

a) Factor the top and bottom and find the roots (zeros of the top) and poles (zeros of the bottom).

b) State the location of holes and vertical asymptotes -- holes happen for x-values where a root and a pole cancel out, asymptotes happen for x-values of poles which don't cancel out

c) State the location of horizontal asymptotes -- take the fraction of the highest degree terms, if the degree of the top and bottom are the same, then that will be the y-value



1) y = 1 / x

2) y = (x - 1) / (x+1)

3) y = (x^2 - x - 6) / (x^2 + 6x + 8)

Answer 1

Quadratic Functions:

 

Quadratic functions are in the form         y = ax² + bx + c

 

Use the discriminant from the quadratic formula:

 

b² - 4ac

 

 

If (b² - 4ac) is negative, then 2 complex solutions.

If (b² - 4ac) is zero, then 1 repeating real solution.

If (b² - 4ac) is positive, the 2 distinct real solutions.

 

 

Rational Functions:

 

I will do the last problem since it has more to it.  Then you can do the other two, since they are much simpler.

 

Factor the top and bottom.

 

(x - 3)(x + 2) / (x + 4)(x + 2)

 

The factor (x + 2) occurs on the top and bottom.  This means that is has a solution, and at the same time undefined.   So the factor (x + 2) indicates a hole.  The hole occurs at x=-2.

 

Now the rational simplifies to (x - 3) / (x + 4).  If we evaluate this expression at x=-2, we get the y value of the hole.

 

y = (-2 - 3) / (-2 + 4)

y = -5/2

 

The coordinate of the hole is (-2, -5/2).

 

The vertical asymptote is the vertical line passing through the x value that makes the rational undefined.  No curve touches this vertical line.   In this case , x=-4 is the vertical asymptote.

 

The horizontal asymptote is the horizontal line passing through the y value that makes gives the range.  No curve touches this horizontal line.  Since the top and bottom are both linear, the horizontal asymptote is the ratio between the top's leading coefficient and bottom's leading coefficient.

 

y = 1  is the horizontal asymptote.

 

 

Note:

 

If degree of top is less than degree of bottom, the horizontal asymptote is y=0.

If degree of top is greater than degree of bottom, the horizontal asymptote is the non-slope line.  You must use long division to divide the top by the bottom.  The quotient part is the asymptote.