Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# Polynomial Functions (high school) ?

Given that the range of a polynomial function is {y | y =< 7, y is real numbers} do you have enough information to make any conclusions about the sign of the leading coefficient and or the degree of the polynomial function? If so, state a possible polynomial function that meets these criteria and explain.

I've tried doing this question myself but it just lead me to the wrong idea and now I'm completely confused. I'd appreciate if anyone can help me out!

Relevance
• After a little thought ...  it can't be linear

.. it can be any even degree

.. it cannot be any odd degree .. think about x^3

one 'end' goes to + infinity and the other goes to - infinity  ...  so the range will include numbers greater than 7 ...

so  y = ax^2 + bx + c  ....  max is 7

...  max occurs at x  =  -b/(2a) = 7

let a = -1 for simplicity  ...  must be neg to make the quadratic have a max not a min...  again think about shapes..

x = -b / (-2)  =  b/2

y = ax^2 + bx + c . <<<  again to make things easy, c=0

y = -x^2 + bx

7 = -(b/2)^2 + b(b/2)  =  -b^2 / 4  +  b^2/2

7  =   b^2/4

b^2 = 28

b = +/- sqrt28  <<<  I will use +

y  =  -x^2 + (sqrt28)x   <<<  this is the polynomial

• If y = 7 is a maximum, then the leading coefficient must be negative with an even degree polynomial.

That way if x is very negative when squared is positive, then when negated is negative again.  And when x is very positive, when squared is positive, then when negated is negative.

If the polynomial was odd, then the global min and max is always -∞ and ∞, which is why the degree must be even to have a non-infinity global min or max.