Question

Dadas as funções

$f(x) = ax^2+4x-3,\ com\ a\geq\ 0\ e\ y=-k,\ onde\ k \neq 0$

a) Para quais valores de a, podemos afirmar que a reta y, não intercepta essa parábola?

b) Encontre as abscissas de intersecção entre a reta e a parábola em função de a e k.

c) Encontre as ordenadas das intersecções entre a reta e a parábola.

Answer 1

$\masthsf{f(x)=ax^2+4x-3}\\\mathsf{}a \geq 0\\\mathsf{y=-k}\\\mathsf{k \neq 0}\\\\\mathsf{Para~que~a~reta~y~n\~ao~intercepte~a~parabola~k~precisa~assumir}\\\mathsf{um~valor~diferente~de~zero.~Para~isso~teremos~que~encontrar}\\\mathsf{as~raizes~reais~da~equac\~ao.}\\\\\mathsf{-k \neq ax^2+4x-3}~~(*-1)\\\mathsf{k \neq -ax^2-4x+3}$ 

$\mathsf{\Delta=(-4)^2-4.(-a).3}\\\mathsf{\Delta=16+12a}\\\\\mathsf{Para~que~haja~raizes~reais~delta~precisa~ser~maior~ou~igual~a~zero}\\\\\maths{16+12a \geq 0}\\\mathsf{12a \geq -16}\\\\\mathsf{a \geq -\frac{16}{12}}\\\\\mathsf{a \geq -\frac{4}{3}}\\\\\\\mathsf{(a)=\boxed{S=\{a\in\mathbb{R}~|~a \geq -\frac{4}{3}\}}}$


$\mathsf{(b)}\\\\\\\mathsf{Vamo~fatorar~a~equac\~ao~sabendo~que~o ~coeficiente~angular~(a)}\\\mathsf{tem~influ\^encia~nos~demais~termos.}\\\\\\\mathsf{ax^2-4x+3=k}\\\mathsf{a.(x-1).(x-3)=k}\\\mathsf{(x-1).(x-3)=\frac{k}{a}}\\\\\mathsf{\boxed{S=(1,3)}}\\\\\\\mathsf{(c)}\\\\\mathsf{x^2-4x+3=\frac{k}{a}}\\\mathsf{0^2-4.0+3=\frac{k}{a}}\\\\\mathsf{\boxed{\frac{k}{a}=3}}$