Question

Determine a imagem de cada função quadrática.

a) y = x² - 3x

b) y = x² - 2x – 3

c) y = - x² + 4x

d) y = 3x² - 4x + 2

e) y = - x² - 2x + 1

f) y = x² - x – 6

Answer 1

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$\boxed{\sf{\underline{Imagem~da~func_{\!\!,}\tilde ao~quadr\acute atica}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf y\geq-\dfrac{\Delta}{4a}~se~a>0}}}}\\\huge\boxed{\boxed{\boxed{\boxed{\boxed{\sf y\leq-\dfrac{\Delta}{4a}~se~a<0}}}}}$

$\tt a)~\sf y=x^2-3x\\\sf\Delta=b^2-4ac\\\sf\Delta=(-3)^2-4\cdot1\cdot0\\\sf\Delta=9\\\sf y_V=-\dfrac{\Delta}{4a}=-\dfrac{\diagup\!\!\!9}{4\cdot\diagup\!\!\!3}\\\sf y_V=-\dfrac{3}{4}\dfrac{}{}\\\sf\left\{Im=y\in\mathbb{R}/y\geq-\dfrac{3}{4}\right\}$

$\tt b)~\sf y=x^2-2x-3\\\sf\Delta=4+12=16\\\sf y_V=-\dfrac{\Delta}{4a}\\\sf y_V=-\dfrac{16}{4\cdot1}=-4\\\sf\left\{Im=y\in\mathbb{R}/y\geq-4\right\}$

$\tt c)~\sf y=-x^2+4x\\\sf\Delta=16\\\sf y_V=-\dfrac{\Delta}{4a}\\\sf y_V=-\dfrac{16}{4\cdot(-1)}=4\\\sf \left\{Im=y\in\mathbb{R}/y\leq4\right\}$

$\tt d)~\sf y=3x^2-4x+2\\\sf\Delta=16-24=-8\\\sf y_V=-\dfrac{-\diagup\!\!\!\!8^2}{\diagup\!\!\!\!4\cdot3}=-\dfrac{2}{3}\\\sf\left\{Im=y\in\mathbb{R}/y\geq-\dfrac{2}{3}\right\}$

$\tt e)~\sf y=-x^2-2x+1\\\sf\Delta=4+4=8\\\sf y_V=-\dfrac{\diagup\!\!\!8^2}{\diagup\!\!\!4\cdot(-1)}=2\\\sf\left\{Im=y\in\mathbb{R}/y\leq2\right\}$

$\tt f)~\sf y=x^2-x-6\\\sf\Delta=1+24=25\\\sf y_V=-\dfrac{\Delta}{4a}=-\dfrac{25}{4\cdot1}=-\dfrac{25}{4}\\\sf\left\{Im=y\in\mathbb{R}/y\geq-\dfrac{25}{4}\right\}$