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Graph of Quadratic Expression

Let y = ax 2+...

Question

Let $y = a x^{2} + b x + c (a \neq 0)$ and $a, b, c \in R$ . If $a b c > 0$ , then which of the following graph(s) satisfy the given condition?

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Solution

The correct options are
A
C
D
Since, all the given graph are downward, therefore
$a < 0$
Let the $x$ coordinate of vertex be $x_{1} = - \frac{b}{2 a}$
$b = - 2 a x_{1}$
$b$ has same sign as of $x_{1}$
When $x = 0 \Rightarrow y = c$
When $a b c > 0 \Rightarrow b c < 0$

Case $: 1$
$c < 0$
$x$ coordinate of vertex $(x_{1})$ is positive.
$b > 0$
So,
$b c < 0$

Case $: 2$
$c < 0$
$x$ coordinate of vertex $(x_{1})$ is negative.
$b < 0$
So,
$b c > 0$

Case $: 3$
$c > 0$
$x$ coordinate of vertex $(x_{1})$ is negative.
$b < 0$
So,
$b c < 0$

Case $: 4$
$c < 0$
$x$ coordinate of vertex $(x_{1})$ is positive.
$b > 0$
So,
$b c < 0$

Suggest Corrections

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