Question

which inequality is less than the quadratic function with zeros (-7) and (1) and includes the point (left(2,\frac{27}{2}
ight)) on the boundary?
( circ y < -\frac{3}{2}x^2 + 9x + \frac{21}{2} )
( circ y < \frac{3}{2}x^2 + 9x - \frac{21}{2} )
( circ y < -\frac{3}{2}x^2 - 9x + \frac{21}{2} )
( circ y < \frac{3}{2}x^2 - 9x - \frac{21}{2} )

Explanation:

Step1: Write factored quadratic form

Given zeros $x=-7$ and $x=1$, the quadratic function is $y=a(x+7)(x-1)$

Step2: Expand the factored form

$y=a(x^2 + 6x -7)$

Step3: Solve for coefficient $a$

Substitute $(2, \frac{27}{2})$ into the equation:
$\frac{27}{2}=a(2^2 + 6\times2 -7)$
$\frac{27}{2}=a(4+12-7)$
$\frac{27}{2}=9a$
$a=\frac{27}{2\times9}=\frac{3}{2}$

Step4: Substitute $a$ and rewrite

$y=\frac{3}{2}(x^2 + 6x -7)=\frac{3}{2}x^2 + 9x - \frac{21}{2}$

Step5: Apply inequality condition

The inequality is "less than" the quadratic function, so $y<\frac{3}{2}x^2 + 9x - \frac{21}{2}$

Answer:

B. $y<\frac{3}{2}x^2 + 9x - \frac{21}{2}$