Question

of the function ( f(x) = 2x^2 + bx + c ) is shown. what are the values of ( b ) and ( c )?

Explanation:

Step1: Identify y-intercept (c)

The y-intercept is where $x=0$. From the graph, when $x=0$, $f(0)=-6$. Substitute into $f(x)=2x^2+bx+c$:
$2(0)^2 + b(0) + c = -6$
$\implies c = -6$

Step2: Use another point to find b

Pick the vertex (3, 0) (lowest point on the parabola). Substitute $x=3$, $f(3)=0$, and $c=-6$ into the function:
$2(3)^2 + b(3) - 6 = 0$
$\implies 2(9) + 3b - 6 = 0$
$\implies 18 + 3b - 6 = 0$
$\implies 12 + 3b = 0$
$\implies 3b = -12$
$\implies b = -4$

Answer:

$b=-4$, $c=-6$