A downward-opening parabola is graphed on a coordinate plane. The x-axis for Width ranges from negative 2 to 10 in increments of 2 and the y-axis for Area ranges from negative 2 to 18 in increments of 2.

Question

The graph represents the area of a rectangle whose perimeter is 16. The graph is modeled by a quadratic equation of the form f(x)=ax2+bx+c
. Given that c=0
, using the points (3,15)
and (6,12)
, which system of linear equations can be used to identify the coefficients a and b?

(1 point)
Responses

9a+3b=15
and 144a+12b=6
9 a plus 3 b equals 15 and 144 a plus 12 b equals 6

9a+3b=15
and 36a+6b=12
9 a plus 3 b equals 15 and 36 a plus 6 b equals 12

225a+15b=3
and 36a+6b=12
255 a plus 15 b equals 3 and 36 a plus 6 b equals 12

225a+15b=3
and 144a+12b=6

Answer 1

To derive the system of linear equations for the coefficients \( a \) and \( b \) in the quadratic equation \( f(x) = ax^2 + bx + c \), given that \( c = 0 \), we can use the points (3, 15) and (6, 12). This means that:

1. When \( x = 3 \), \( f(3) = 15 \): \[ a(3^2) + b(3) + 0 = 15 \implies 9a + 3b = 15 \]

2. When \( x = 6 \), \( f(6) = 12 \): \[ a(6^2) + b(6) + 0 = 12 \implies 36a + 6b = 12 \]

Thus, the system of linear equations to solve for \( a \) and \( b \) is: \[ \begin{align*} 9a + 3b &= 15 \ 36a + 6b &= 12 \end{align*} \]

Therefore, the correct response is: 9a + 3b = 15 and 36a + 6b = 12