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Question

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.

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)=ax^2+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

225a+15b=3
and 144a+12b=6
255 a plus 15 b equals 3 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

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

225a+15b=3
and 36a+6b=12
255 a plus 15 b equals 3 and 36 a plus 6 b equals 12
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Answer 1

To derive the system of linear equations from the given points (3, 15) and (6, 12) using the quadratic function \( f(x) = ax^2 + bx + c \) with \( c = 0 \), we first substitute the points into the equation.

1. For the point (3, 15): \[ f(3) = 15 \ a(3^2) + b(3) + 0 = 15 \ 9a + 3b = 15 \]

2. For the point (6, 12): \[ f(6) = 12 \ a(6^2) + b(6) + 0 = 12 \ 36a + 6b = 12 \]

So, the system of equations we get is:

• \( 9a + 3b = 15 \)
• \( 36a + 6b = 12 \)

This matches with the third response option:

9a + 3b = 15 and 36a + 6b = 12.