Question

Use the image to answer the 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)=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

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

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 144a+12b=6

Answer 1

To find the coefficients \( a \) and \( b \) for the quadratic equation \( f(x) = ax^2 + bx + c \) where \( c = 0 \), we can use the points given: \( (3, 15) \) and \( (6, 12) \).

We start by substituting the points into the equation:

1. For the point \( (3, 15) \):
\[
f(3) = a(3^2) + b(3) + 0 = 15
\]
This simplifies to:
\[
9a + 3b = 15 \quad \text{(Equation 1)}
\]

2. For the point \( (6, 12) \):
\[
f(6) = a(6^2) + b(6) + 0 = 12
\]
This simplifies to:
\[
36a + 6b = 12 \quad \text{(Equation 2)}
\]

Thus, the resulting system of linear equations is:
- \( 9a + 3b = 15 \)
- \( 36a + 6b = 12 \)

Now let's match this with the provided response options:

1. **Response Option**:
- \( 9a + 3b = 15 \)
- \( 144a + 12b = 6 \) (Incorrect)

2. **Response Option**:
- \( 225a + 15b = 3 \) (Incorrect)
- \( 36a + 6b = 12 \)

3. **Response Option**:
- \( 9a + 3b = 15 \)
- \( 36a + 6b = 12 \) (Correct)

4. **Response Option**:
- \( 225a + 15b = 3 \) (Incorrect)
- \( 144a + 12b = 6 \)

Hence, the correct answer is:

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