Question

Use the information to answer the question.

Information
A quadratic function passes through the point negative 1 comma 9, the point 0 comma 0, and the point 2 comma 6 on the coordinate plane.

Question
What is the equation of the quadratic function written in general form?

Answer options with 4 options
A.
y equals 4 x squared minus 5 x

B.
y equals negative x squared plus 5 x

C.
y equals 12 x squared plus 3 x

D.
y equals StartFraction 24 over 5 EndFraction x squared minus StartFraction 21 over 5 EndFraction

Answer 1

To find the equation of the quadratic function in general form \(y = ax^2 + bx + c\), we can use the three points given: (-1, 9), (0, 0), and (2, 6).

1. **Using the point (0, 0)**:
\[
0 = a(0)^2 + b(0) + c \implies c = 0
\]
So the equation simplifies to:
\[
y = ax^2 + bx
\]

2. **Using the point (-1, 9)**:
\[
9 = a(-1)^2 + b(-1) \implies 9 = a - b \quad \text{(1)}
\]

3. **Using the point (2, 6)**:
\[
6 = a(2)^2 + b(2) \implies 6 = 4a + 2b \quad \text{(2)}
\]

Now we have two equations from points (1) and (2):
1. \( a - b = 9 \)
2. \( 4a + 2b = 6 \)

Next, we can solve these equations simultaneously.

From equation (1):
\[
b = a - 9
\]

Substituting \(b\) into equation (2):
\[
4a + 2(a - 9) = 6
\]
Simplifying this:
\[
4a + 2a - 18 = 6 \implies 6a - 18 = 6 \implies 6a = 24 \implies a = 4
\]

Now substituting \(a = 4\) back into the equation for \(b\):
\[
b = 4 - 9 = -5
\]

Thus, we have:
\[
a = 4, \quad b = -5, \quad c = 0
\]

The equation of the quadratic function is:
\[
y = 4x^2 - 5x
\]

Looking at the answer choices, we find:
**A. \(y = 4x^2 - 5x\)** is the correct answer.