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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

1 answer

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.