Question

use a system of equations to find the parabola of the form $y = ax^2 + bx + c$ that goes through the three given points.
$(4, -58), (-3, -16), (2, -16)$

the parabola fitting these three points is $y = \square$.

Explanation:

Step1: Substitute points into equation

For point \((4, -58)\): \( -58 = a(4)^2 + b(4) + c \) → \( 16a + 4b + c = -58 \)
For point \((-3, -16)\): \( -16 = a(-3)^2 + b(-3) + c \) → \( 9a - 3b + c = -16 \)
For point \((2, -16)\): \( -16 = a(2)^2 + b(2) + c \) → \( 4a + 2b + c = -16 \)

Step2: Subtract equations to eliminate \(c\)

Subtract the second from the third:
\((4a + 2b + c) - (9a - 3b + c) = -16 - (-16)\)
\(-5a + 5b = 0\) → \( -a + b = 0 \) → \( b = a \)

Subtract the third from the first:
\((16a + 4b + c) - (4a + 2b + c) = -58 - (-16)\)
\(12a + 2b = -42\)

Step3: Substitute \(b = a\) into \(12a + 2b = -42\)

\(12a + 2a = -42\) → \(14a = -42\) → \(a = -3\)

Since \(b = a\), \(b = -3\)

Step4: Find \(c\) using \(4a + 2b + c = -16\)

Substitute \(a = -3\), \(b = -3\):
\(4(-3) + 2(-3) + c = -16\) → \(-12 - 6 + c = -16\) → \(-18 + c = -16\) → \(c = 2\)

Step5: Write the parabola equation

Substitute \(a = -3\), \(b = -3\), \(c = 2\) into \(y = ax^2 + bx + c\):
\(y = -3x^2 - 3x + 2\)

Answer:

\(y = -3x^2 - 3x + 2\)