Answer:
a = 1, b = -2, c = -3
Explanation:
The equation in factored form shows the roots with the "r" and "s" variables. Note that they are written negative "r" and negative "s" in the formula.
y = a(x - r)(x - s)
The vertex is always halfway between the two roots.
<------[root]--------[vertex]--------[root]---->
The vertex is at x = 1, and one root is at x = 3. The other root is at x = -1.
Write the equation in factored form to show the two roots.
y = a(x - 3)(x + 1)
Substitute the vertex (1, -4) into the equation. The point is written (x, y).
-4 = a(1 - 3)(1 + 1)
Solve for "a".
-4 = a(-2)(2) Simplify within the brackets
-4 = -4a Multiply -2 and 2. Divide both sides by -4
a = 1 Solved for "a" (and kept the variable on the left side).
Substitute a = 1 back into the formula.
y = a(x - 3)(x + 1)
y = 1(x - 3)(x + 1)
We don't have to write multiplied by 1 though, because anything multiplied by 1 is itself.
y = (x - 3)(x + 1)
Since we are looking for "a", "b", and "c", find the equation in standard form.
y = ax² + bx + c
To change factored form into standard form, expand the brackets.
y = (x - 3)(x + 1)
y = x² + x - 3x - 3
y = x² - 2x - 3
Compare the expanded equation to y = ax² + bx + c
∴ a = 1, b = -2, c = -3