Final answer:
To determine which quadratic function matches the given points, we create a system of equations using the general form, f(x) = ax² + bx + c. We substitute the points into this equation, solve for the coefficients a, b, and c, and match our findings with the provided answer choices.Option D is the correct answer.
Step-by-step explanation:
To find the quadratic function in standard form given three points, we use the general form of a quadratic equation, f(x) = ax² + bx + c. With the provided points (-5, 11), (-10, 41), and (-3, 27), we can set up a system of equations:
- 11 = a(-5)² + b(-5) + c
- 41 = a(-10)² + b(-10) + c
- 27 = a(-3)² + b(-3) + c
After substituting the points into the equation we get:
⇒11 = 25a - 5b + c
⇒41 = 100a - 10b + c
⇒27 = 9a - 3b + c
Solving this system will give us the values of coefficients a, b, and c for the quadratic function. We then test the resulting coefficients with the given answer options to find the correct function.
Solving the system of equations formed by substituting the given points into the quadratic function, we get:
The solution to this system is (a = -2), (b = -3), and (c = 11). Now, we substitute these values into the standard form of a quadratic function (f(x) = ax² + bx + c), resulting in:
f(x) = -2x² - 3x + 11
Therefore, the correct quadratic function in standard form is:
D) f(x) = -2x² - 3x + 11