Question

Y=x²-11x+24 A(2;6) M(-1;36)

Answer 1

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Answer 2

To find the equation of the parabola defined by the points A(2, 6) and M(-1, 36), we can use the general form of a quadratic equation, which is:

y = ax² + bx + c

Substituting the coordinates of point A into the equation, we get:

6 = a(2)² + b(2) + c ---> Equation 1

Substituting the coordinates of point M into the equation, we get:

36 = a(-1)² + b(-1) + c ---> Equation 2

We now have a system of two equations with three variables (a, b, and c). To solve this system, we need a third equation. Since the parabola is defined by the quadratic equation y = x² - 11x + 24, we can substitute the general equation y = ax² + bx + c into this equation to get the third equation:

x² - 11x + 24 = ax² + bx + c

Simplifying, we get:

x² - 11x + 24 - ax² - bx - c = 0 (1 - a)x² + (-11 - b)x + (24 - c) = 0 ---> Equation 3

Since the equation must hold for all values of x, the coefficients of Equation 3 must be equal to zero. Therefore, we have:

1 - a = 0 ---> Equation 4 -11 - b = 0 ---> Equation 5 24 - c = 0 ---> Equation 6

From Equation 4, we find that a = 1. From Equation 5, we find that b = -11. From Equation 6, we find that c = 24.

Now we have determined the values of a, b, and c. Substituting these values back into Equation 1, we can solve for a, b, and c:

6 = (1)(2)² + (-11)(2) + 24 6 = 4 - 22 + 24 6 = 6

The equation is satisfied, indicating that the points A(2, 6) lie on the parabola defined by y = x² - 11x + 24.

Similarly, we can substitute the values of a, b, and c into Equation 2 to check if point M(-1, 36) lies on the parabola:

36 = (1)(-1)² + (-11)(-1) + 24 36 = 1 + 11 + 24 36 = 36

Again, the equation is satisfied, indicating that the point M(-1, 36) also lies on the parabola defined by y = x² - 11x + 24.

Therefore, the equation of the parabola passing through points A(2, 6) and M(-1, 36) is:

y = x² - 11x + 24