The gradient of the normal to the function with the equation 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 1 at the point (1, -2) is 1.  Find the values of a and b.a = b =

The gradient of a function at a given point is given by the derivative of the function at that point.

Step 1: Find the derivative of the function f(x) = ax^2 + bx + 1.

The derivative f'(x) = 2ax + b.

Step 2: Substitute the x-coordinate of the given point into the derivative to find the gradient of the function at that point.

Substitute x = 1 into f'(x) to get f'(1) = 2a + b.

Step 3: The gradient of the normal to the function at a given point is the negative reciprocal of the gradient of the function at that point.

Given that the gradient of the normal is 1, the gradient of the function at that point is -1.

Therefore, 2a + b = -1.

Step 4: Substitute the coordinates of the given point into the equation of the function to find another equation in a and b.

Substitute x = 1 and f(x) = -2 into the equation to get -2 = a + b + 1.

Therefore, a + b = -3.

Step 5: Solve the system of equations 2a + b = -1 and a + b = -3 to find the values of a and b.

Subtract the second equation from the first to get a = 2.

Substitute a = 2 into the second equation to get 2 + b = -3, so b = -5.

Therefore, the values of a and b are 2 and -5, respectively.