The curve with equation y = ax^2 + bx + c passes through the points P(2,6) abd Q(3,16), and has a gradient of 7 at the point P. Find the values of the constants a,b and c.
Thanks!
These conditions must be satisfied:
6 = a*2^2 + b*2 + c = 4a + 2b + c
16 = 9a + 3b + c
dy/dx @ P = 2a*2 + b = 4a + b = 7
Solve those three equations in 3 unknowns. It does not require integral calculus.
4a + 2b + c = 6
4a + b = 7
Combine these two to get b + c = -1
36a + 12 b + 4c = 64
36a + 18 b + 9c = 54
Combine the last 2 to get
6b + 5c = -10
Now you have two equations in two unknowns. Use substitution to eliminate one of the variables.
6b + 5(-1 -b) = -10
b = -5
4a = 7-b = 12 ; a = 3
5c = -10 -6b = -10 + 30 = 20
c = 4
y = 3x^2 -5x + 4
To find the values of the constants a, b, and c, we need to use the given information and solve a system of equations.
1. Use the given points to form two equations:
For point P(2,6):
6 = a(2^2) + b(2) + c
For point Q(3,16):
16 = a(3^2) + b(3) + c
2. Take the derivative of the equation y = ax^2 + bx + c to find the gradient (slope) at point P.
The derivative is dy/dx = 2ax + b.
Since the gradient at point P is given as 7, we have:
7 = 2a(2) + b
Now we have a system of three equations:
Equation 1: 6 = a(2^2) + b(2) + c
Equation 2: 16 = a(3^2) + b(3) + c
Equation 3: 7 = 2a(2) + b
We can solve this system of equations to find the values of a, b, and c.
Let's start solving the system:
From Equation 3, we have: 7 = 4a + b
We can rewrite this as: b = 7 - 4a
Now substitute this expression for b into Equations 1 and 2:
Equation 1: 6 = a(2^2) + (7 - 4a)(2) + c
Simplifying: 6 = 4a + 14 - 8a + c
Equation 2: 16 = a(3^2) + (7 - 4a)(3) + c
Simplifying: 16 = 9a + 21 - 12a + c
Rearranging both equations to isolate c:
Equation 1: c = 6 - 4a
Equation 2: c = 16 - 9a
Since both equations are equal to c, we can set them equal to each other:
6 - 4a = 16 - 9a
Simplifying: 9a - 4a = 16 - 6
5a = 10
Dividing both sides by 5:
a = 2
Now substitute the value of a back into Equation 3 to find b:
b = 7 - 4(2)
b = 7 - 8
b = -1
Finally, substitute the values of a and b into Equation 1 (or 2) to find c:
c = 6 - 4(2)
c = 6 - 8
c = -2
Therefore, the values of the constants are a = 2, b = -1, and c = -2.
To find the values of the constants a, b, and c, we can use the information given about the curve.
Step 1: Write the equation of the curve using the given points and the general equation of a quadratic curve.
The general equation of a quadratic curve is y = ax^2 + bx + c, where a, b, and c are constants.
Using the information given, we can substitute the x and y values of point P(2,6) into the equation:
6 = a(2)^2 + b(2) + c ...(1)
Similarly, we can substitute the x and y values of point Q(3,16) into the equation:
16 = a(3)^2 + b(3) + c ...(2)
Step 2: Find the derivative of the curve equation.
Taking the derivative of the curve equation will give us the equation for the gradient at each point on the curve. In this case, we are given that the gradient at point P(2,6) is 7.
Differentiating the curve equation y = ax^2 + bx + c with respect to x, we get:
dy/dx = 2ax + b
Step 3: Use the derivative to find the gradient at point P.
Since the gradient at point P is given as 7, we can substitute the x-coordinate of P (which is 2) into the derivative equation and set it equal to 7:
2(2)a + b = 7 ...(3)
Step 4: Solve the system of equations.
Now we have three equations (equations (1), (2), and (3)) with three unknowns (a, b, and c). We can solve this system of equations to find the values of a, b, and c.
By solving equations (1) and (2) simultaneously, we can find the values of a and b:
6 = 4a + 2b ...(4)
16 = 9a + 3b ...(5)
Next, we can substitute the value of b from equation (3) into equations (4) and (5):
6 = 4a + 2(7 - 4a)
16 = 9a + 3(7 - 4a)
Expanding and simplifying these equations, we get:
6 = 4a + 14 - 8a
16 = 9a + 21 - 12a
Further simplification yields:
-8a + 4a = 14 - 6
-3a = -5
a = -5 / -3
a = 5/3
Now, substitute the value of a into equation (3) to find the value of b:
2(2)(5/3) + b = 7
20/3 + b = 7
b = 21/3 - 20/3
b = 1/3
Finally, substitute the values of a and b into equation (1) to find the value of c:
6 = (5/3)(2)^2 + (1/3)(2) + c
6 = 20/3 + 2/3 + c
6 = 22/3 + c
c = 18/3 - 22/3
c = -4/3
Therefore, the values of the constants a, b, and c are:
a = 5/3, b = 1/3, and c = -4/3.