Determining the Values of a, b, and c for a Parabola Passing Through Specific Points and Having a Minimum Value
In this article, we will solve a problem involving a parabolic equation that passes through the origin and has a specific minimum point. We will step through the process of finding the coefficients a, b, and c in the equation yax^2 bx c.
Problem Statement
The curve yax^2 bx c passes through the origin and has a minimum point at -2, -4. The goal is to find the values of a, b, and c.
Step-by-Step Solution
Let's first examine the given information and break down the steps systematically:
Step 1: Determine the Value of c
The curve passes through the origin (0,0). This implies that when x0, y0. Therefore:
0 a(0)^2 b(0) c
Which simplifies to:
c 0
Step 2: Use the Point (-2, -4) to Form an Equation
Since the point -2, -4 lies on the curve, we can substitute x-2 and y-4 into the equation:
-4 a(-2)^2 b(-2) c
Substituting c 0, we get:
-4 4a - 2b
Step 3: Use the Minimum Point Condition
The minimum point condition (first derivative equal to zero) at -2, -4 gives:
[ frac{dy}{dx} 2ax b ]
At the minimum point, x-2, we have:
0 2a(-2) b
Which simplifies to:
0 -4a b
Step 4: Solve the System of Equations
We now have two equations:
-4 4a - 2b
0 -4a b
From the second equation, we can solve for b:
b 4a
Substitute b 4a into the first equation:
-4 4a - 2(4a)
-4 4a - 8a
-4 -4a
a 1
Using a 1 to find b:
b 4a 4(1) 4
Thus, we have a 1, b 4, and c 0.
Conclusion
The equation of the parabola is:
y x^2 - 4x
Summary of Key Points
The equation of the parabola is y ax^2 bx c. The curve passes through the origin, which gives us c 0. The point -2, -4 lies on the curve, providing the equation -4 4a - 2b. The minimum point condition gives [ frac{dy}{dx} 2ax b 0 ], which helps to formulate another equation. Solving the system of equations yields a 1, b 4, and c 0.Therefore, the values of a, b, and c are 1, 4, 0 respectively.