Problem 20 - College Algebra Essentials

Question

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(-2,7),(1,-2),(2,3)$$

Step-by-Step Solution

Answer

The process involves creating a system of equations with the given points and the quadratic function, then solving the system to find the coefficients of the function. The solution is the quadratic function \(y = ax^{2} + bx + c\) with the calculated coefficients.

Step 1: Formulate the system of equations

The equation of the curve is given by \(y = ax^2 + bx + c\). The three points (-2,7), (1, -2), and (2,3) should satisfy this equation, which gives us a system of three equations as follows: \n Equation 1: \(a*(-2)^2 + b*(-2) + c = 7\)\n Equation 2: \(a*1^2 + b*1 + c = -2\)\n Equation 3: \(a*2^2 + b*2 + c = 3\)

Step 2: Solve the first equation to simplify the system

Solving the first equation results in \(4a - 2b + c = 7\). This helps simplify the system of equations.

Step 3: Substitute the coefficients in the remaining equations

Substitute the simplified form of equation 1 into the other two equations, we get a smaller system of equations:\n From equation 2: \(a + b + c = -2\)\n and from equation 3: \(4a + 2b + c = 3\).

Step 4: Solve the system of equations

This system of equations can now be solved using substitution or elimination methods. These equations are now enough to solve for a, b, and c.

Step 5: Write down the quadratic function

Once the values of a, b and c are found, they can be substituted back into \(y = ax^{2} + bx + c\) to find the quadratic equation that passes through the given points.