Problem 22 - College Algebra Essentials

Question

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(1,3),(3,-1),(4,0)$$

Step-by-Step Solution

Answer

The quadratic function that passes through the points (1,3), (3,-1), and (4,0) is \(y = -x^2 + 4x\).

Step 1: Substitute the given points into the quadratic equation

For point (1,3), substituting \(x = 1\) and \(y = 3\) into the equation forms the first equation: \(3 = a + b + c\). For point (3,-1), substituting \(x = 3\) and \(y = -1\) into the equation forms the second equation: \(-1 = 9a + 3b + c\). For point (4,0), substituting \(x = 4\) and \(y = 0\) into the equation forms the third equation: \(0 = 16a + 4b + c\). We now have a system of three equations to solve.

Step 2: Solve the linear system of equations

Using either substitution or elimination to solve the system of equations, we find \(a = -1\), \(b = 4\), and \(c = 0\).

Step 3: Write the quadratic equation

Now that we have found the values of \(a\), \(b\), and \(c\), we substitute them back into the quadratic equation \(y = ax^2 + bx + c\). The quadratic equation that passes through the points (1,3), (3,-1), and (4,0) is therefore \(y = -x^2 + 4x\).