Explanation:
Certainly! To express the quadratic function \(f(x) = 3x^2 - 12x + 19\) in vertex form, you can complete the square. Here are the step-by-step instructions:
Step 1: Begin with the given function:
\[f(x) = 3x^2 - 12x + 19\]
Step 2: Factor out the leading coefficient, which is 3:
\[f(x) = 3(x^2 - 4x) + 19\]
Step 3: Complete the square inside the parentheses. To do this, take half of the coefficient of the linear term (\(-4x\)) and square it. Half of \(-4x\) is \(-2x\), and when you square that, you get \(4x^2\). Add and subtract this value inside the parentheses:
\[f(x) = 3(x^2 - 4x + 4x^2) + 19 - 12\]
Step 4: Simplify the expression inside the parentheses:
\[f(x) = 3(x^2 - 4x + 4) + 19 - 12\]
Step 5: Rewrite the expression by factoring the perfect square trinomial:
\[f(x) = 3[(x - 2)^2] + 7\]
Step 6: Distribute the 3 back into the perfect square trinomial:
\[f(x) = 3(x - 2)^2 + 7\]
Now, the function \(f(x)\) is in vertex form, where the vertex is at the point \((2, 7)\), and the "a" value (coefficient of \(x^2\)) is 3.