Final answer:
To write the quadratic function in vertex form by completing the square, we first rewrite the function with a completed square. Then we factor the quadratic within the parentheses and simplify to obtain the vertex form. The vertex of the parabola is (4, -95).
Step-by-step explanation:
To write the given quadratic function in vertex form by completing the square, we will first rewrite the function with a completed square in the form:
f(x) = a(x - h)² + k
Where (h, k) represents the vertex of the parabola.
Given function: f(x) = 3x² - 24x - 47
To complete the square, divide the coefficient of x by 2, square the result, and add and subtract it within the parentheses:
f(x) = 3(x² - 8x) - 47
To complete the square, add and subtract 16 within the parentheses:
f(x) = 3(x² - 8x + 16 - 16) - 47
Now, factor the quadratic within the parentheses and simplify:
f(x) = 3((x - 4)² - 16) - 47
Finally, distribute the 3 to simplify further:
f(x) = 3(x - 4)² - 48 - 47
f(x) = 3(x - 4)² - 95
The vertex of the parabola is given by the values (h, k). From the equation, we can determine that the vertex is (4, -95).