Final Answer:
The given expression f(x) = 3x² - 18x - 4 can be written in the form f(x) = 3(x - 3)² - 31, and the identified vertex is (3, -31).
Step-by-step explanation:
To express f(x) = 3x² - 18x - 4 in the form f(x) = a(x - h)² + k, we'll complete the square. Start by factoring out the coefficient of x² from the quadratic term:
f(x) = 3(x² - 6x) - 4
Now, complete the square within the parentheses by taking half of the coefficient of x, squaring it, and adding/subtracting the result:
f(x) = 3(x² - 6x + 9 - 9) - 4
Combine the constant terms:
f(x) = 3(x - 3)² - 31
Now, the expression is in the desired form, f(x) = a(x - h)² + k, where a = 3, h = 3, and k = -31. The vertex form provides a clear understanding of the parabolic function, where the vertex is located at the coordinates (h, k). In this case, the vertex is (3, -31), indicating that the parabola opens upwards, and the minimum value occurs at x = 3. This form facilitates easy interpretation of the graph and analysis of key features such as the vertex and the direction of the parabola.