Final Answer:
The vertex of the quadratic function y = x² + 12x - 5 is at the point (-6, -41). Plotting the points displays the parabolic shape on a graph, confirming the vertex coordinates.
Step-by-step explanation:
To find the vertex of the quadratic function y = x² + 12x - 5 using the "long method" (completing the square), we begin by rewriting the function in the form y = a(x - h)² + k. First, complete the square by halving the coefficient of the x-term (12), squaring it (36), and adding it to both sides of the equation:
y = x² + 12x - 5
(adding and subtracting 36 to complete the square)
y = (x² + 12x + 36) - 5 - 36
y = (x + 6)² - 41
Comparing this equation to the standard form y = a(x - h)² + k, the vertex is at the point (-h, k), so the vertex of this parabola is (-6, -41).
Plotting the points for the function y = x² + 12x - 5 allows visualization of the parabolic shape on a coordinate plane. Knowing that the vertex occurs at (-6, -41), this point serves as the lowest point (minimum) of the parabola, illustrating how the "long method" enables determination of the vertex by transforming the quadratic equation into its completed square form and extracting the vertex from that form.