The turning point of a curve represents the vertex or the highest or lowest point on the graph of a quadratic function. To find the coordinates of the turning point, we can complete the square.Given the quadratic function: y = x^2 + 12x - 5 .
Step 1: Group the terms containing x^2 and x separately from the constant term: y = (x^2 + 12x) - 5
Step 2: To complete the square, we need to add and subtract a value that allows us to factor a perfect square trinomial from the first two terms. The value we need to add and subtract is half the coefficient of x, squared. In this case, half of 12 is 6, and 6 squared is 36. So we add and subtract 36 within the parentheses: y = (x^2 + 12x + 36 - 36) - 5
Step 3: Now, we can factor the perfect square trinomial and simplify: y = ((x + 6)^2 - 36) - 5 Step 4: Combine like terms: y = (x + 6)^2 - 36 - 5 y = (x + 6)^2 - 41 The equation is now in vertex form: y = (x - h)^2 + k, where (h, k) represents the coordinates of the vertex. Comparing our equation to the vertex form, we can see that the vertex is (-6, -41).
Therefore, the coordinates of the turning point of the curve y = x^2 + 12x - 5 are (-6, -41).
