Final answer:
To find the input that produces the largest or smallest output for the equation y = 2x^2 + 4x - 11, one must find the vertex of the parabola. The vertex is (-1, -12), and since the parabola opens upwards, x = -1 produces the smallest output, which is y = -12.
Step-by-step explanation:
The student is asking about finding the input that produces the largest or smallest output for a quadratic equation. Specifically, the equation in question is y = 2x2 + 4x - 11. To determine the input that produces the largest or smallest output, we will need to find the vertex of the parabola represented by this equation. The vertex form of a quadratic equation is y = a(x - h)2 + k, where (h, k) is the vertex of the parabola.
To convert the given equation to vertex form, we can complete the square:
1. Write the equation in the standard form: y = ax2 + bx + c.
2. Factor out the coefficient of x2, if necessary.
3. Take half of the coefficient of x (which is 4 in this case), square it (which gives us 4), and add it to both sides of the equation.
4. Write the complete square and the constant term separately.
5. Express the equation in vertex form.
After completing the square, the vertex of the parabola is (-1, -12). Since a is positive (2 in this case), the parabola opens upwards, so the vertex represents the smallest value of the function. Therefore, the input that produces the smallest output is x = -1, and the smallest output is y = -12.