Refer back to Example 1 in the Learning Activity titled “Minimum and Maximum Values.” A quadratic equation is provided, along with the coordinates of its vertex. In your own words, describe the logic that allows one to conclude that: 1) there must be a maximum or minimum value, and 2) the location for that maximum or minimum value must be at the vertex. Also, describe how you would be able to tell, from just the equation and the coordinates of the vertex, whether the quadratic had a maximum value or a minimum value. In formulating your response, consider both a geometric viewpoint (“From the perspective of the graph, what’s going on?”) and an algebraic one (“What do the numbers say, and why?”).
In Example 1 of the Learning Activity titled “Minimum and Maximum Values,” a quadratic function is given in the standard form of ax^2 + bx + c. The vertex of the parabola is also given as (h, k).
One can conclude that there must be a maximum or minimum value for the quadratic equation because the parabola is a U-shape curve that opens either upwards or downwards. The curve can either increase or decrease indefinitely, or reach a highest point and start decreasing or reach a lowest point and start increasing. The vertex of the parabola is the point where the curve changes direction, from increasing to decreasing or vice versa. So, the vertex is either the minimum or maximum point of the parabola.
To identify whether the vertex represents a minimum value or a maximum value, one must look at the coefficient of the x^2 term. If it is positive, then the parabola opens upwards, and the vertex represents the minimum value. If it is negative, then the parabola opens downwards, and the vertex represents the maximum value. This can be seen geometrically as well since the vertex is the lowest or highest point on the curve.
The logic behind concluding that the location for the maximum or minimum value must be at the vertex is both algebraic and geometric. Algebraically, the vertex is the point where the slope of the curve is zero, which means the derivative of the function is zero at that point. Since the slope is either positive or negative on both sides of the vertex, and the derivative changes sign at the vertex, it must be the point of maximum or minimum.
Geometrically, the vertex of the parabola is the reflection point of the axis of symmetry. The axis of symmetry divides the parabola into two congruent halves, and since the maximum or minimum point must be equidistant from these two halves, it must lie on the axis of symmetry, which is through the vertex of the parabola.
In summary, the shape of the parabola and its vertex determine whether it has a minimum or maximum value. The vertex is the point where the function changes direction and is therefore the location of the minimum or maximum value. The algebraic and geometric properties of the parabola help to deduce these conclusions.