Sketch or Interpret a Curve with Known Maximum and Minimum Points

Sketching or Interpreting Curves with Known Maximum and Minimum Points

Understanding where maximum and minimum points lie can help with sketching the curve. If you know these points, you can plot them first to give you an idea of the shape of the curve.
Remember, curves can be symmetrical or asymmetrical, so don’t assume symmetry.
For any function, a maximum point is where the function switches from an increasing to a decreasing function. It’s where the function reaches its highest y-value locally.
Conversely, a minimum point is where the function switches from a decreasing to an increasing function. It’s where the function reaches its lowest y-value locally.
Maximum and minimum points are also known as extremum points. Another type of extremum point is a saddle point, which is a flat region on a curve; the derivative here is zero but unlike a maxima or minima, the curve does not change direction.
In the context of a story problem, the maximum and minimum points of a curve could represent the highest and lowest values of something, such as profit, cost, or number of goods produced. Understanding where these points lie on the curve can give valuable insight into the problem.
When sketching a curve, plot other known points along with the maximum and minimum. Remember the relationship between the function, its derivative and its second derivative at these points — generally, the function will be zero where its derivative is maximum, minimum or a point of inflection.
Even when sketching, aim for precision. Use a graphing tool when one is available to make sure your plot is accurate.