When you're looking for the highest and lowest points on a graph, there are some easy ways to find them. Let's go over the main methods!

1. Looking at the Graph

The easiest way is just to look at the graph itself.

• The highest points are called maximums.
• The lowest points are the minimums.

For example, if you draw the graph of the function (f(x) = -x^2 + 4), it makes a shape like an upside-down U. The very top point, called the vertex, is the maximum point. In this case, it happens at the point (0, 4).

2. Using Derivatives

If you're learning calculus, you can use something called derivatives to find maximums and minimums.

• First, find the derivative of a function, called (f'(x)).
• Set this derivative equal to zero to find critical points. These are where you might find maximums or minimums.
• To check if a critical point is a max or a min, use the second derivative. If (f''(x) < 0), it's a max, and if (f''(x) > 0), it's a min.

3. Making a Table of Values

Another handy method is to create a table of values. This approach works well for functions like quadratics.

For example, using (f(x) = x^2 - 4x + 5), you can calculate the function's values for (x = 0, 1, 2, 3, 4). By doing this, you can see that the minimum value happens at the point (2, 1).

4. Finding Zeros

Also, don't forget about zeros, which are the points where the graph crosses zero. These zeros can give you clues about where the maximum or minimum points are.

If there are zeros on either side of a point, then that point could be the highest or lowest value on the graph.

By using these techniques, you'll be able to easily find the maximum and minimum values on graphs!