Lesson 23

In a previous lesson, students recalled that a quadratic expression in vertex form can help us identify the vertex of a graph of a quadratic function. They then used completing the square to rewrite expressions from both standard and factored forms into vertex form. In this lesson, they use the vertex form to determine the maximum or minimum value of a function and to solve problems.

This is not the first time that students find a maximum or minimum value of a quadratic function. In an earlier unit on quadratic functions, students had a brief encounter with this idea, including with the use of the vertex form to determine a maximum or minimum.

At that time, however, students did not yet know how to rewrite expressions in vertex form, so they could only use an expression to determine maximum or minimum if the given expression is already in vertex form. (Otherwise, students would have had to graph the expression or analyze a table of values.) Now that they can rewrite a given expression into vertex form, students can find a maximum or minimum of a function regardless of form and solve new kinds of problems.

An increased emphasis on using the structure of the vertex form to explain maximums and minimums also distinguishes the work in this lesson from earlier work. Previously, students may have relied on their observation of graphs, or recalling that the graph of an equation \(y=a(x-h)^2+k\) opens upward when \(a\) is positive and downward when \(a\) is negative. Here, they reason that:

• \((x-h)^2\) is always zero or positive.
• When \(x=h\), the expression \((x-h)^2\) is 0 because \(0^2=0\). When it \(x\) is any other value, the expression has a value greater than 0.
• When \(a\) is positive, \(a(x-h)^2\) is positive except when \(x=h\) (at which point it is 0). This means 0 is the lowest possible value.
• When \(a\) is negative, \(a(x-h)^2\) is negative except when \(x=h\) (at which point it is 0). This means 0 is the highest possible value.

As students reason about and explain why a vertex is a maximum or a minimum, they practice constructing logical arguments (MP3) and being precise in their communication (MP6).