In an earlier lesson, students saw that a quadratic expression in vertex form can reveal the location of the vertex of the graph of the function defined by the expression. It can also suggest the direction of the opening of the graph (that is, whether the parabola opens up or down). Here, students think about how the structure of the form helps to explain these connections and helps to show whether the vertex of a graph represents the minimum or the maximum value of the function (MP7). They informally use the symmetry of the graph to locate two additional points on the graph to help them make a sketch.
Students begin to see that we can anticipate what happens on the graph on either side of the vertex by reasoning about the parts of an expression in vertex form. The squared expression \(a(x-h)^2\) is 0 when \(x = h\). For all other values of \(x\), the value of \((x-h)^2\) is positive because squaring any number gives a positive number. This suggests that, if the coefficient \(a\) is positive, the expression \(a(x-h)^2\) will be positive (or greater than the value at the vertex, so the vertex is the minimum). If \(a\) is negative, \(a(x-h)^2\) will be negative (or less than the value at the vertex, which means the vertex is the maximum).
Students are not assessed on this line of reasoning, but are prompted to apply their new insights to sketch the graphs of quadratic functions and to match a set of quadratic functions to the graphs that represent them.
- Create a graph of a quadratic function written in vertex form, showing a maximum or minimum and the $y$-intercept.
- Use an equation in vertex form to identify the maximum or minimum of a quadratic function.
- Let’s graph equations in vertex form.
Print and cut up cards from the blackline master. Prepare 1 copy for every 2 students.
- I can graph a quadratic function given in vertex form, showing a maximum or minimum and the $y$-intercept.
- I know how to find a maximum or a minimum of a quadratic function given in vertex form without first graphing it.
vertex form (of a quadratic expression)
The vertex form of a quadratic expression in \(x\) is \(a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants, and \(a\) is not 0.