Students just explored the connections between quadratic functions expressed in factored form and their graphs. In this lesson, they experiment with the graphs of quadratic functions expressed in standard form and reason about how the parameters of the expressions—specifically the coefficient of the squared term and the constant term—relate to features of the graphs. Students use technology to change these values and produce the graphs. They study the effects and generalize their observations (MP8).
Then, students practice identifying equivalent quadratic expressions in standard and factored form and their corresponding graph. To do so, they look for and make use of structure (MP7). The work here strengthens students’ understanding of the ties across various representations of quadratic functions.
Note that when students graphed equations in factored form earlier, they dealt mostly with monic quadratic expressions, in which the coefficient \(a\) in \(ax^2 +bx+c\) is 1, and the factored expression is in the form of \((x+m)(x+n)\). Limiting the examples to such expressions enables students to see more easily the connections between the numbers in the factors and the \(x\)-intercepts of the graphs. When graphing from the standard form, that limitation is not necessary, as the connections between the coefficient \(a\) and the features of the graph can be studied and revealed more straightforwardly.
(In a later unit, students will learn to use the zero product property to solve quadratic equations. At that time, they will revisit the factored form and how it reveals the \(x\)-intercepts of the graph, including for non-monic quadratic expressions that can be written as \((px+m)(qx+n)\).)
- Comprehend (orally and in writing) how the $a$ and $c$ in $y=ax^2 + bx + c$ are visible on the graph.
- Coordinate (orally and in writing) different representations of quadratic functions (expressions, tables, and graphs).
- Use technology to explore how the parameters of quadratic expressions in standard form are visible on the graph.
- Let’s see how the numbers in expressions like \(\text-3x^2+4\) affect their graph.
Print and cut up cards from the blackline master. Prepare 1 copy for every 2 students.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can explain how the $a$ and $c$ in $y=ax^2+bx+c$ affect the graph of the equation.
- I understand how graphs, tables, and equations that represent the same quadratic function are related.