Lesson 10

Graphs of Functions in Standard and Factored Forms

Lesson Narrative

This lesson serves two goals. The first is to relate the work in the past couple of lessons on quadratic expressions back to the quadratic functions that represent situations. Now students have additional insights that enable them to show (algebraically) that two different expressions can define the same function.

The second goal is to prompt students to notice connections between different forms of quadratic expressions and features of the graphs that represent the expressions. Students are asked to identify the $$x$$- and $$y$$-intercepts of graphs representing expressions in standard and factored form. They observe that some numbers in the expressions are related to the intercepts and hypothesize about the patterns they observe (MP7). This work sets the foundation for upcoming lessons, in which students look more closely at how the parameters of quadratic expressions are related to their graphs.

Learning Goals

Teacher Facing

• Coordinate (orally and in writing) a quadratic expression given in factored form and the intercepts of its graph.
• Interpret (orally and in writing) the meaning of $x$-intercepts and $y$-intercepts on a graph of a quadratic function that represents a context.

Student Facing

• Let’s find out what quadratic expressions in standard and factored forms can reveal about the properties of their graphs.

Student Facing

• I can explain the meaning of the intercepts on a graph of a quadratic function in terms of the situation it represents.
• I know how the numbers in the factored form of a quadratic expression relate to the intercepts of its graph.

CCSS Standards

Building On

A quadratic expression that is written as the product of a constant times two linear factors is said to be in factored form. For example, $$2(x-1)(x+3)$$ and $$(5x + 2)(3x-1)$$ are both in factored form.
The standard form of a quadratic expression in $$x$$ is $$ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants, and $$a$$ is not 0.