Lesson 6

Symmetry in Equations

Lesson Narrative

This is the second of two lessons where students learn to identify odd and even functions. Previously, students learned to identify characteristics of graphs of odd and even functions. They then focused on specific points, and concluded the lesson by summarizing that a function $$f$$ is even if $$f(x)=f(\text-x)$$ and a function $$g$$ is odd if $$g(x)=\text-g(\text-x)$$. In this lesson, the focus is on equations and justifying if a function is even, odd, or neither using only the equation. This lesson also marks a transition to working primarily with functions written in terms of $$x$$ and not in terms of another function.

Students are given opportunity to make use of structure as they complete the graphs of an even function, an odd function, and a function that is neither (MP7). They also learn to use an input of $$\text-x$$ when determining if a function is even, odd, or neither from the structure of its equation.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Learning Goals

Teacher Facing

• Explain (orally) how to sketch the second half of an even or odd function given the first half.
• Justify (orally and in writing) why a function is even, odd, or neither from an equation.

Student Facing

• Let’s use equations to decide if a function is even, odd, or neither.

Required Preparation

Provide access to tracing paper for students who need support visualizing the transformations.

Student Facing

• I can complete graphs of even and odd functions if I know what half the graph looks like.
• I can identify even and odd functions by their equations.

CCSS Standards

A function $$f$$ that satisfies the condition $$f(x) = f(\text-x)$$ for all inputs $$x$$. You can tell an even function from its graph: Its graph is symmetric about the $$y$$-axis.
A function $$f$$ that satisfies $$f(x) = \text-f(\text-x)$$ for all inputs $$x$$. You can tell an odd function from its graph: Its graph is taken to itself when you reflect it across both the $$x$$- and $$y$$-axes. This can also be seen as a 180$$^\circ$$ rotation about the origin.