In the previous lesson, students learned to graph the reflection of a function across an axis. They paid close attention to the signs of inputs and outputs when reflecting over the \(x\)- and \(y\)-axes. Building on that experience, this is the first of two lessons where students learn to identify even functions and odd functions. A function \(f\) is even if the outputs for \(x\) and \(\text-x\) are the same. Visually, the graph of \(f\) appears symmetric across the vertical axis. Algebraically, we say that \(f(x)=f(\text-x)\) for any input \(x\). A function \(g\) is odd if the output for \(x\) is the opposite of the output for \(\text-x\). Visually, the graph of \(g\) has a type of symmetry defined by successive reflections across both the \(x\)- and \(y\)-axes taking the graph of \(g\) to itself. Algebraically, we say that \(g(x)=\text-g(\text-x)\) for any input \(x\).
In this lesson, students first identify key features of each type of function by sorting graphs into two groups. They then match tables of values to the graphs and refine their ideas about what makes a function odd or even, giving students the opportunity to use repeated reasoning as they establish definitions for even and odd functions (MP8). In the next lesson, students formalize these ideas using function notation established in the Lesson Synthesis and learn to identify a function as even, odd, or neither from an equation.
- Generalize (orally and in writing) what is true for coordinate pairs of even and odd functions.
- Identify (orally) features that graphs of even functions have in common and features that graphs of odd functions have in common.
- Let's look at symmetry in graphs of functions
- I can identify even and odd functions by their graphs.
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.
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