Lesson 6

Symmetry in Equations

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

Problem 1

Classify each function as odd, even, or neither.

  1. \(f(x)=3x^4+3\)
  2. \(f(x)=x^3-4x\)
  3. \(f(x)=\frac{1}{x^2+1}\)
  4. \(f(x)=x^2+x-3\)

Problem 2

Here is a graph of a function \(f\) for \(0 \leq x \leq 5\).

Function on coordinate plane.
  1. The function \(g\) is even and takes the same values as \(f\) for \(0 \leq x \leq 5\). Sketch a graph of \(g\).
  2. The function \(h\) is odd and takes the same values as \(f\) for \(0 \leq x \leq 5\). Sketch a graph of \(h\).

Problem 3

The linear function \(f\) is given by \(f(x) = mx + b\). If \(f\) is even, what can you conclude about \(m\) and \(b\)? 

Problem 4

Here are the graphs of \(y = f(x)\) and \(y = f(x-1)\) for a function \(f\).

Graph of 2 functions.

Which graph corresponds to each equation? Explain how you know.

(From Unit 5, Lesson 2.)

Problem 5

Write an expression for two of the graphs in terms of \(f(x)\).

Graph of function f, a, b, c, and d.
(From Unit 5, Lesson 3.)

Problem 6

Here is a graph of the function \(f\) given by \(f(x) = x^3\).

Function on coordinate plane.
  1. What happens if you reflect the graph across the \(x\)-axis and then across the \(y\)-axis?
  2. Is \(f\) even, odd, or neither?
(From Unit 5, Lesson 5.)