Lesson 6

Symmetry in Equations

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

6.1: Notice and Wonder: Same and Different

What do you notice? What do you wonder?

A. Graph of \(g(x)=x^5+x\)

Graph of polynomial. Crosses the x-axis once, at 0 comma 0. Also passes through points negative 1 comma negative 2 and 1 comma 2.

B. Graph of \(\text-g(x)=\text-(x^5+x)\)

Graph of polynomial. Crosses the x-axis once, at 0 comma 0. Also passes through points negative 1 comma 2 and 1 comma negative 2.

C. Graph of \(g(\text-x)=(\text-x)^5+(\text-x)\)

Graph of polynomial. Crosses the x-axis once, at 0 comma 0. Also passes through points negative 1 comma 2 and 1 comma negative 2.

D. Graph of \(\text-g(\text-x)=\text-\left((\text-x)^5+(\text-x)\right)\)

Graph of polynomial. Crosses the x-axis once, at 0 comma 0. Also passes through points negative 1 comma negative 2 and 1 comma 2.

 

6.2: Finish the Graph

Here is a graph of \(y = f(x)\) for \(\text-5 \le x \le 0\). Draw the graph for \(0 \le x \le 5\) and be prepared to explain your reasoning if:

  1. \(f\) is even

    A graph of function f on a coordinate plane.

  2. \(f\) is odd

    A graph of function f on a coordinate plane.

  3. \(f\) is neither even nor odd

    A graph of function f on a coordinate plane.

6.3: Odd and Even Equations

Take turns with your partner to decide if the function is even, odd, or neither. If it’s your turn, explain to your partner how you decided. If it’s your partner’s turn, listen carefully to their reasons and decide if you agree. If you disagree, discuss your thinking and work to reach an agreement.

  1. \(f(x) = 3x^4 - 2x^2 + 1\)
  2. \(g(x) = x^3 - x\)
  3. \(h(x) = (x^2 -1)(x^2 - 4)\)
  4. \(j(x) = 2^x + 2^{\text-x}\)
  5. \(k(x) = (x^3 - 1)x\)
  6. \(m(x) = (x-0.9)x(x+1.1)\)
  7. \(n(x) = x(x^2 -1)(x^2 - 4)\)
  8. \(p(x) = (x^2 + 4)(x^2 - 3)\)
  9. \(q(x) = \frac{1}{x} + x\)
  10. \(r(x) = \frac{1}{x} - x\)


Write three equations with at least three terms each where one represents an even function, one an odd function, and one is neither even nor odd. Swap equations with your partner and identify which equations represent which type of function.

Summary

Graph of function f. F is a parabola that opens up with vertex at 0 comma 1. Also passes through points negative 1 comma 4 and 1 comma 4.

Remember the even function \(f\) with this graph from earlier?

An equation for \(f\) is \(f(x)=3x^2+1\). Since we already know \(f\) is even, we also know that the output at \(x\) and \(\text-x\) is the same for any value of \(x\) in the domain of \(f\). Said another way, \(f(x)=f(\text-x)\) for all inputs \(x\). If we didn't know \(f\) was even, we could check by using \(\text-x\) as the input.

For example, since \(f(x) = 3x^2 + 1\),

\(\displaystyle \begin{align} f(\text-x) &= 3(\text-x)^2 + 1 \\ f(\text-x) &= 3x^2 + 1 \\ f(\text-x)&= f(x) \end{align}\)

which shows the function \(f\) is even.

Let's look at a different function. Consider the function \(g\) defined as \(g(x) = e^x - e^{\text-x}\). Using \(\text-x\) as the input, we have:

\(\displaystyle \begin{align} g(\text-x) &= e^{\text-x} - e^{\text-(\text-x)} \\ g(\text-x)&= e^{\text-x}- e^{x} \\ g(\text-x)&= \text-e^x+e^{\text-x}\\ \text-g(\text-x)&=e^x-e^{\text-x} \\ \text-g(\text-x)&=g(x) \end{align}\)

This means \(g\) is odd since \(g(x) = \text-g(\text-x)\).

Glossary Entries

  • even function

    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.

  • odd function

    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.