Odd and Even Functions

1. Odd and Even Functions MCC9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

2. Even Functions • A function is even if: • f(x) = f(-x) • Is f(x) = x2 an even function? • Does f(x) = f(-x)? Let’s see. • f(x) = x2 • f(-x) = (-x)2 • f(-x) = x2 • f(x) does equal f(-x), therefore, f(x) = x2 is an even function.

3. Even Functions • Geometrically speaking, the graph of an even function is symmetric with respect to the y-axis, meaning that f(x) = f(-x) (opposite x-values yield the same y-value). y = x2

4. Odd Function • A function is odd if: • -f(x) = f(-x) • Is f(x) = x3 an even function? • Does -f(x) = f(-x)? Let’s see. • f(x) = x3 • -f(x) = f(-x) • -f(x) = -(x)3 f(-x) = (-x)3 • -f(x) = -x3 f(-x) = (-1)3(x)3 • -f(x) = -x3 f(-x) = -x3 • -f(x) does equal f(-x), therefore, f(x) = x3 is an odd function.

5. Odd Function • Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. • Opposite x-values yield opposite y-values.

6. Even or Odd • Determine if the function is even or odd: • f(x) = x5

7. Even or Odd • Determine if the function is even or odd: • f(x) = x4 + x2

8. Even or Odd • Determine if the function is even or odd: • f(x) = x4 + x