Name: ______________________________________ Date: __________________ Period: ______ Integrated Advanced Algebra Notes: Symmetry with Even and Odd Functions Textbook: Lesson 3.1, Pages 126 – 127 Homework: Worksheet on Graphs of Even and Odd Function

Essential Question: How do you determine if a function is even, odd, or neither from looking at a graph?

You learned about line and rotational symmetry in previous math classes. Remember, that a figure has line symmetry if there is a line that divides the figure into two parts that are mirror images of each other. A figure has rotational symmetry if, the figure aligns with itself when rotated about its center by an angle of 180° or less. A figure can also have point symmetry; a figure is symmetric about a single point if, when rotated about that point 180°, it aligns with itself. Rotational symmetry of 180° is point symmetry about the center. Example 1: For each logo shown, identify and explain any symmetry that you see.

1. 7. 13. 2. 8. 14. 3. 9. 15. 4. 10. 16. 5. 11. 17. 6. 12.

Name: ______________________________________ Date: __________________ Period: ______

x f(x)

x g(x)

Now, let’s look at the graphs of the parent functions for symmetry. A function is an even function when:

f(–x) = f(x) Graph is symmetric about the y-axis.

Example: y= x

Trick: Can you fold the graph in half along the y-axis and it aligns perfectly? Then it is even.

A function is an odd function when: f(–x) = –f(x) Graph is symmetric about the origin.

(or rotated 180° about the origin) Example: y = x3

Trick: Can you rotate the graph upside

down (OR, flip the graph over the y-axis and then the x-axis) and it still looks the same? Then it is odd.

A function is neither even nor odd if it doesn’t meet the requirements to be an even function or an odd function. Example 2: Decide if the following graphs are even, odd, or neither. Explain.

a) y = x2 b) y = x c) y = x d) y= x

1

Example 3: Graph the following functions. Then, determine if the function is even, odd, or

neither. Justify your answer by stating how the graph is or isn’t symmetrical. a) f(x) = 2x2 – 4x b) g(x) = -½ x3 + 4x

Name: ______________________________________ Date: __________________ Period: ______

x f(x)

x g(x)

Integrated Advanced Algebra Worksheet: Graphs of Even and Odd Function Determine whether each of the following functions is even, odd, or neither. Justify your answer by stating how the graph is or isn’t symmetrical.

1. 24 3xx21

f(x) 2. g(x) = -x2 + x 3. h(x) = x5 + x3 – 8x

4. f(x) = x3 – 4x + 6 5. x4

g(x) 6. 53h(x) x

Graph the each of the following functions. Then, determine if the function is even, odd, or neither. Justify your answer by stating how the graph is or isn’t symmetrical. 7. f(x) = -½ x2 + 5 8. g(x) = x3 – 3x